FIXED POINT PROBLEMS, EQUIVARIANT
STABLE HOMOTOPY, AND A TRACE MAP FOR
THE ALGEBRAIC K-THEORY OF A POINT
Manos Lydakis
CONTENTS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
1
Introduction
Preliminaries
Fiberwise euclidean neighborhood retracts
The K-theory of fiberwise ENRs
A cyclic category
Fixed point problems
Fixed point problems in euclidean space
A topological Pontryagin construction
Fixed point problems and equivariant stable homotopy
Split fixed point problems
Filtered fixed point problems
Separating the fixed points of the identity map of a filtered space
Appendix: Pushouts of fiberwise ENRs
References
1
4
7
9
12
15
18
23
25
27
29
32
36
38
Introduction
Write Q(∗) for the colimit over n of Ωn S n and let A(∗) be the algebraic K-theory of a
point. We construct a map tr from A(∗) to Q(∗) by “counting” fixed points of identity
maps. The map has equivariant properties reminiscent of the cyclotomic trace of [1],
and related to “counting” periodic points of identity maps. Before we give an outline of
this paper, we illustrate our construction by looking at the induced map tr∗ : π0 (A(∗)) →
π0 (Q(∗)):
Let C be a CW -complex which is a subspace of Rn , let f be a self-map of C, and
suppose that the fixed point set F of f is compact. Choose a retraction r: U → C for
some neighborhood U of C in Rn and let z: U → Rn be given by z(x) = x − f r(x). View
S n as Rn ∪ ∞ (with ∞ as the basepoint). Choose a (pointed) map g: S n → S n such that
g(x) = 0 ⇐⇒ x ∈ F and g = z near F , and say that “g corresponds to f ”. Then:
• Such a g always exists, its homotopy class [g] is independent of the choices above,
and depends only on the germ of f along F .
• This construction can be done even if f is defined only near its fixed point set.
If the fixed point set of f is a disjoint union, say F = F1 q F2 , and if fi denotes
the restriction of f near Fi , then [g] = [g1 ] + [g2 ] in π0 (Ωn S n ) (where g, resp gi ,
1
corresponds to f , resp. fi ).
• π0 (A(∗)) is a group generated by all finite CW -complexes in (some) Rn (subject
to some relations), and tr∗ (C) = [g] − [1S n ] in π0 (Ωn S n ), where g corresponds to
the identity map of C and 1S n is the identity map of S n .
• The relations among all C in π0 (A(∗)) are:
– If C is homotopy equivalent to C 0 then C = C 0 .
– If C is a subcomplex of C 0 then C + C 0 /C = C 0 .
From this it easily follows that the reduced Euler characteristic χ
e defined by C 7→
χ(C)−1 is an isomorphism from π0 (A(∗)) to Z. The degree of a self-map of S n gives
another isomorphism, call it deg, from π0 (Q(∗)) to Z. In these terms, deg ◦ tr∗ = χ
e.
Let g: S n → S n correspond to f : C → C under the construction above. If we call
the integer deg(g) the “number of fixed points of f ” we obtain the following classical
statement: The number of fixed points of the identity map of a finite CW complex C
equals the Euler characteristic of C.
We now indicate how the equivariant properties of tr are related to “counting” periodic points. Let G be the cyclic group of order k. Let G act on Rkn = Rn × . . . × Rn by
permuting the copies of Rn , and let it act on S kn by fixing ∞ and so that the inclusion
Rkn ⊂ S kn is equivariant. Finally, G acts on Ωkn S kn by (cg)(x) = cg(c−1 x) for c ∈ G,
x ∈ S kn , and g: S kn → S kn . Let QG (∗) be the G-space which is the colimit over n of
Ωkn S kn . Thus if we forget that QG (∗) is a G-space, then we may identify it with Q(∗).
Let QG (∗)G ⊂ QG (∗) denote the fixed points of G in QG (∗).
f1
f2
fk−1
fk
Now let C1 → C2 → . . . → Ck → C1 be a diagram of CW complexes in Rn
and continuous maps, and suppose that the fixed point set Fi ⊂ Ci of the composition
fi−1 fi−2 . . . f1 fk . . . fi+1 fi is compact. Choose neighborhood retractions ri for each Ci .
Define z: U → Rkn by
z(x) = (x1 − fk rk (xk ), x2 − f1 r1 (x1 ), x3 − f2 r2 (x2 ), . . . , xk − fk−1 rk−1 (xk−1 ))
(where U is an open set in Rkn depending on the domains of the ri ). Choose a pointed
map g: S kn → S kn such that g(x) = 0 ⇐⇒ z(x) = 0 and g = z near their vanishing set.
Then:
• Such a g always exists, its stable homotopy class [g] in π0 (QG (∗)) is independent
of the choices above, and depends only on the germs of the fi along Fi .
• If f1 = f2 = . . . = fk = f then we can choose g in QG (∗)G and the class [g] in
π0 (QG (∗)G ) is independent of the choices above, with the restriction that we must
choose r1 = r2 = . . . = rk , and depends only on the germ of f near F1 = F2 =
. . . = Fk = F . In other words, a self map f of a CW complex C determines an
element [g] ∈ π0 (QG (∗)G ) that depends only on the germ of f along its k-periodic
point set.
• All this can be done if the fi are only defined near Fi , and union of disjoint fixed
point sets, resp. k-periodic point sets, corresponds to addition in π0 (QG (∗)), resp.
π0 (QG (∗)G ).
• Further, tr∗ (C) is the class [g]−[1] ∈ π0 (QG (∗)) where g corresponds to the identity
map of C, in particular tr∗ lifts to π0 (QG (∗)G ).
We introduce some definitions suggested by the above results: Fix an integer k ≥ 1
f1
f2
fk−1
fk
and consider all circular diagrams E1 → E2 → . . . → Ek → E1 where the Ei ’s are
reasonable spaces (more about this later), fi is only defined on an open set in Ei and the
2
fixed point set of some (and therefore any) composition obtained by “starting at some Ei
and going once around the circle” is compact. A “fixed point problem” is an equivalence
class of such diagrams, where the equivalence relation is “germ along the fixed point
set”. Note that the cyclic group Ck of order k acts on fixed point problems by rotating
them, and it is reasonable to call a fixed point problem which is invariant under Ck a
“k-periodic point problem”. Thus our previous remarks suggest that there is a Ck -space
of fixed point problems having the Ck -homotopy type of QCk (∗).
This paper is organized as follows: In §2 we review some facts that we need, mostly
without proofs. In §3 we introduce the spaces on which fixed point problems are defined
(fiberwise euclidean neighborhood retracts, as defined by Dold in [6]), and in §4 we
construct a model for A(∗) based on compact fiberwise EN Rs. In §8 we construct a
model for QCk (∗) based on “germs of self-maps of spheres along their vanishing set”.
The equation f (x) = 0 ⇐⇒ x = x − f (x) is used in §9 to construct a model for QCk (∗)
based on fixed point problems. In §7 we assemble these models into a single SO(2) space
(in fact a cyclic simplicial set) which for all k has the Ck -homotopy type of QCk (∗). This
space is based on fixed point problems in (some) Rn , and in §6 we enlarge it (without
changing its Ck -homotopy type) to a space of fixed point problems in arbitrary EN Rs.
For this we need a general construction, which is introduced in §5. The main idea in
§6 is that if we use a neighborhood retraction of Rn onto E to produce a fixed point
problem in Rn from a fixed point problem in E, then the result does not depend on the
neighborhood retraction. In §10 we construct a Γ-space of fixed point problems with the
“addition up to homotopy” given by union of disjoint fixed point sets, and in §11 we
enlarge it (without changing its Ck -homotopy type) so that it looks more like our model
for A(∗). The construction in §11 is a formal consequence of the properties of homotopy
limits, i. e. it can be applied to any simplicial category and it produces a larger simplicial
category of the same homotopy type. In §12 we construct a space with a circle action
(in fact a cyclic bisimplicial set), whose loop space has the Ck -homotopy type of A(∗)
with the trivial action, and an SO(2)-map from it to the construction of §11. This last
model for A(∗) is based on “homotopies from the identity map of a filtered space to a
map whose fixed point set splits compatibly with the filtration”.
3
The following diagram might help in orienting the reader:
|Sk ∆ |
§8
?
|S k ∆ |
§8
?
|Zk ∆ |
§9
?
|Θ0k−1 F ix∆ |
§7
?
|ΘF ix∆ |
6
§6
Ω|wSCW|
§4
?
Ω|wSE∆0 |
|H ∆↓ΘF ix∆ |
§4
?
Ω|wSE∆ |
6
§4
§6
?
|H∆ |
§6
?
|ΘF∆ |
Ω|sE∆ |
6
§12
§10
?
Ω|ΘsF∆ |
Ω|Θ × sE∆ |
6
§12
§11
?
tr Ω|f Θ F∆ |
Ω|hΘ E∆ |
This is the chain of maps we construct from A(∗) to QCk (∗) (we obtain such a diagram
for every k ≥ 1, but notice that most of the diagram does not depend on k). Above, all
vertical maps are Ck -homotopy equivalences, where the top four spaces on the left have
the trivial action, and every map is labeled by the section in which it is proven to be a
Ck -homotopy equivalence. The top space on the left is (one of the standard definitions
of) A(∗), and the top space on the right is (the realization of) the singular complex of
QCk (∗). Finally, the map tr is defined in §12.
This paper is a slightly modified version of the major part of the author’s thesis
([11]).
2
2.1
Preliminaries
Notation and terminology
It will be convenient to view Rn as a subset of Rn+1 by identifying x and (x, 0) and to
view the n-sphere S n as Rn ∪ ∞ with ∞ as the basepoint. We denote the union over n of
Rn by R∞ . We identify SO(2) with R/Z and write Ck for the cyclic subgroup of order
k in SO(2). The generator 1/k ∈ R/Z of Ck will be denoted by ck or c. We make Rk
into a Ck -space by defining ck (x0 , x1 , . . . , xk−2 , xk−1 ) = (xk−1 , x0 , x1 , . . . , xk−2 ). Thus
Ck acts on Rkn since it is a subgroup of Ckn .
We write T OP for the category of topological spaces, SET for the category of sets,
and CAT for the category of small categories. If A and B are categories and we mention
the category B A of functors from A to B (and natural maps between them) then A will be
4
small even if we do not explicitly say it. A functor f : A → B gives a functor f] : AC → B C
taking the map φ: g → h to the map f φ: f g → f h, and a functor f ] : C B → C A taking
the map φ: g → h to the map φf : gf → hf . If a and b are objects of A, the set of maps
in A from a to b will be denoted by A(a, b).
Finally, here is some terminology: We will sometimes refer to the objects of B A as
“A-objects in B”, and to the morphisms of B A as “A-maps in B”. We will also say
“A-set” instead of A-object in SET , “A-functor” instead of A-map in CAT , and so on.
2.2
Simplicial objects
We now prove two short propositions and briefly review simplicial path objects.
Let EN R be the category of compact euclidean neighborhood retracts (compact subspaces of some Rn whose identity map extends to one of their neighborhoods in Rn ) and
continuous maps. We denote the standard embedding ∆ → EN R again by ∆. If X is
an EN Rop -object we denote by X∆ the simplicial object given by X ◦ ∆op . We denote
by XI×∆ the simplicial object X ◦ (I×)op ◦ ∆op , where I is the unit interval, which we
identify with ∆1 . The maps i0 , i1 : 1EN R → I× induce maps i∗0 , i∗1 : XI×∆ → X∆ .
Proposition 2.1 For any X: EN Rop → SET , i∗0 and i∗1 are simplicially homotopic.
Proof: Define the map (of sets) ∆([q], [1]) × XI×∆q → X∆q as the adjoint of
∆([q], [1]) → EN R(∆q , ∆1 × ∆q ) →
f
7→
(∆(f ), 1∆q )
SET (XI×∆q , X∆q )
A computation shows that this defines a simplicial homotopy ∆1 × XI×∆ → X∆ between
i∗0 and i∗1 , QED.
The above is of course true for many subcategories of T OP and not just EN R, but
this is the only case we need.
Let X and Y be spaces. We say that two maps from X to Y are weakly homotopic,
provided that they are homotopic when restricted to any compact subspace of X. Thus
a map weakly homotopic to an identity map is a weak homotopy equivalence.
Proposition 2.2 Let X be a simplicial space and suppose that any two face maps Xq →
Xq−1 are weakly homotopic. Then for any map f in ∆, X(f ) is a weak homotopy
equivalence.
Proof: For ∂i : Xq → Xq−1 with i 6= q we have si ∂i = ∂i si+1 ∼ ∂i+1 si+1 = 1 where
“∼” means “weakly homotopic”. Since ∂i si = 1, both si and ∂i are weak homotopy
equivalences for 0 ≤ i ≤ q − 1. For ∂q the conclusion follows from ∂q sq−1 = 1, and for sq
it follows from ∂q sq = 1, QED.
We now recall some facts about simplicial path objects. Let P : ∆ → ∆ be the functor
taking f : [n] → [m] to f∗ : [n + 1] → [m + 1], where f∗ (0) = 0 and f∗ (i + 1) = f (i) + 1. The
maps ∂ 0 : [n] → [n+1] give a map 1∆ → P . If X is a simplicial object, its path object P X
is X ◦ P op . Thus the maps ∂0 : Xn+1 → Xn give a map P X → X. Let c: ∆ → ∆ be the
constant functor [n] 7→ [0]. We abbreviate X ◦ cop by X0 . There are maps P → c → P
given by [n + 1] → [0] → [n + 1], k 7→ 0 7→ 0, whose composition in the other order is the
identity of c. Thus there are maps P X → X0 → P X whose composition in the other
order is the identity of X0 . If X is a simplicial set the composition in the order above
is simplicially homotopic to the identity of P X. See [18, Lemma I.5.1] for the simplicial
homotopy.
2.3
Γ-objects
For n a non-negative integer, let < n >= {0, 1, . . . , n}. Let Γ be the category with
objects all < n >, and morphisms all maps of sets between its objects that fix 0. There
is a functor ∆op → Γ taking f : [n] → [m] in ∆ to f ∗ : < m >→< n > in Γ, where f ∗ fits in
a commutative diagram:
5
∂∆1m
incl.- 1
∆m
λ <m>
f]
?
∂∆1n
f]
incl.- ?1
∆n
f∗
λ - ?
<n>
Above, ∆1m = ∆([m], [1]), ∂∆1m = constant maps in ∆1m , and λ(a) = 0 if a is constant,
else λ(a) = min a−1 (1). Note that λ induces a bijection ∆1m /∂∆1m →< m > which shows
that f ∗ is well defined and that f 7→ f∗ is functorial. This functor gives, for every
Γ-object X, a simplicial object which we also denote by X.
Let X be a Γ-space. We say that X is very special if it satisfies the following three
conditions:
Γ1. X0 is contractible.
Γ2. The map (p1 , p2 , . . . , pn ): Xn → X1 × X1 × . . . × X1 is a homotopy equivalence,
where pk : Xn → X1 is the map induced by the map < n >→< 1 > in Γ that maps all
elements of < n > except k to 0.
(p1 ,p2 )
Γ3. The set π0 (X1 ) is a group with multiplication induced by X1 ×X1 ←
where m is induced by the map < 2 >→< 1 > in Γ that maps only 0 to 0.
m
X2 → X1 ,
Proposition 2.3 Let X be a very special Γ-space which is the realization of a Γ-simplicial
set. Then the following diagram is homotopy-cartesian:
X1 = (P X)0
- PX
?
X0
?
-X
∂0
For a proof see [14, Prop. 1.5]. Together with the previous results on path objects, this
gives a homotopy equivalence from X1 to the loop space of (the realization of) X.
2.4
Cyclic objects
Recall the category Λ defined by Connes in [4]: It is generated by ∆ ⊂ Λ together with
maps cp : [p] → [p] satisfying:
(cp )
p+1
cp ∂
0
=
1[p]
= ∂p
cp ∂ i
= ∂ i−1 cp−1 , 1 ≤ i ≤ p
cp s0
= sp (cp+1 )
cp si
= si−1 cp+1 , 1 ≤ i ≤ p
2
The cyclic objects in a category are by definition the Λop -objects in that category. Recall
from [8] that the realization of the simplicial set determined by a cyclic set has an SO(2)action and that the realization of the simplicial map determined by a map of cyclic sets is
SO(2)-equivariant. It follows that a cyclic multisimplicial set determines a multisimplicial
SO(2)-space, which in turn determines an SO(2)-space. Suppose now that G is a finite
subgroup of SO(2), that X and Y are G-multisimplicial sets, resp. cyclic multisimplicial
sets, and that f : X → Y is a G-multisimplicial, resp. cyclic multisimplicial, map. Then
we say f is a G-homotopy equivalence if and only if |f | is one (in this paper “G-homotopy
equivalence” means “strong G-homotopy equivalence”; see [3, §1] for an introduction to
equivariant homotopy theory).
The main example of a cyclic set in this paper will be the cyclic nerve N cy C of a
(small) category C, where:
Npcy C = {C0 → C1 → . . . → Cp → Cp+1 ∈ Np+1 C|C0 = Cp+1 },
6
γ0
γ1
γp−1
γp
γ1
γp−1
γ2
γ0 γp
∂0 (C0 → C1 → . . . → Cp → C0 ) = C1 → C2 → . . . → Cp → C1 ,
the maps ∂i , 1 ≤ i ≤ p, and si , 0 ≤ i ≤ p, are the restrictions of the corresponding maps
of Np+1 C to Npcy C, and finally cp : Npcy C → Npcy C is given by
γ0
γ1
γp−1
γp
γp
γ0
γp−2
γp−1
cp (C0 → C1 → . . . → Cp → C0 ) = Cp → C0 → . . . → Cp−1 → Cp .
If C is a groupoid then N C embeds in N cy C via
γ0
γ1
γp−2
γp−1
γ0
γ1
γp−2
γp−1
γp
(C0 → C1 → . . . → Cp−1 → Cp ) 7→ (C0 → C1 → . . . → Cp−1 → Cp → C0 )
where γp is defined by γp γp−1 . . . γ0 = 1C0 . If T denotes the trivial groupoid with
two objects then the embedding N T → N cy T is an isomorphism, therefore N cy T is
contractible. It follows that an equivalence C → C 0 of categories induces a homotopy
equivalence N cy C → N cy C 0 : If ij : C → C × T denotes one of the two inclusions induced
by the inclusions [0] → T , then a natural isomorphism φ: F0 → F1 of functors C → C 0
may be written as a functor φ̂: C × T → C 0 with φ̂ij = Fj , j = 0, 1.
Fix an integer k ≥ 1. The k th subdivision functor sdk : ∆ → ∆ is defined by
[n] 7→ [n]qk
f
7→ f qk .
f
g
More precisely, sdk maps [n − 1] → [m − 1] to [kn − 1] → [km − 1] where for i ∈ [kn − 1],
say i = nq + r with 0 ≤ q ≤ k − 1 and 0 ≤ r ≤ n − 1, we define g(i) as nq + f (r).
For any simplicial object X let sdk X denote X ◦ (sdk )op .
Lemma 2.1 If X is a simplicial set there is a natural homeomorphism hk from |X|
to |sdk X|. If further X is a cyclic set, then the action of (ckp+k−1 )p+1 on Xkp+k−1 =
sdkp X gives a Ck -simplicial set and hk is then a map of Ck -spaces.
See [1] for a proof.
This, together with some elementary facts about G-CW -complexes (see for example
§1 of [3]), implies:
Proposition 2.4 Let f : X → Y be a map of cyclic sets. Then f is a Ck -homotopy
equivalence for all k ≥ 1 if and only if the induced map (sdk X)Ck → (sdk Y )Ck is a
homotopy equivalence for all k ≥ 1.
Note that for any small category C there is an isomorphism ψ: (sdk N cy C)Ck → N cy C
whose inverse sends a circular diagram to its “k-fold covering”. This implies:
Proposition 2.5 An equivalence C → C 0 of small categories induces a Ck -homotopy
equivalence N cy C → N cy C 0 for all k ≥ 1.
Let M denote the multiplicative submonoid of Z consisting of 0 and 1. There is a splitting
of cyclic sets
N cy M = N cy {1} q Θ
where Θp = {(m0 , . . . , mp ) ∈ M p+1 = Npcy M |mi = 0 for some i}. An easy computation
shows that Θ is simply connected and its homology groups vanish. Further, the isomorphism ψ: (sdk N cy M )Ck → N cy M restricts to an isomorphism from (sdk Θ)Ck to Θ. Thus
Prop. 2.4 implies that Θ has the Ck -homotopy type of a point, for all positive integers k.
3
Fiberwise euclidean neighborhood retracts
We say that a subspace A of Rn is a euclidean neighborhood retract (EN R) if the
identity map of A extends to a neighborhood of A in Rn . We say that a space B is EN R
if it is homeomorphic to such an A.
7
The two definitions of EN R are equivalent for subspaces of Rn . If A ⊂ B are both
EN Rs then A has a neighborhood U in B deformable in B to A relative to A, i. e., maps
ht : U → B exist for t ∈ I with ht |A = A ⊂ B, h0 = U ⊂ B, and h1 (U ) ⊂ A (where
h: I → M ap(U, B) is continuous). For proofs see [5, IV.8].
Let EN R0 be the full subcategory of T OP 0 with objects the (pointed) compact
EN Rs. If A → B is a map in EN R0 we say it is a cofibration if it is injective. This
is equivalent to the standard definition of the term (see [15]). Choose a one-point space
∗ in EN R0 . We claim EN R0 is a category with cofibrations in the sense of [18]: For
any diagram of the form C ← A → B in EN R0 , define C ∪A B as usual (i. e. pushout
in T OP). Then the only non-trivial part in proving that EN R0 is a category with
cofibrations amounts to showing C ∪A B is EN R if A → B is injective, and this is a
special case of [16, Prop. II.6.5].
Now let B be a compact EN R and T OP/B the category of spaces over B. For any
space X, we view subspaces of B × X as spaces over B using the projection B × X → B.
We say that a space E over B is EN RB (euclidean neighborhood retract over B) if there
i
r
is an open subset U of some B × Rn and maps E → U → E over B whose composition
p
is 1E . Dold proves in [6] that if E is a compact EN R then E → B is EN RB if and only
if it is a Hurewicz fibration.
Proposition 3.1 If X is a normal space over B, Y is a closed subspace, and E is
f
EN RB , then any map Y → E over B extends over B to a neighborhood of Y in X.
i
r
Proof: Let E → U → E be as in the definition of EN RB and extend if to all of X
using the Tietze Theorem. If V is the inverse image of U , then the retraction composed
with the extension gives a map V → E which extends f , QED.
Corollary 3.1 If F ⊂ E are both EN RB s and F is closed in E, then F is a retract
over B of one of its neighborhoods in E.
Corollary 3.2 If F ⊂ E is an inclusion of EN RB ’s and C is a closed subset of E containing F , then any retraction over B of C onto F extends to a neighborhood retraction
of E onto F .
Corollary 3.3 If A, B, and C are compact EN Rs, C = A ∪ B, and E is a space over
C such that EA is EN RA and EB is EN RB , then E is EN RC .
Remark. If Y ⊂ X is an inclusion over B and f, g: X → E are maps over B which
agree on Y , and if E is EN RB , then f and g are homotopic over B relative to Y when
restricted to some neighborhood of Y in X. The proof is essentially the same as in the
special case B = ∗, as found in [5, IV.8].
Proposition 3.2 If F ⊂ E are both EN RB s and F is closed in E then there is a
neighborhood of F in E deformable over B in E to F relative to F .
Proof: By Corollary 3.1 there is a neighborhood U of F in E and a retraction r: U → F .
Now use the preceding remark with X = F , Y = U , f = r, and g = U ⊂ E, QED.
Let B be a compact EN R. We define a category EB . The objects of EB are the
compact EN RB ’s E which are subspaces over B of some B × Rn , together with a
section s: B → E. The morphisms from E to E 0 are the maps over and under B. We
make EB a pointed category by choosing the zero object B × R0 .
We claim that EB is a category with cofibrations if we define E → E 0 to be a cofibration if and only if it is injective. Note that this is equivalent to the natural definition
of “cofibration over B” as given in [10] for example (this is essentially a consequence of
Proposition 3.2). Again the nontrivial part amounts to verifying Axiom Cof 3 of [18],
and this follows from Prop. A.1 of the Appendix.
8
If C is a compact EN R and ∂: B → C is a continuous map, we define a functor
s
∂ ∗ : EC → EB as follows: If C → E is an object of EC , the corresponding object of EB is
∗
∂ s
B → ∂ ∗ E, where ∂ ∗ E = (∂ × 1)−1 (E) (here 1 denotes the identity map of R∞ ), and
∂ ∗ s(b) = (b, P s∂(b)) ∈ ∂ ∗ E ⊂ B × R∞
where P : C × R∞ → R∞ is the projection. If f : E → E 0 is a morphism in EC , then
∂ ∗ f : ∂ ∗ E → ∂ ∗ E 0 is given by:
(∂ ∗ f )(b, x) = (b, P f (∂(b), x))
Then ∂ ∗ is a pointed functor preserving cofibrations and the pushout diagrams of Axiom
Cof 3. In fact, E is a functor from EN Rop to (small) categories with cofibrations. Recall
from §2.2 the simplicial category with cofibrations E∆ it determines. We will show in the
next section that E∆ can be made into a simplicial category with cofibrations and weak
equivalences whose K-theory is A(∗).
4
The K-theory of fiberwise ENRs
Recall from [18] the definition of the K-theory associated to a category with cofibrations
and weak equivalences. In this section we show that the K-theory of the simplicial category with cofibrations and weak equivalences E∆ is A(∗) (up to homotopy), where for any
compact EN R B, EB is viewed as a category with cofibrations and weak equivalences by
letting the weak equivalences be the isomorphisms. We use the notation and terminology
of [18] in what follows.
Let CW be the (small) category of pointed finite CW -complexes in (some) Rn and
pointed cellular maps considered as a category with cofibrations and weak equivalences
as follows: A cofibration is (a map isomorphic to) a cellular inclusion and a weak equivalence is a weak homotopy equivalence of spaces. The algebraic K-theory of a point,
written A(∗), is defined to be the K-theory of CW. The category EB is also a category with cofibrations and weak equivalences with the same definition of “weak equivalence”. Because the projection p associated to an object E of EB is a fibration, and
∂
because the induced functor ∂ ∗ of a map C → B in EN R is given by pullback along
op
∂, E is in fact a functor from EN R to categories with cofibrations and weak equivalences. We denote by wEB the subcategory of weak equivalences in EB . The inclusion
of CW in E∆0 is an exact functor. Note that CW and EB have cylinder functors, where
T (X → Y ) = ∗ ∪∗×I X × I ∪X×1 Y , and the Cylinder Axiom of [18] is satisfied in both
cases. Let Σ denote the suspension functor on any of those categories.
(0)
Let CW (0) (resp. E∆0 ) be the full subcategory of CW (resp. E∆0 ) consisting of
the connected objects. These are subcategories with cofibrations and weak equivalences. It follows from Prop. 1.6.2 of [18] that the maps wSΣ: wSCW → wSCW and
wSΣ: wSCW (0) → wSCW (0) are homotopy equivalences. Note that Σ: CW → CW facΣ0
tors as CW → CW (0) ⊂ CW. We have a commutative diagram
wSCW H wSΣ- wSCW
HHwSΣ0 6
6
i
i
H
HH
j
wSCW (0) wSΣ- wSCW (0)
where i is the inclusion. It follows that both i ◦ wSΣ0 and wSΣ0 ◦ i are homotopy
equivalences and therefore both i and wSΣ0 are homotopy equivalences. Analogous
(1)
remarks apply to E∆0 . Further, if we consider the subcategories CW (1) and E∆0 given
by the simply connected objects, the same argument gives that the induced maps in
(0)
(1)
K-theory of CW (1) ⊂ CW (0) and E∆0 ⊂ E∆0 are homotopy equivalences.
Theorem 4.1 The inclusion of CW in E∆0 induces a homotopy equivalence in K-theory.
9
(1)
Proof: By the remarks above, it suffices to show that the inclusion of CW (1) in E∆0 induces a homotopy equivalence in K-theory. We prove this by verifying that the inclusion
(1)
CW (1) → E0 has the Approximation Property (see pp. 352–353 of [18]). The conclusion
then follows from the Approximation Theorem (Thm. 1.6.7 of [18]). Condition App1
clearly holds, and we verify App2.
Lemma 4.1 If X is a simply connected compact EN R then X has the homotopy type
of a finite CW -complex.
Proof: Say X ⊂ U ⊂ Rn , and X is a retract of U , where U is open in Rn and its closure
is compact. We can write U as a union of a locally finite collection of cubes C1 , C2 , . . . as
follows: Consider all cubes in Rn with vertices in (Z[1/2])n ⊂ Rn and with side length
= 1/2k , k = 1, 2, . . .. Choose a maximal cube C1 ⊂ U . Proceeding inductively, choose a
maximal cube Ck ⊂ U − int(C1 ∪ . . . ∪ Ck−1 ). Only finitely many C ∈ {Ck } intersect X
since X is compact and {Ck } is locally finite. Their union has an obvious CW -structure
and retracts on X. By [20], X has the homotopy type of a CW -complex K, which is then
simply connected and dominated by a finite CW -complex. It follows from [19, Thm. F]
that K, and therefore X, has the homotopy type of a finite CW -complex, QED.
f
(1)
Now fix an object K of CW (1) and a map K → X in E∆0 . By the previous lemma,
a
b
we can find an object K0 of CW (1) and maps X → K0 , K0 → X which are homotopy
inverses of each other. Find a cellular map g: K → K0 homotopic to a ◦ f . In CW (1)
form the cylinder of g to obtain a diagram as in p. 348 of [18]:
- T (g) ∼
K0
K
HH
∼ p
H
gHH
1
j ?
H
K0
p
b
The map T (g) → K0 → X equals bg ∨ b when restricted to K ∨ K0 . But
g ∼ af ⇒ bg ∼ baf ∼ f ⇒ bg ∨ b ∼ f ∨ b
and since bg ∨ b extends to T (g), so does f ∨ b:
- T (g) ∼
K0
K
HH
h
∼
HH
f H
b
j ?
H
X
The triangle on the right shows that h is a weak equivalence and then the left triangle
shows that App2 is satisfied, QED.
Theorem 4.2 The canonical map wSE∆0 → wSE∆ is a homotopy equivalence.
F
Proof: We show that for fixed q the degeneracy E∆0 → E∆q has the Approximation
Property. Notice that F is given by ∆q ×. It is clear that F satisfies App1. We show
that it satisfies App2. Fix an object X of E∆0 and a map f : ∆q × X → E in E∆q .
π
Choose a point δ ∈ ∆q . Since E → ∆q is a fibration and ∆q is contractible, E is fiberhomotopy-equivalent to ∆q × π −1 (δ). Let a: E → ∆q × π −1 (δ) and b: ∆q × π −1 (δ) → E
be maps over ∆q which are homotopy inverses over ∆q .
f
a
Consider the composition ∆q × X → E → ∆q × π −1 (δ). We write (af )(d, x) =
c
(d, (af )d (x)). Define X → π −1 (δ) by x 7→ (af )δ (x). Using a contraction of ∆q to δ, we
see that af is homotopic over ∆q to F (c). Form the cylinder of c in E∆0 so that we have
a commutative diagram
10
X
HH
∼
- T (c) ∼ p
H
cHH
1
j ?
H
π −1 (δ)
π −1 (δ)
where π −1 (δ) has the basepoint induced by c. Now apply F to the diagram above and let
F (p)
b
g be the composition F (T (c)) → F (π −1 (δ)) → E. We obtain a commutative diagram
of maps over ∆q :
-F (T (c)) ∼
F (X)
H
HH
∼ g
∼ b ◦ F (c) HH
b
j ?
H
E
F (π −1 (δ))
Now F (X) ∨ F (π −1 (δ)) → F (T (c)) is a cofibration over ∆q and
b ◦ F (c) ∨ b ∼ b ◦ a ◦ f ∨ b ∼ f ∨ b
where “∼” means “homotopic over ∆q ”. Thus there is h: F (T (c)) → E making the
following commutative:
-F (T (c)) ∼
F (X)
HH
f
h
∼
H
HH
b
j ?
H
E
F (π −1 (δ))
The triangle on the right shows that h is a weak equivalence in E∆q and then the triangle
on the left shows that F satisfies App2, QED.
For any category with cofibrations C we write iC for the category of weak equivalences
in C consisting of the isomorphisms of C. It is shown in the Cor. to the Lemma 1.4.1 of
[18] that the inclusion of sC in iSC is a homotopy equivalence, where sn C is the set of
objects of iSn C. For any category with cofibrations C, if wC is a subcategory of weak
equivalences in C, then C w denotes the full subcategory with cofibrations of C with objects
all C such that the canonical map ∗ → C is in wC.
Theorem 4.3 The inclusion of sE∆ in wSE∆ is a homotopy equivalence.
w
w
Proof: We show that sE∆
∼ iSE∆
is contractible. The conclusion then follows from
the Fibration Theorem (Thm. 1.6.4 of [18]), since sE∆ ∼ iSE∆ . Here we need the
fact that “the realization of a simplicial fibration up to homotopy is a fibration up to
homotopy provided that the base is connected in every simplicial dimension” (see Lemma
5.2 of [17] for a more precise statement and a proof). The Fibration Theorem applies to
iE∆q ⊂ wE∆q , although wE∆q does not satisfy the extension axiom (see p. 327 of [18]),
because we can consider only simply connected EN R∆q and weak equivalences between
simply connected spaces satisfy the extension axiom (a homology isomorphism between
simply connected spaces is a weak homotopy equivalence).
p
σ
Let E → B be a compact EN RB with section B → E, and suppose σ is a homotopy
equivalence. Let F = I × E ∪1×E B.
Lemma 4.2 F is EN RI×B .
Proof: We may assume that for some disk D in Rn we have an inclusion E ⊂ B × D
over B. In the commutative diagram below
11
1 E
E
incl.
p
? proj. ?
-B
B ×D
the inclusion is a cofibration and p is a fibration and a homotopy equivalence. It follows
that E is a retract (over B) of B × D, thus F is a retract of
I × B × D ∪1×B×D B = B × (I × D ∪1×D ∗)
which is EN RI×B , since I × D ∪1×D ∗ is easily seen to be EN RI , QED.
w
w
Let CB be the exact functor from EB
to EI×B
given by
CB (f )
f
CB (E → E 0 ) = CB (E) → CB (E 0 ), where
CB (E) = {(t, b, ty) : (b, y) ∈ E ⊂ B × R∞ }
with section I × B → CB (E) given by (t, b) 7→ (t, b, ts(b)) where the section of E is given
w
by b 7→ (b, s(b)). By the previous lemma, CB (E) is indeed an object of EI×B
. The map
CB (f ) is defined by CB (f )(t, b, ty) = (t, b, tfb (y)), where f (b, y) = (b, fb (y)).
w
w
In the notation of Prop. 2.1, C gives a map from E∆
to EI×∆
with i∗0 C = ∗ and
∗
w
w
∗
i1 C = 1E∆w . Note that (sn E )I×∆ = sn (EI×∆ ) and by Prop. 2.1 sn i0 is homotopic to sn i∗1 .
w
It follows that sn E∆
is contractible for each n, since sn i∗0 ◦ sn C = ∗ and sn i∗1 ◦ sn C = 1,
QED.
5
A cyclic category
In this section we show that a certain “almost cyclic” object which we will study in §6
(namely Θ0 F) which does not have degeneracies, can be traded for another one that does.
We begin by giving a summary of the problem. Fix a category C and consider the object
Θ0 C which is “almost a cyclic category”: The objects of Θ0p C are precisely the elements
f0
f1
fp−1
fp
f0
f0
0
fp−1
fp0
0
1
of Npcy C. A map from C0 → C1 → . . . → Cp → C0 to C00 →
C10 →
. . . → Cp0 → C00
is a pair (s, r), where s = (s0 , s1 , . . . , sp ) and r = (r0 , r1 , . . . , rp ) fit into commutative
diagrams in C:
si
fi
- Ci0
- Ci+1
Ci
Ci
@
6
ri
ri
si+1
1@
0
?
R ?
@
0 fi0
Ci
Ci
Ci+1
The reason that Θ0 C is not a cyclic category is that there are no degeneracy maps
Θ0p C → Θ0p+1 C, since given a map (s, r) as above, it is not true that the following
commutes
1
- Ci
Ci
6
ri
si
?
0 10
Ci
Ci
unless ri is an isomorphism instead of just a retraction in C. However, simplicial objects
without degeneracies do have realizations, so the homotopy type of Θ0 C is defined. We
will construct a cyclic category ΘC which has that homotopy type.
Let R be the full subcategory of Γ with objects < 0 > and < 1 >. The non-identity
r
s
maps of R are a retraction < 1 >→< 0 >, a section < 0 >→< 1 >, and sr: < 1 >→< 1 >.
Let C be a small category. We identify the R-maps in C with diagrams
f0
C0 - C00
s 6
r
s0 6
r0
? f1
?
C1 - C10
12
in C with rs and r0 s0 identities, s0 f0 = f1 s, and f0 r = r0 f1 . In addition to the usual
composition of R-maps, which we call “horizontal”, there is a “vertical” composition of
R-maps. In fact, there is a “bicategory” (see [17]) RC with bimorphisms the R-maps
in C, vertical morphisms the R-objects in C, horizontal morphisms the maps in C, and
objects the objects of C.
View the multiplicative monoid {0, 1} = M as a bicategory with trivial vertical
direction. Call an R-map b, as in the previous diagram, special, if f1 = s0 f0 r. Note
that the horizontal composition of two bimorphisms is special if so is one of them. Thus
M × RC has a sub-bicategory rC whose bimorphisms are:
{(m, b) : m = 0 ⇒ b is special}
For any bicategory B let N cy B be the (cyclic category) “cyclic nerve of B in the horizontal
direction”. The restriction of M × RC → M to rC induces a map from N cy rC to N cy M .
We define ΘC as the pullback of Θ ⊂ N cy M ← N cy rC, where Θ is defined at the end of
§2.4.
We describe ΘC more explicitly: ob Θp C = Θp × Npcy C and a map in Θp C from
fi
f0
0
{mi , Ci → Ci+1 }i∈Z/p+1 to {m0i , Ci0 →i Ci+1
}i∈Z/p+1 exists only if mi = m0i (all i) in
s
r
which case it is given by section-retraction pairs {Ci →i Ci0 , Ci0 →i Ci }i∈Z/p+1 in C such
that for each i ∈ Z/p + 1 the diagram
fi
- Ci+1
Ci
6
si ri si+1 6
ri+1
? f0
?
i0
0
Ci
Ci+1
is a bimorphism in RC, special if mi = 0.
Let Φ be the subcategory of ∆ consisting of the injective maps. Thus any simplicial
object determines a Φop -object. A Φop -space X has a realization kXk with the following
properties (see [14]): X 7→ kXk is a functor from Φop -spaces to spaces. If X → Y is a
map of Φop -spaces with Xn → Yn a weak equivalence for all n, then kXk → kY k is a
weak equivalence. Finally, if X is a simplicial space arising from the partial realization of
a bisimplicial set, then there is a natural weak equivalence kXk → |X|. Thus Proposition
2.4 implies the following:
Proposition 5.1 Let f : X → Y be a map of cyclic sets. If the induced map from
k(sdk X)Ck k to k(sdk Y )Ck k is a homotopy equivalence for all k ≥ 1 then f is a Ck homotopy equivalence for all k ≥ 1.
Observe Θ has a Φop -subset Θ0 , Θ0p = {(0, 0, . . . , 0)}. If X is any Φop -category, there
is a natural isomorphism X ≈ Θ0 × X. There are Φop -subcategories Θ0 C ⊂ ΘC s of
ΘC defined as follows: Θ0 C is the pullback of Θ0 ⊂ Θ ← ΘC. The objects of ΘC s
are the objects of ΘC and its morphisms are all morphisms of ΘC that have all their
associated bimorphisms special. There is a natural isomorphism Θ × Θ0 C ≈ ΘC s , which,
together with the isomorphism Θ0 × Θ0 C ≈ Θ0 C, shows that the inclusion Θ0 C ⊂ ΘC s
is isomorphic to Θ0 × Θ0 C ⊂ Θ × Θ0 C. In particular, the inclusion Θ0 C ⊂ ΘC s is a
homotopy equivalence.
Suppose now that F is a cyclic subset of N cy C, and that I is a subcategory of the
vertical morphisms of RC containing all objects of C. Let Θp (F, I) be the subcategory
of Θp C with objects Θp × Fp and morphisms all morphisms of Θp C between elements
of Θp × Fp with all the associated vertical morphisms of RC in I. Then Θ(F, I) is a
cyclic subcategory of ΘC with Φop -subcategories Θ0 (F, I) ⊂ Θ(F, I)s defined as follows:
Θ0 (F, I) is the pullback of Θ0 C ⊂ ΘC ⊃ Θ(F, I) and Θ(F, I)s is the pullback of ΘC s ⊂
ΘC ⊃ Θ(F, I). The inclusion Θ0 (F, I) ⊂ Θ(F, I)s is a homotopy equivalence for the same
reason Θ0 C ⊂ ΘC s is one. If I contains all the vertical morphisms of RC we write ΘF
(resp. Θ0 F ) for Θ(F, I) (resp. Θ0 (F, I)).
13
We say that a subcategory A of B is a right ideal in B if the following condition is
β
satisfied: For any a → b in B, if a is an object of A then β is a morphism of A.
Proposition 5.2 If for all p ≥ 0 the category Θp (F, I) is a right ideal in Θp C, then
Θ0 (F, I) ⊂ Θ(F, I) is a homotopy equivalence.
Proof: We show that Θ(F, I)s ⊂ Θ(F, I) is a homotopy equivalence. For each n ≥ 0
we give a retraction Nn Θ(F, I) → Nn Θ(F, I)s such that the composition Nn Θ(F, I) →
Nn Θ(F, I)s ⊂ Nn Θ(F, I) is a homotopy equivalence.
For any bicategory B we denote by N B the (simplicial category) “nerve of B in the
vertical direction”, and we identify N N cy B with N cy N B whenever no confusion seems
likely. Define T0 : Nn RC → Nn RC by
f0
C0
s0 6
r0
? f1
C1
s1 6
r1
?
..
.
- C00
s00 6
r00
?
- C10
s01 6
r10
?
..
.
0
sn−2 6
rn−2 s0n−2 6
rn−2
? fn−1 ?
- C0
Cn−1
n−1
0
0
sn−1 6
rn−1 sn−1 6
rn−1
? fn
?
- Cn0
Cn
f0
C0
s0 6
r0
?
C1
s1 6
r1
?
7→
- C0
0
s00 6
r00
?
- C10
s01 6
r10
?
s00 f0 r0
..
.
..
.
0
sn−2 6
rn−2
s0n−2 6
rn−2
0
0 0
?
?
s
. . . s1 s0 f0 r0 r1 . . . rn−2 - 0
Cn−1 n−2
Cn−1
0
sn−1 6
rn−1
s0n−1 6
rn−1
0 0
?s0 s0
?
. . . s1 s0 f0 r0 r1 . . . rn−2 rn−1 0
- Cn
Cn n−1 n−2
T0 is not a functor because it does not take identities to identities. However, it does
“preserve composition”, i. e., (if f f 0 is defined then) T0 (f )T0 (f 0 ) (is defined and) equals
T0 (f f 0 ). This is enough to produce a map of Φop -sets from N cy Nn RC to N cy Nn RC. Let
T1 denote the identity of Nn RC. Then T : M × Nn rC → Nn rC defined by T (m0 , m, C) =
(m, Tm0 (C)) also preserves composition, and therefore induces a Φop -map
N cy M × N cy Nn rC → N cy Nn rC.
The last map restricts to a map from N cy M × Nn ΘC to Nn ΘC (this is clear), which in
turn restricts to a map from N cy M × Nn Θ(F, I) to Nn Θ(F, I) (we check this below).
We have to check that the map N cy M × Nn ΘC → Nn ΘC takes N cy M × Nn Θ(F, I)
in Nn Θ(F, I). The image of (m0 , m, C) ∈ Npcy M × Nn Θp C under that map, say (m, C 0 ),
has the same vertical morphisms in RC as C. Thus it remains to show that the rows of
C 0 are in Fp . This follows from the fact that Θp (F, I) is a right ideal in Θp C.
G0
We claim the restriction Θ × Nn Θ(F, I) → Nn Θ(F, I) is a homotopy equivalence:
Consider the diagram
proj
G0Nn Θ(F, I) Θ × Nn Θ(F, I)
Nn Θ(F, I)
Y
H
*
HH 1
1
6
diag
HH
H Nn Θ(F, I) where diag is induced by the map rC → M × rC, (m, b) 7→ (m, m, b). The left triangle
clearly commutes. The right triangle commutes because Tm (C) = C if (m, C) ∈ Nn rC.
Finally proj is a homotopy equivalence since Θ is contractible, showing that G0 is a
homotopy equivalence.
G0
Now observe that Θ0 × Nn Θ(F, I) ⊂ Θ × Nn Θ(F, I) → Nn Θ(F, I) factors as the
14
G
composition Θ0 × Nn Θ(F, I) → Nn Θ(F, I)s ⊂ Nn Θ(F, I) since all the bimorphisms of
T0 (C) are special for any map C in Nn RC.
≈
G
We claim Nn Θ(F, I) → Θ0 × Nn Θ(F, I) → Nn Θ(F, I)s is the map we want: It is a
retraction since T0 (C) = C for any map C in Nn RC consisting of special bimorphisms,
≈
G
and the composition Nn Θ(F, I) → Θ0 × Nn Θ(F, I) → Nn Θ(F, I)s ⊂ Nn Θ(F, I) equals
G0
≈
Nn Θ(F, I) → Θ0 × Nn Θ(F, I) ⊂ Θ × Nn Θ(F, I) → Nn Θ(F, I) which is a homotopy
equivalence since so is Θ0 ⊂ Θ and G0 , QED.
Finally we note that these constructions are natural in the following sense: If C is
a D-category, F is a D-cyclic subset of N cy C, and I is a D-subcategory of the vertical morphisms of RC, then Θ(F, I) is a D-cyclic category with a D × Φop -subcategory
Θ0 (F, I).
6
Fixed point problems
Let B be a compact EN R. We define a category GB : The objects are all (E, F ) with E
an EN RB in some B × Rn and F a compact subset of E. The morphisms from (E, F )
to (E 0 , F 0 ) are germs along F of maps of pairs f : (U, F ) → (E 0 , F 0 ) where U is an open
subset of E, f is over B, and the restriction F → F 0 is a homeomorphism. Note that G
is an EN Rop -category for the same reason E is one.
We call an endomorphism of (E, F ) in N0cy GB a k-periodic point problem if F is the
k-periodic point set of one of its representatives. For p ≥ 0 we call a p-simplex of N cy GB
a k-periodic point problem if so is one of its vertices. The k-periodic point problems
form a cyclic subset of N cy GB which is preserved by the induced maps N cy GB → N cy GC
of maps C → B. Further, if we write Pk p B = k-periodic point problems in Npcy GB then
Θp PkB ⊂ Θp GB is a right ideal, in fact:
• For any p and σ ∈ Npcy GB , if a vertex of σ is in Pk 0B then σ is in Pk p B (thus
N cy G = Pk q (N cy G − Pk )).
• For all R-objects (s, r) in GB and all f in N0cy GB with sf r defined, sf r ∈ Pk 0B if
and only if f ∈ Pk 0B (thus ΘG = ΘPk q (ΘG − ΘPk )).
We will say “fixed point problem” instead of “1-periodic point problem” and we will
write “F” instead of “P1 ”. Note that the isomorphism ψ: (sdk N cy G)Ck → N cy G restricts
to an isomorphism from (sdk F)Ck to Pk .
In this section we construct maps ΘF ix∆ ← H ∆ ↓ΘF ix∆ → H∆ ⊂ ΘF∆ which are
Ck -homotopy equivalences for all k ≥ 1, and we shall analyze the Ck -homotopy type of
ΘF ix in later sections. We first construct H∆ ⊂ ΘF∆ . Let IB be the subcategory of
the vertical morphisms of RGB defined as follows: An R-object (s, r) in GB is in IB if
and only if s is (the germ of) an inclusion. Let H = Θ(F, I) ⊂ ΘF.
Lemma 6.1 The inclusion H∆ ⊂ ΘF∆ is a Ck -homotopy equivalence for all k ≥ 1.
Proof: We show that Nn HB ⊂ Nn ΘFB is a Ck -homotopy equivalence for all k ≥ 1,
n ≥ 0, and B ∈ ob EN R. We use the following simple fact:
If X, Y1 , and Y2 are G-spaces, f : X → Y1 q Y2 is a G-homotopy equivalence, and
X1 = f −1 (Y1 ), then the restriction f : X1 → Y1 is a G-homotopy equivalence.
Observe that N cy rG = ΘG q (N cy {1} × N cy RG) and ΘG = ΘF q (ΘG − ΘF). Let
0
HB
be the “horizontally full” sub-bicategory of RGB with vertical morphisms IB . Let
00
0
HB
= M × HB
∩ rGB . Then H = N cy H00 ∩ ΘF. Thus it suffices to show that the
00
inclusion of Nn N cy HB
in Nn N cy rGB is a Ck -homotopy equivalence. This will follow
00
from Proposition 2.5 if we show that the inclusion Nn HB
⊂ Nn rGB is an equivalence of
categories. This is true for essentially the same reason that the category of n composable
injections is equivalent to the category of n composable inclusions, QED.
n
For B a compact EN R and n a non-negative integer let GB
be the full subcategory of
n
GB with objects all (E, F ) ∈ ob GB such that E = B × Rn and let F ixnB = N cy GB
∩ FB .
15
Let ΘF ixn = Θ × F ixn . There is a stabilization map s: ΘF ixn → ΘF ixn+1 defined as
follows:
For m ∈ M = {0, 1} let m̂ denote the map from R to R given by multiplication by
m. Given (m, C) ∈ Θp F ixnB with m = (m0 , m1 , . . . , mp−1 , mp ) ∈ Θp and C ∈ F ixnp B
given by
)
f0
fp−1
f1
fp
C0 → C1 → . . . → Cp → C0
with Ci = (B × Rn , Fi )
(6.1)
0
let s(m, C) = (m, C 0 ) ∈ Θp F ixn+1
B , where C is the diagram:
C00
f0 ×m̂0
→
fp−1 ×m̂p−1
f1 ×m̂1
fp ×m̂p
→
Cp0 →
C10 → . . .
n+1
0
with Ci = (B × R
, Fi )
C00
)
(6.2)
Let ΘF ix be the colimit over n of ΘF ixn . In order to compare ΘF ix and H we need
an intermediate object H ↓ΘF ix.
Fix ([p], B) ∈ ob Λ × EN R and view (the set) Θp F ixB as a (discrete) category. There
is an inclusion in : Θp F ixnB → Hp B . We claim that there is a map φp,B : in → in+1 s in
n
(Hp B )Θp F ixB :
n+1
Let π: R
→ Rn be the (orthogonal) projection for which i: Rn ⊂ Rn+1 is a section.
For (m, C) ∈ Θp F ixnB with C as in (6.1) and C 0 as in (6.2) let φp,B;m,C be the following
morphism of Hp B :
m0 , f0
C0
6
B×i
B×π
m1 , f1
- C1
6
B×i
B×π
m0 , f0 × m̂0 - ?
C10
?
C00
- ...
m1 , f1 × m̂1 - . . .
mp , fp
- C0
6
B×i
B×π
mp , fp × m̂p - ?
C00
(6.3)
Every square above is a bimorphism in RGB , special if mi = 0. Further, B × i is (the
germ of) an inclusion. This shows that φp,B;m,C is a map in Hp B . Since Θp F ixnB is
discrete, the assertion that φp,B; is natural is vacuously true. Note that for any map
∂: ([p], B) → ([p0 ], B 0 ) in Λ × EN R we have ∂ ∗ φp0 ,B 0 ;m,C = φp,B;∂ ∗ (m,C) (in the notation
of [12], φ ∈ Hom(ΘF ixn , H)).
We are now ready to define H ↓ΘF ix: For ([p], B) ∈ ob Λ × EN R, let Hp B ↓Θp F ixnB
have objects all maps Ĉ → in (C) in Hp B with Ĉ ∈ ob Hp B and C ∈ Θp F ixnB . A map
in Hp B ↓Θp F ixnB from Ĉ → in (C) to Ĉ 0 → in (C 0 ) exists only if C = C 0 , in which case
it is given by a map Ĉ → Ĉ 0 making the obvious triangle commute. There are forgetful
functors Θp F ixnB ← Hp B↓Θp F ixnB → Hp B .
Lemma 6.2 The map Hp B↓Θp F ixnB → Θp F ixnB is a homotopy equivalence.
Proof: For any C in (the set) Θp F ixnB , the connected component of (the category)
1
1
Hp B↓Θp F ixnB containing C → C has C → C as a terminal object, QED.
All this is sufficiently natural to give a Λop × EN Rop -category H ↓ΘF ixn and maps
ΘF ixn ← H ↓ΘF ixn → H. In fact, we have a commutative diagram
-H
ΘF ixn H ↓ΘF ixn
s
ŝ
1
?
?
?
-H
ΘF ixn+1 H ↓ΘF ixn+1
where the map ŝ evaluated at the object Ĉ → in (C) of Hp B↓Θp F ixnB is the composition
φ
Ĉ → in (C) → in+1 s(C). The map φ is defined in (6.3). Let H ↓ΘF ix be the colimit
(with respect to ŝ) of H ↓ΘF ixn . We obtain a diagram ΘF ix ← H ↓ΘF ix → H.
16
Theorem 6.1 In the diagram ΘF ix∆ ← H ∆ ↓ ΘF ix∆ → H∆ ⊂ ΘF∆ all maps are
Ck -homotopy equivalences for all k ≥ 1.
Proof: Fix k ≥ 1 and let D = Φop × EN Rop . There are D-categories
ΘP erkn = ψ(sdk ΘF ixn )Ck
Θ
0
P erkn
k
H = Θ(Pk , I)
n Ck
0
H 0 = Θ0 (Pk , I)
= ψ(sd Θ F ix )
as well as a map of diagrams αn → αn+1 , where αn is the diagram
∼
- H0
Θ0 P erkn H 0↓Θ0 P erkn
∼
?
∼
ΘP erkn ∼
?
-H
?
H ↓ΘP erkn
where the definitions of H ↓ΘP erkn and H 0 ↓Θ0 P erkn are completely analogous to the
definition of H ↓ΘF ixn , and, as before, the map from αn to αn+1 is given by s on the
left, ŝ on the center, and the identity on the right. Above, the inclusion of Θ0 P erkn
in ΘP erkn is a homotopy equivalence because Θ is contractible, the inclusion of H 0 in
H is a homotopy equivalence by Prop. 5.2, and the left horizontal maps are homotopy
equivalences by the same argument as in Lemma 6.2.
Consider the colimit of the diagrams αn : It is a diagram of the form:
∼
- H0
H 0↓Θ0 P erk
Θ0 P erk ∼
?
∼
ΘP erk ∼
?
-H
?
H ↓ΘP erk
Note that (sdk (H ↓ΘF ix))Ck is isomorphic to H ↓ΘP erk . In fact, the bottom row of the
diagram above is isomorphic to the diagram obtained from
ΘF ix ← H ↓ΘF ix → H
by applying sdk , taking fixed points of Ck , applying ψ, and forgetting the degeneracies
0
is
in the Θ-direction. Thus it suffices to prove that the map H 0∆ ↓ Θ0 P erk ∆ → H∆
a homotopy equivalence: It would then follow that all maps of the above diagram are
homotopy equivalences (after restricting from EN Rop to ∆op ), and the conclusion of
the Theorem would follow from Proposition 5.1. We show that in fact for [m] ∈ ob ∆
and [p] ∈ ob Φ the map Nm (Hp0 ∆↓Θ0p P erk ∆ ) → Nm Hp0 ∆ is a fibration and a homotopy
equivalence.
A typical element of Nm Hp0 B is a diagram
C00
60
f00 -
r0
C10
60
f01 -
...
f0p−1
-
r0
f10
r1
- C1
1
61
rm−1
0
fm
-
- ...
f1p−1
- Cp
1
6p
f0p-
r0
- C0
1
60
1
fm
-
r1
..
.
6p
rm−1
rm−1
...
p−1
fm
-
p
Cm
C00
60
f1p
r1
..
.
61
1
Cm
C0p
6p
r0
f11
r1
..
.
60
0
Cm
C01
61
p
fm
-
..
.
60
rm−1
0
Cm
(6.4)
j
where Cij = (Eij , F j ) ∈ ob GB , such that Eij ⊂ Ei+1
(on the level of germs), rij is (the
j
j
germ of) a retraction for which Ei ⊂ Ei+1 is a section, the k-periodic point set of each row
is F 0 , and each square represents a special bimorphism. This last condition implies that
the maps {fij }i>0 are determined by (f00 , f01 , . . . , f0p ). We abbreviate such a diagram by
j∈Z/p+1
((f00 , f01 , . . . , f0p ), {rij }0≤i≤m−1 ). Further, Nm−1 (Hp0 B ↓Θ0p P erkn B ) may be identified with
17
j
a subset of Nm Hp0 B , namely the set of all diagrams as above with Em
= B × Rn for all
0
0
j ∈ Z/p + 1. With these identifications, the map from Nm (Hp B↓Θp P erk B ) to Nm Hp0 B
is:
j∈Z/p+1
j∈Z/p+1
((f 0 , f 1 , . . . , f p ), {rij }0≤i≤m ) 7→ ((f 0 , f 1 , . . . , f p ), {rij }0≤i≤m−1 )
Now fix a diagram
τ̂Nm (Hp0 ∆↓Θ0p P erk ∆ )
∂∆q
∩
σ̂
∆q
?
- Nm H 0
p∆
Since ∂∆q is finite, τ̂ lifts to Nm (Hp0 ∆ ↓Θ0p P erkn ∆ ) for some n. Let σ̂ be represented by
an element σ of Nm Hp0 ∆q , say σ is given by diagram (6.4). We may assume that all Eij
are subsets of ∆q × Rn by increasing n if necessary.
Let τ̂ be represented by τ ∈ Nm (Hp0 ∂∆q ↓Θ0p P erkn ∂∆q ) (that such a representation
j∈Z/p+1
exists is a consequence of Cor. 3.3), say τ = ((g 0 , g 1 , . . . , g p ), {tji }0≤i≤m ). Since the
diagram above commutes, g j = f0j |∂∆q and tji = rij |∂∆q for 0 ≤ i ≤ m − 1 and all
j ∈ Z/p + 1.
j∈Z/p+1
We must find ρ ∈ Nm Hp0 ∆q with ρ|∂∆q = τ and ρ = ((f00 , f01 , . . . , f0p ), {rij }0≤i≤m ).
We have all the ingredients for ρ except the retractions {rj }j∈Z/p+1 . These should
m
j
j
satisfy rm
|∂∆q = tjm . That such rm
exist is implied by Corollary 3.2, QED.
7
Fixed point problems in euclidean space
We prove in this section that ΘF ix∆ has the Ck -homotopy type of Θ0k−1 F ix∆ . The proof
will be an application of Proposition 2.2, i. e. we will show that certain face maps are
weakly homotopic. This is done in the proof of Lemma 7.5 where we write down a lot
of homotopies. To illustrate the ideas involved, we give below a sketch of that proof for
the special case k = p = 1.
Given a fixed point problem
f0
f1
A = (Rn , F0 ) → (Rn , F1 ) → (Rn , F0 )
we will construct a “path of fixed point problems”
gt
{Bt = (Rn × Rn , Gt ) → (Rn × Rn , Gt )}a≤t≤b
between
Ba = (Rn × Rn , F0 × 0)
f0 f1 ×0
→
(Rn × Rn , F0 × 0)
f0 f1
(thus Ba “stably equals” ∂0 A = (Rn , F0 ) → (Rn , F0 )) and
Bb = (Rn × Rn , F1 × 0)
f1 f0 ×0
→
(Rn × Rn , F1 × 0)
(thus Bb “stably equals” ∂1 A). The path {Bt } will be a concatenation of two paths
meeting at
˜ 0
×f
˜ 0 ) f1→
˜ 0)
C = (Rn × Rn , F1 ×F
(Rn × Rn , F1 ×F
where
˜ 0 = {(x, f1 (x))|x ∈ F1 } = {(f0 (x), x)|x ∈ F0 }
F1 ×F
and
˜ 0 (x, y) = (f0 (y), f1 (x))
f1 ×f
(we abuse notation by identifying a map with its germ). Notice that the fixed point set
˜ 0 is indeed F1 ×F
˜ 0 , which is compact (in fact homeomorphic, via the projections,
of f1 ×f
to both F1 and F0 ), thus C is indeed a fixed point problem.
18
We give below the path from Ba to C only, the path from C to Bb being similar. It
will again be a concatenation of two paths, namely
h
˜ 0 ) →t (Rn × Rn , F1 ×F
˜ 0 )}0≤t≤1
{(Rn × Rn , F1 ×F
with
ht (x, y) = ((1 − t)f0 f1 (x) + tf0 (y), f1 (x))
and
kt
˜ 0 )}0≤t≤1
˜ 0) →
(Rn × Rn , F1 ×tF
{(Rn × Rn , F1 ×tF
with
˜ 0 = {(x, tf1 (x))|x ∈ F1 }
F1 ×tF
and
kt (x, y) = (f0 f1 (x), tf1 (x)).
˜ 0 is the fixed point set of ht and that F1 ×tF
˜ 0
The easy verification of the facts that F1 ×F
is the fixed point set of kt is left to the reader.
Theorem 7.1 For all k ≥ 1 the composition
Θ0k−1 F ix∆ ⊂ Θk−1 F ix∆ = sdk0 ΘF ix∆ → sdk ΘF ix∆
is a Ck -homotopy equivalence.
Proof: We show that the inclusion of Θ0k−1 F ix∆ in sdk Θ0 F ix∆ is a Ck -homotopy equivalence. Let Θ1 ⊂ Θ denote the simplicial set generated by Θ0 . Thus Θ1p is the singleton
containing (1, 1, . . . , 1, 0) ∈ M p+1 and the inclusion Θ1 ⊂ Θ is a homotopy equivalence.
Let Θ1 P erkn denote the pullback of the following diagram:
ψ(sdk ΘF ixn )Ck → ψ(sdk Θ)Ck = Θ ⊃ Θ1
Thus Θ1 P erkn = Θ1 × ψ(sdk F ixn )Ck , in particular the canonical map from Θ1 P erkn to
(sdk ΘF ixn )Ck is a homotopy equivalence.
The stabilization map ΘF ixn → ΘF ixn+1 induces a map Θ1 P erkn → Θ1 P erkn+1 . Let
1
Θ P erk = colim Θ1 P erkn . Thus Θ1 P erk is the pullback of
ψ(sdk ΘF ix)Ck → ψ(sdk Θ)Ck = Θ ⊃ Θ1
and the canonical map Θ1 P erk → (sdk ΘF ix)Ck is a homotopy equivalence.
Recall that Θ0 P erkn = (sdk Θ0 F ixn )Ck and Θ0 P erk = (sdk Θ0 F ix)Ck (these do not
have degeneracies in the Θ-direction). Fix l dividing k and let m = k/l. Observe that
for each n there is a commutative diagram
0
n
(Θ0k−1 F ixn )Cl ≈ sdm
0 Θ P erl
≈- m 1
sd0 Θ P erln
?
(sdk Θ0 F ixn )Cl ≈ sdm Θ0 P erln
≈- m 1? n
sd Θ P erl
m
where the isomorphisms on the left are given by sdm
0 ψ and sd ψ and the isomorphisms
on the right are induced by the (unique) isomorphism between the Φop -sets Θ0 and Θ1 .
Lemma 7.1 For all positive integers k and non-negative integers p and n the diagram
Θ0p P erkn ∆
≈- 1
Θp P erkn ∆
?
Θ0p P erkn+1
∆
≈- 1 ?n+1
Θp P erk ∆
commutes up to homotopy.
19
Proof: We have to show that two maps from Θ1p P erkn ∆ to Θ1p P erkn+1
∆ are homotopic.
Define α: Θ1p P erkn ∆ → Θ1p P erkn+1
by
mapping
I×∆
fi
{(∆q × Rn , Fi ) → (∆q × Rn , Fi+1 )}i∈Z/p+1
to
f0
0
{(I × ∆q × Rn+1 , Fi0 ) →i (I × ∆q × Rn+1 , Fi+1
)}i∈Z/p+1
where Fi0 = I × Fi and
fi0 (t, d, x, y)
=
(t, fi (d, x), 0), i = p
(t, fi (d, x), ty), 0 ≤ i ≤ p − 1
The conclusion now follows from Proposition 2.1, QED.
Lemma 7.2 For all positive integers k and non-negative integers n the diagram
kΘ0 P erkn ∆ k
≈kΘ1 P erkn ∆ k
?
kΘ0 P erkn+1
∆ k
?
≈kΘ1 P erkn+1
∆ k
commutes up to homotopy.
Proof: We have to show that two maps from kΘ1 P erkn ∆ k to kΘ1 P erkn+1
∆ k are homotopic.
Call these maps kα1 k and kα2 k. Let β1 : Θ1 P erkn ∆ → ΘP erkn ∆ be the inclusion and
≈
incl.
β2 : Θ1 P erkn ∆ → ΘP erkn ∆ be the composition Θ1 P erkn ∆ → Θ0 P erkn ∆ → ΘP erkn ∆ .
Since the projection π of ΘP erkn ∆ on Θ1 P erkn ∆ is a homotopy equivalence, and since
πβ1 = πβ2 = 1Θ1 P erkn ∆ , kβ1 k and kβ2 k are homotopic. The conclusion follows from the
commutative diagrams
αj - 1
Θ P erkn+1
∆
6
π
stab.
- ΘP ern+1
k∆
Θ1 P erkn ∆
βj
?
ΘP erkn ∆
for j = 1, 2, QED.
We state the following elementary fact as a lemma:
Lemma 7.3 Let {Xi , Yi }i=1,2 be CW -complexes which are the union of subcomplexes
f
g
{Xin , Yin }n=1,2,...
and let X1 → X2 and Y1 → Y2 be filtration preserving maps. Suppose
i=1,2
hn
i
that there are homotopy equivalences {Xin →
Yin }n=1,2,...
such that for each n we have
i=1,2
n
ghn1 = hn2 f . Suppose also that hni is homotopic to hn+1
|X
i for i = 1, 2. Then if g is a
i
homotopy equivalence, so is f .
Thus the following lemma completes the proof of Theorem 7.1:
Lemma 7.4 For each k ≥ 1, the simplicial space [p] 7→ |Θ1p P erk ∆ | is constant up to
homotopy.
Proof: It suffices by Proposition 2.2 to show that any two face maps of |Θ1 P erk ∆ | are
weakly homotopic. This will follow if we show that for every n any two compositions
∂
(k+1)n
∂j
(k+1)n
Θ1p P erkn ∆ →i Θ1p−1 P erkn ∆ → Θ1p−1 P erk ∆
and
Θ1p P erkn ∆ → Θ1p−1 P erkn ∆ → Θ1p−1 P erk ∆
are homotopic. Using Lemma 7.1 we obtain Lemma 7.4 from the next lemma:
20
Lemma 7.5 For all positive integers n and p, and for all i with 0 ≤ i ≤ p − 1 the two
compositions
∂
(k+1)n
Θ0p P erkn ∆ →i Θ0p−1 P erkn ∆ → Θ0p−1 P erk ∆
Θ0p P erkn ∆
∂i+1
→
Θ0p−1 P erkn ∆
→
and
(k+1)n
Θ0p−1 P erk ∆
are homotopic.
Remark. The reason for going back and forth between Θ1 P erk ∆ and Θ0 P erk ∆ is that
the former has degeneracies (these are needed in order to apply Prop. 2.2) and the latter
is the fixed point set of Ck in sdk Θ0 F ix∆ (note that Θ1 P erk ∆ is not the fixed point set
of Ck in sdk Θ1 F ix∆ , in fact sdk Θ1 F ix∆ is not invariant under Ck in sdk ΘF ix∆ ).
Proof: If p ≥ 2 then using the action of Cp+1 on Θ0p P erkn ∆ and the action of Cp
(k+1)n
on Θ0p−1 P erk ∆
we may assume that i = 0. Let ∂00 (resp. ∂10 ) denote the first (resp.
second) composition in the statement of the lemma. Both of these will be compared to
(k+1)n
α: Θ0p P erkn ∆ → Θ0p P erk ∆
defined as follows:
For any space X and any m ∈ Z, m ≥ 0, we write P for the projection from X × Rm
to Rm . Recall the action of Ck+1 on R(k+1)n defined in §2.1, and let c be the self-map
of ∆q × R(k+1)n given by c(d, x) = (d, ck+1 (x)). Fix an element τ of Θ0p P erkn ∆q given by
fi
τ = {(∆q × Rn , Fi ) → (∆q × Rn , Fi+1 )}i∈Z/p+1
(7.1)
For 0 ≤ i ≤ p define gi : Fi → F0 by gi = fp . . . fi+1 fi . For 0 ≤ i ≤ p − 1 let Fi0 be the
following compact subset of ∆q × R(k+1)n :
Fi0
= {(d, x, P g0k−1 gi+1 (d, x), . . . , P g0 gi+1 (d, x), P gi+1 (d, x))|(d, x) ∈ Fi+1 }
= {(d, P fi . . . f0 g0k−1 (d, x), P g0k−1 (d, x), . . . , P g0 (d, x), x)|(d, x) ∈ F0 }
Define
Ci0 = (∆q × R(k+1)n , Fi0 )
fi0 : Ci0
→
0
Ci+1
,
fi0 (d, x0 , . . . , xk )
= (fi+1 (d, x0 ), x1 , . . . , xk )
0≤i≤p−2
0≤i≤p−2
0
0
0
fp−1
: Cp−1
→ C00 , fp−1
(d, x0 , . . . , xk ) = (f0 (d, xk ), P fp (d, x0 ), x1 , . . . , xk−1 )
f0
f0
0
0
We define α(τ ) to be {Ci0 →i Ci+1
}i∈Z/p . We have to check that {Ci0 →i Ci+1
}i∈Z/p
(k+1)n
is in Θ0p−1 P erk ∆q , i. e. we need to check that Fi0 is the k-periodic point set of
0
0
fi−1
. . . f00 fp−1
. . . fi0 . This follows for all i if it is true for i = 0 and if fi0 maps Fi0
0
homeomorphically onto Fi+1
. We indicate how the hardest of the two statements may
0
be checked, i. e. that F0 is the fixed point set of:
(fp0 . . . f00 )k = (c ◦ ((fp ◦ . . . ◦ f1 ) × 1R(k−1)n ×∆q f0 ))k
It follows from the equations
• c ◦ (G0 ×∆q . . . ×∆q Gk−1 ×∆q Gk ) = (Gk ×∆q G0 ×∆q . . . ×∆q Gk−1 ) ◦ c
• The fixed point set of c−1 ◦ (Kk ×∆q . . . ×∆q K1 ×∆q K0 ) is
{(d, x0 , . . . , xn ) | (d, xn ) ∈ f. p. set of Kn−1 ◦ . . . ◦ K0
and (d, xi−1 ) = Ki (d, xi ) for 1 ≤ i ≤ n}
where Gi and Ki are maps ∆q × Rn → ∆q × Rn over ∆q .
21
(k+1)n
We will define maps H (j) : Θ0p P erkn ∆ → Θ0p−1 P erk I×∆ for 1 ≤ j ≤ 6 by mapping τ
as in (7.1) to τ (j) given by
f
(j)
(j)
(j)
i
(I × ∆q × R(k+1)n , Fi+1 )}i∈Z/p
{(I × ∆q × R(k+1)n , Fi ) →
Define H (1) by
(1)
= I × Fi0 , 0 ≤ i ≤ p − 1
(1)
= I × fi0 ,
Fi
fi
0≤i≤p−2
(1)
fp−1 (t, d, x0 , x1 , . . . , xk )
=
(t, d, (1 − t)P f0 fp (d, x0 ) + tP f0 (d, xk ), P fp (d, x0 ), x1 , . . . , xk−1 )
In the notation of Prop. 2.1, i∗1 H (1) = α.
Define
T1 : I × ∆q × R(k+1)n −→
I × ∆q × R(k+1)n
7→
(t, d, x0 , x1 , . . . , xk )
(t, d, x0 , tx1 , . . . , txk )
and define H (2) by
(2)
Fi
(1)
= T1 (Fi ), 0 ≤ i ≤ p − 1
(2)
fi
(1)
0≤i≤p−2
= fi ,
(2)
fp−1 (t, d, x0 , x1 , . . . , xk ) = (t, d, P f0 fp (d, x0 ), tP fp (d, x0 ), x1 , . . . , xk−1 )
Then i∗1 H (2) = i∗0 H (1) . Note that the difference between i∗0 H (2) and ∂00 is that the former
has some “×1 instead of ×0”. Replacing “×1” by “×t”, as in the proof of Lemma 7.1,
we obtain a homotopy between i∗0 H (2) and ∂00 .
Define H (3) by
(3)
(1)
Fi = Fi , 0 ≤ i ≤ p − 1
(3)
fi
(1)
1≤i≤p−1
= fi ,
(3)
f0 (t, d, x0 , x1 , . . . , xk ) = (t, d, (1 − t)P f1 f0 (d, x1 ) + tP f1 (d, x0 ), x1 , . . . , xk )
Then i∗1 H (3) = α.
Define
T2 : I × ∆q × R(k+1)n
−→
I × ∆q × R(k+1)n
(t, d, x0 , x1 , . . . , xk )
7→
(t, d, tx0 , x1 , . . . , xk )
and define H (4) by
(4)
Fi
(3)
1≤i≤p−1
= Fi ,
(4)
(3)
F0 = T2 (F0 )
(4)
(3)
fi = fi ,
0≤i≤p−2
(4)
fp−1 (t, d, x0 , x1 , . . . , xk ) = (t, d, tP f0 (d, xk ), P fp (d, x0 ), x1 , . . . , xk−1 )
Then i∗1 H (4) = i∗0 H (3) .
Define
T3 : I × ∆q × R(k+1)n
(t, d, x0 , x1 , . . . , xk )
−→
I × ∆q × R(k+1)n
7→
(t, d, x0 , x1 , tx2 , . . . , txk )
and define H (5) by
(5)
(4)
= T1 (Fi ),
1≤i≤p−1
(5)
(4)
F0 = T3 (F0 )
(5)
(4)
fi = fi ,
1≤i≤p−1
Fi
22
(5)
f0 (t, d, x0 , x1 , . . . , xk ) = (t, d, P f1 f0 (d, x1 ), tx1 , x2 , . . . , xk )
Then i∗1 H (5) = i∗0 H (4) .
The map
R0 : Rn × Rn
−→
Rn × Rn
(x, y)
7→
(y, (−1)n x)
is in SO(2n). Choose a path Rt in SO(2n) with R0 = 1R2n and R1 = R0 .
Define H (6) by
(6)
(5)
Fi = Fi , 1 ≤ i ≤ p − 1
(6)
fi
(6)
F0
(5)
= fi ,
1≤i≤p−2
(5)
= {(t, d, Rt (x0 , x1 ), x2 , . . . , xk )|(t, d, x0 , x1 , x2 , . . . , xk ) ∈ F0 }
(6)
fp−1 (t, d, x0 , x1 , . . . , xk ) = (t, d, Rt (0, P fp (d, x0 )), x1 , . . . , xk−1 )
(6)
f0 (t, d, x0 , x1 , . . . , xk ) = (t, d, (P f1 f0 × 0)(d, Rt−1 (x0 , x1 ), x2 , . . . , xk )
Then i∗0 H (6) = i∗0 H (5) and the difference between i∗1 H (6) and ∂10 is that the former has
some “×1 instead of ×0”. Replacing “×1” by “×t”, as in the proof of Lemma 7.1, we
obtain a homotopy between i∗1 H (6) and ∂10 , QED.
8
A topological Pontryagin construction
Let G be a finite group acting by linear isometries on some Rn . Recall that we view S n
as Rn ∪ ∞ with ∞ as the basepoint. We define a pointed action of G on S n by requiring
that Rn ⊂ S n is a G-map. As usual, Ωn X will denote the space of pointed maps from
S n to the (pointed) space X. Thus Ωn S n is a (pointed) G-space via the mapping space
action.
If X is a space, define an equivalence relation on T OP(X, S n ) by “germ along the
vanishing set”, i.e.
f ∼ g ⇐⇒ f −1 (0) = g −1 (0) and ∃V open in X with
f −1 (0) ⊂ V and f |V = g|V .
Let S∆ = singular complex of Ωn S n . We identify
S∆q = T OP((∆q × S n , ∆q × ∞), (S n , ∞))
Thus S∆ is a G-simplicial set. Let S ∆q = S∆q / ∼ and π: S∆q → S ∆q be the canonical
map. Then S ∆ is a G-simplicial set, and π: S∆ → S ∆ a G-simplicial map. As the notation
S∆ and S ∆ suggests, there are obvious functors S and S from EN Rop to G-sets, whose
restriction to ∆op gives S∆ and S ∆ , and the same is true for π.
Proposition 8.1 π is a G-homotopy equivalence.
H
Proof: We show that for any subgroup H of G the restriction S∆
→ SH
∆ is a homotopy
H
H
equivalence. Since S∆ and S ∆ are fibrant simplicial sets, it suffices to show that the
relative simplicial homotopy groups vanish. Fix a commutative diagram:
∂∆q
∩
∆q
F- H
S∆
π
f- ?
SH
∆
H
We construct homotopies F 0 : I × ∂∆q → S∆
and f 0 : I × ∆q → S H
∆ satisfying:
• F00 = F and f00 = f
23
• f 0 |I × ∂∆q = πF 0
H
• For some g: ∆q → S∆
we have πg = f10 and g|∂∆q = F10 .
Let F correspond to an H-map τ : ∂∆q × S n → S n and f correspond to π(σ) where σ is
a map from ∆q × S n to S n which is an H-map near its vanishing set. Using Prop. 2.1 we
see that the construction of F 0 , resp. f 0 , is reduced to the construction of τ 0 : (I × ∂∆q ×
S n , I × ∂∆q × ∞) → (S n , ∞), resp. σ 0 : (I × ∆q × S n , I × ∆q × ∞) → (S n , ∞) satisfying:
• τ 0 is an H-map and σ 0 is an H-map near its vanishing set.
• τ 0 |{0} × ∂∆q × S n = τ and σ 0 |{0} × ∆q × S n = σ
• σ 0 |I × ∂∆q × S n ∼ τ 0
• σ 0 |{1} × ∆q × S n is an H-map, and σ 0 |{1} × ∂∆q × S n = τ 0 |{1} × ∂∆q × S n .
Let σ̇ = σ|∂∆q × S n . Find U open in ∂∆q × S n with τ −1 (0) = σ̇ −1 (0) ⊂ U and σ̇|U =
τ |U . We may assume that both τ and σ̇ carry the complement of U outside {x ∈ Rn :
|x| < } for some positive real number . We may also assume that σ is an H-map on
σ −1 ({x ∈ Rn : |x| < }).
Let h: (S n − 0, ∞) → (Rn , 0) be defined by x 7→ (1/|x|2 )x and observe that h is a
G-homeomorphism since G acts by linear isometries on Rn . Choose a (continuous) map
λ from R to I so that λ(x) = 0 for |x| ≤ 1/ and λ(x) = 1 for |x| ≥ 2/. Define
α: I × S n → S n by
x,
x ∈ Rn and |x| ≤ /2
αt (x) =
−1
h ((tλ(|h(x)|) + 1 − t)h(x)), otherwise
so that α has the following properties:
• All αt are pointed G-maps.
• α0 = 1S n
• For all t in I and x in Rn with |x| ≤ /2 we have αt (x) = x.
• For all x in Rn with |x| ≥ we have α1 (x) = ∞.
Thus we may take τ 0 , resp. σ 0 , to be (1I × τ ) ◦ α, resp. (1I × σ) ◦ α, QED.
Define the functor Z from EN Rop to G-sets by


[


ZB = 
{f : U → Rn |f −1 (0) is compact} / ∼
U open in B × Rn
where ∼ is again “germ along the vanishing set”:
f ∼ g ⇐⇒ f −1 (0) = g −1 (0) and ∃W open in B × Rn with
f −1 (0) ⊂ W and f and g defined and equal on W .
The action is again the mapping space action, where cf is defined on cU if f is defined
on U .
Proposition 8.2 The functors from EN Rop to G-sets Z and S are isomorphic.
Proof: There is an injective map S → Z given by restriction to the inverse image of
Rn . We show it is surjective.
Fix a germ in ZB , say represented by f : U → Rn . Let V be open in U with
f −1 (0) ⊂ V and V compact. Use the Tietze Theorem to extend f |∂V to a map g
from (B × S n ) − V to S n − 0 ≈ Rn by mapping B × ∞ to ∞. The maps f |V and g fit
together to define h: B × S n → S n and h|h−1 (Rn ) ∼ f , QED.
24
9
Fixed point problems and equivariant stable homotopy
Fix a positive integer k and recall the action of Ck on Rkn defined in §2.1. The inclusion
of Rkn in Rk(n+1) is not a Ck -map, but the embedding α: Rkn → Rk(n+1) given by
α(x0 , . . . , xkn−1 ) = (x0 , . . . , xn−1 , 0, xn , . . . , x2n−1 , 0, . . . , x(k−1)n , . . . , xkn−1 , 0)
is a Ck -map, as is the orthogonal projection β: Rk(n+1) → Rkn for which α is a section.
Recall from the previous section the associated action of Ck on Ωkn S kn and the simplicial Ck -set Zkn∆ that has the same Ck -homotopy type as Ωkn S kn . Define a suspension
map Σ: Ωkn S kn → Ωk(n+1) S k(n+1) by
( ∞,
x=∞
(Σf )(x) =
∞,
x ∈ Rk(n+1) and f β(x) = ∞
αf β(x) + x − αβ(x), otherwise
Let QCk (∗) denote the colimit over n of Ωkn S kn . Note that Σ induces a map from Zkn
to Zkn+1 , thus if we let Zk be the colimit over n of Zkn then the results of the previous
section imply the following:
Theorem 9.1 The realization of the simplicial Ck -set Zk ∆ has the Ck -homotopy type
of QCk (∗).
We now define a map γ: Θ0k−1 F ixn → Zkn . Fix an element
[fi ]
f = {(B × Rn , Fi ) → (B × Rn , Fi+1 )}i∈Z/k
fi
of Θ0k−1 F ixnB , where [fi ] is the germ along Fi of a map (Ui , Fi ) → (B × Rn , Fi+1 ) over B
(Ui is an open subset of B×Rn ) that restricts to a homeomorphism from Fi to Fi+1 . Write
P for the projection of B × Rm on Rm (for any m). Define g: U0 ×B . . . ×B Uk−1 → Rkn
by g(b, x) = x − ck (P f0 (b, x0 ), . . . , P fk−1 (b, xk−1 )) where we write x = (x0 , . . . , xk−1 )
for x ∈ Rkn = Rn × . . . × Rn . An easy computation shows that the zero set of g is the
following compact subset F of B × Rkn :
{(b, x0 , . . . , xk−1 ) ∈ B × Rkn |(b, x0 ) ∈ F0 and (b, xi ) = fi−1 (b, xi−1 ) for positive i}
Thus if [g, F ] denotes the germ of g along F , then [g, F ] ∈ ZknB and we define γ(f ) =
[g, F ]. Note that γ is a Ck -map compatible with stabilization maps, and therefore induces
a map γ: Θ0k−1 F ix → Zk .
Theorem 9.2 The map γ: Θ0k−1 F ix∆ → Zk ∆ is a Ck -homotopy equivalence.
Proof: For [g, F ] ∈ ZknB write g = (g0 , . . . , gk−1 ) by identifying Rkn and Rn × . . . × Rn
in the usual way. Define δ: Zkn → Θ0k−1 F ixkn by
[fi ]
δ([g, F ]) = {(B × Rkn , Fi ) → (B × Rkn , Fi+1 )}i∈Z/k
i
where fi (b, x) = (b, c−1
k (x)−gi+1 (b, ck (x)) (here “i+1” should be taken modulo k and the
n
kn
i
subtraction of gi+1 (b, ck (x)) ∈ R from c−1
is meaningful by our identification
k (x) ∈ R
n
kn
of R with a subspace of R ), and Fi = (1B × c−i
k )(F ). Note that δ is a Ck -map,
although it is not compatible with the stabilization maps (unless k = 1, in which case γ
and δ are inverse isomorphisms, thus we may assume k ≥ 2). The conclusion now follows
from the next two lemmas.
γ
δ
Lemma 9.1 The composition Θ0k−1 F ixn∆ → Zkn∆ → Θ0k−1 F ixkn
∆ is Ck -homotopic to the
stabilization map.
25
Proof: We will define Ck -maps H (l) : Θ0k−1 F ixn∆ → Θ0k−1 F ixkn
I×∆ for l = 1, 2, by mapping
f to H (l) (f ), where:
[fi ]
f = {(∆q × Rkn , Fi ) → (∆q × Rkn , Fi+1 )}i∈Z/k
(l)
[f
(l)
]
(l)
i
H (l) (f ) = {(I × ∆q × Rkn , Fi ) →
(I × ∆q × Rkn , Fi+1 )}i∈Z/k
Define H (1) by
(1)
fi (t, d, x) = (t, d, P fi (d, x0 ), x2 , x3 , . . . , xk−1 , tx0 )
(1)
Fi
= {(t, d, x0 , tx1 , tx2 , . . . , txk−1 )|(d, x0 ) ∈ Fi and (d, xj ) = fi+j−1 (d, xj−1 )}
and observe that, in the notation of Prop. 2.1, i∗1 H (1) is the composition in the statement
of the lemma.
Define H (2) by
(2)
fi (t, d, x) = (t, d, P fi (d, x0 ), tx2 , tx3 , . . . , txk−1 , 0)
(2)
Fi
i∗1 H (2)
= {(t, d, x0 , 0, 0, . . . , 0)|(d, x0 ) ∈ Fi }
i∗0 H (1)
and observe that
=
and that i∗0 H (2) is the stabilization map. The conclusion now follows from Prop. 2.1.
γ
δ
kn
Lemma 9.2 The composition Zkn∆ → Θ0k−1 F ixkn
∆ → Zk ∆ is Ck -homotopic to the stabilization map.
2
Proof: For x = (x0 , . . . , xk−1 ) ∈ Rk n we write xi = (xi 0 , . . . , xi k−1 ) ∈ Rkn . We will
define Ck -maps H (l) : Zkn∆ → ZkknI×∆ for l = 1, 2, 3, by H (l) ([g, F ]) = [g (l) , F (l) ] where
(l)
(l)
(l)
(l)
g (l) (t, d, x) = (g0 (t, d, x), . . . , gk−1 (t, d, x))
(l)
gi (t, d, x) = (gi 0 (t, d, x), . . . , gi k−1 (t, d, x))
Define H (1) by
(1)
gi j (t, d, x) = xi j − xi−1 j+1 , j 6= 0
(1)
gi 0 (t, d, x) = xi 0 − xi−1 1 + tgi (d, ci−1
k (xi−1 )) + (1 − t)gi (d, x0 0 , x1 0 , . . . , xk−1 0 )
F (1) = {(t, d, x)|(d, cik (xi )) ∈ F }
where the operations on i and j should be taken modulo k. Note that i∗1 H (1) is the
composition in the statement of the lemma.
Define H (2) by
(2)
gi j (t, d, x) = xi j − xi−1 j+1 , j 6= 0, k − 1
(2)
gi k−1 (t, d, x) = xi k−1 − txi−1 0
(2)
gi 0 (t, d, x) = txi 0 − xi−1 1 + gi (d, x0 0 , x1 0 , . . . , xk−1 0 )
F (2) = {(t, d, Tt (x0 ), . . . , Tt (xk−1 ))|(d, cik (xi )) ∈ F }
where Tt : Rkn → Rkn is defined by Tt (y0 , . . . , yk−1 ) = (y0 , ty1 , ty2 , . . . , tyk−1 ). Note that
i∗1 H (2) = i∗0 H (1) .
Define H (3) by
(3)
gi j (t, d, x) = xi j − txi−1 j+1 , j 6= 0, k − 1
(3)
gi k−1 (t, d, x) = xi k−1
(3)
gi 0 (t, d, x) = −txi−1 1 + gi (d, x0 0 , x1 0 , . . . , xk−1 0 )
F (3) = {(t, d, T0 (x0 ), . . . , T0 (xk−1 ))|(d, cik (xi )) ∈ F }
Note that i∗1 H (3) = i∗0 H (2) and i∗0 H (3) is the stabilization map. The conclusion now
follows from Prop. 2.1.
26
10
Split fixed point problems
Let B be a compact EN R and < n > an object of Γ. We define a category sn GB : The
objects are all (E, F, α) with (E, F ) an object of GB , α: F →< n > continuous (thus for
0 ≤ i ≤ n, F i = α−1 (i) is a (possibly empty) union of connected components of F ),
and α−1 (0) = ∅. We will occasionally write (E, F 1 , . . . , F n ) instead of (E, F, α). The
morphisms in sn GB from (E, F, α) to (E 0 , F 0 , α0 ) are the maps f in GB from (E, F )
to (E 0 , F 0 ) such that α = α0 ◦ fF , where fF denotes the continuous function from F to F 0
determined by f . Thus f determines maps f i : (E, F i ) → (E 0 , F i0 ) in GB .
Note that sn G is in the usual way an EN Rop -category. Now let β: < n >→< m > be
a map in Γ. We define a functor β∗ : sn GB → sm GB : For any object (E, F, α) of sn GB
let β∗ (E, F, α) = (E, β∗ F, β ◦ (α|β∗ F ) where β∗ F = α−1 β −1 (< m > − < 0 >). If f is any
map in sn GB with domain (E, F, α) we define β∗ (f ) to be the germ along β∗ F determined
by f . A computation shows that β∗ is a map of EN Rop -categories and that β 7→ β∗ is
functorial. Therefore sG is a Γ × EN Rop -category.
Fix < n >∈ ob Γ. There is a forgetful map sn G → G taking (E, F, α) to (E, F ). Let
sn F be the inverse image of F under the induced map N cy sn G → N cy G (in particular,
s0 F = N cy s0 G). By its definition sn F is a Λop × EN Rop -subset of N cy sn G. In fact, sF
is a Γ × Λop × EN Rop -subset of N cy sG (forgetting some of the components of the fixed
point set preserves the property “fixed point problem”). Further, ΘsF is a right ideal
in ΘsG since so is ΘF in ΘG and Θsn F is the inverse image of ΘF by Θsn G → ΘG.
These definitions extend naturally to k-periodic point problems: Let
sn Pk = ψ(sdk sn F)Ck = pullback of Pk ⊂ N cy G ← N cy sn G
Note that Θsn Pk = ψ(sdk Θsn F)Ck = pullback of ΘPk ⊂ ΘG ← Θsn G, in particular
ΘsPk is a right ideal in ΘsG.
Recall the definition of a very special Γ-space given in §2.3.
Lemma 10.1 For all positive integers k, the Γ-space ΘsPk∆ satisfies Γ1.
Proof: Note that Θs0 Pk = Θs0 G is independent of k. For any B ∈ ob EN R, let CB be
the following object of s0 GB :
CB = (B × R0 , ∅, αB )
where αB is the (unique) map ∅ →< 0 >. For m = (m0 , . . . , mp ) ∈ Θp , let CmB be the
following object of Θp s0 GB :
CmB = {CB
(mi ,1CB )
→
CB }i∈Z/p+1
Let Θ0pB = {CmB |m ∈ Θp }. Then Θ0 is a Λop × EN Rop -(discrete) subcategory of Θs0 G
which is isomorphic to Θ (Θ is viewed constant in the EN Rop -direction), in particular
Θ0 is contractible. But the inclusion of Θ0 in Θs0 G is a homotopy equivalence, even if we
fix ([p], B) ∈ ob Λop × EN Rop , since the component of Θp s0 GB containing CmB is the
trivial groupoid on its objects, QED.
Lemma 10.2 For all positive integers k, the Γ-space ΘsPk∆ satisfies Γ2.
Proof: Fix < n >∈ ob Γ. It suffices by Prop. 5.2 to show that
(p1 , . . . , pn ): Θ0 sn Pk∆ → Θ0 s1 Pk∆ × . . . × Θ0 s1 Pk∆
is a homotopy equivalence. We show this for fixed ([p], B) ∈ ob Λop × EN Rop .
A typical object of Θ0p sn PkB is
fi
1
n
{(Ei , Fi1 , . . . , Fin ) → (Ei+1 , Fi+1
, . . . , Fi+1
)}i∈Z/p+1
27
such that for each j with 1 ≤ j ≤ n
fj
j
i
{(Ei , Fij ) →
(Ei+1 , Fi+1
}i∈Z/p+1
is a k-periodic point problem. Fix an object (C 1 , . . . , C n ) of Θ0p s1 PkB × . . . × Θ0p s1 PkB ,
say
fj
j
j
i
}i∈Z/p+1
, Fi+1
(Ei+1
C j = {(Eij , Fij ) →
It suffices by Quillen’s Theorem A (see [13, p.93]) to construct a final object of the
category (p1 , . . . , pn )/(C1 , . . . , Cn ).
For each i ∈ Z/p + 1 and each j with 1 ≤ j ≤ n choose a homeomorphism Eij ≈ Ẽij ,
such that:
• Ẽij ⊂ B × Rn for some n.
• For each i ∈ Z/p + 1, {Ẽij |1 ≤ j ≤ n} is pairwise separated by open sets in B × R∞
(i. e. if all Eij are in B × Rn , choose Ẽij in B × Rn × {j} ⊂ B × Rn+1 ).
Use these homeomorphisms to construct objects C̃ j of Θ0p s1 PkB isomorphic to C j , say
f˜j
j
j
i
C̃ j = {(Ẽij , F̃ij ) →
(Ẽi+1
, F̃i+1
}i∈Z/p+1
Let C̃ be the following object of Θ0p sn PkB
[
f˜i [ j
1
n
, F̃i+1
, . . . , F̃i+1
}i∈Z/p+1
{( Ẽij , F̃i1 , . . . , F̃in ) → ( Ẽi+1
j
j
There is a canonical map κ from (p1 , . . . , pn )(C̃) to (C1 , . . . , Cn ) and it is not hard to see
that (C̃, κ) is a final object of (p1 , . . . , pn )/(C1 , . . . , Cn ), QED.
Lemma 10.3 For all positive integers k, the Γ-space ΘsPk∆ satisfies Γ3.
Proof: If X is any Γ-space satisfying Γ1 and Γ2, then π0 (X1 ) is an abelian monoid (this
follows from certain commutative diagrams in Γ; for example, the map
π0 (X1 ) × π0 (X1 ) × π0 (X1 ) →
(a, b, c)
7→
π0 (X1 )
(ab)c
0112
011
is isomorphic to π0 (X3 ) → π0 (X2 ) → π0 (X1 ) induced by < 3 > → < 2 > →< 1 >). Thus
it remains to prove the existence of inverses.
Let J be the interval [−1, 1] in R and define g: J × R → J × R by:
(t, t),
x≥0
g(t, x) =
(t, 2x + t), x ≤ 0
Then the k-periodic point set of g is G = {(t, ±t)|0 ≤ t ≤ 1}.
f
The vertices (E, α) → (E, α) ∈ ob Θ0 s1 Pk∆0 of Θs1 Pk∆ may be identified with the
f
set of all (E, F ) → (E, F ) where E is an EN R in some Rn , F is a compact subset of
f
E, and (E, F ) → (E, F ) is a k-periodic point problem. Identify J with ∆1 , say via an
increasing homeomorphism, so that the following is an object of Θ0 s1 Pk∆1 :
g×f
C̃ = (J × R × E, G × F ) → (J × R × E, G × F )
Since g has no k-periodic points over t = −1, ∂1 C̃ = 0 on π0 . Thus it remains to show
that ∂0 C̃ = C + C 0 on π0 . Now the k-periodic point set of ∂0 C̃ splits into two separated
pieces, i. e. ∂0 C̃ = C1 + C2 on π0 . Letting C1 be the germ along {1} × F ⊂ G|1 × F and
using the nerve direction, we see that C1 = C on π0 , QED.
28
Lemma 10.4 For all positive integers k, the map Θs1 F∆ → ΘF∆ is a Ck -homotopy
equivalence.
G
Proof: Fix k ≥ 1. The induced map Θs1 Pk → ΘPk is an isomorphism of categories for
fixed ([p], B) ∈ ob Λop × EN Rop . The conclusion now follows from Prop. 2.4, QED.
We summarize the results of this section:
Theorem 10.1 For all positive integers k, the loop space of (the realization of ) ΘsF∆
has the Ck -homotopy type of ΘF∆ .
Proof: By Prop. 2.3 and Lemmas 10.1, 10.2, and 10.3, the loop space of ΘsF∆ has the
Ck -homotopy type of Θs1 F∆ , i. e. that of ΘF∆ (by Lemma 10.4), QED.
11
Filtered fixed point problems
In this section we enlarge ΘsF without changing its Ck -homotopy type to obtain (the
∆op ×Λop ×EN Rop -category) f Θ F that looks more like sE. For fixed n, p and B, we will
define the category fnΘp FB as the homotopy limit of a diagram en : In → CAT with In
having an initial object in and en (in ) = Θp sn FB . We use the notation and terminology
of [12] in what follows. In particular, for us the homotopy limit of a diagram of categories
will again be a category (see [12] for the definition). We remark however that if A is
a diagram of categories and N A is the diagram of simplicial sets obtained from A by
using the nerve functor, then N holim A is naturally isomorphic to holim N A. Here we
have in mind the version of the homotopy limit given in [2, XI.3.2] which functorially
yields a simplicial set from a diagram of simplicial sets. It might not have the “correct”
homotopy type if the simplicial sets involved are not fibrant, but it follows from [12,
Thm. 2] that the homotopy limit over a category I with initial object i of a diagram X
of (nerves of) categories does have the correct homotopy type, namely that of Xi .
For [n] ∈ ob ∆, write Inop for the full subcategory of [n]op × [n] with objects (i, j)
op
such that i ≤ j. Clearly I op is a ∆-category and we write f∗op : Im
→ Inop for the functor
induced by f : [m] → [n] in ∆.
+i−i0
0 0
0
0
op
There is a functor eop
n : In → ∆ taking (i, j) → (i , j ) to [j − i] → [j − i ]. Further,
op op
op op
if f : [m] → [n] then there is a map ef : em → en f∗ , given for each 0 ≤ i ≤ j ≤ m by
op
op
eop
is a functor from
f ij : [j − i] → [f (j) − f (i)], where ef ij (k) = f (i + k) − f (i). In fact, e
∆ to Dir(∆) (see [12] for the definition of the category Dir(C) associated to a category C),
op
taking [n] to the object eop
n : In → ∆ of Dir(∆) and taking f : [m] → [n] to the morphism
op op
op op op
op op
(f∗ : Im → In , ef : em → en f∗ ) of Dir(∆).
We give a second description of eop which might be easier to visualize (but is longer
to describe). Recall the functors A/: A → CAT and /A: Aop → CAT (the objects of
A/a are the morphisms of A with target a, and the objects of a/A are the morphisms
of A with source a). Clearly there is a similar functor /A/: Aop × A → CAT (the
objects of a/A/a0 are the diagrams a → a00 → a0 in A). Further, a functor f : A → B
gives a map fˆ: /A/ → /B/ ◦ (f op × f ), given for a, a0 ∈ ob A by the obvious functor
a/A/a0 → f (a)/B/f (a0 ). All this gives a functor CAT → Dir(CAT ) taking A to /A/
and taking f : A → B to (f op × f, fˆ).
op
Let IA
be the full subcategory of Aop × A with objects (a, a0 ) such that A(a, a0 ) 6= ∅.
op
op
op
→ IB
for the map induced
Then I is a functor CAT → CAT and we write f∗op : IA
op
op
by a functor f : A → B. Denote by eA the restriction of /A/ to IA
and by eop
f the
op
restriction of fˆ to eA . We described a functor eop : CAT → Dir(CAT ).
Now let ∆0 be the full subcategory of CAT with objects the finite non-empty totally
ordered sets. Thus ∆ ⊂ ∆0 and the inclusion is an equivalence of categories. In fact,
there is a unique retraction ∆0 → ∆ (because the automorphism group of each object of
∆ is trivial and ∆ has precisely one object for each isomorphism class of ∆0 ).
Note that if T ∈ ob ∆0 then, for all t, t0 ∈ T , t/T /t0 is again an object of ∆0 , i. e.
eop : CAT → Dir(CAT ) when restricted to ∆0 takes values in Dir(∆0 ). Because ∆ is a
29
retract of ∆0 , we obtain eop : ∆ → Dir(∆) and a computation shows that it equals the
functor described in the beginning of this section.
We now describe an embedding εop : ∆ → Dir(∆) and a map φop : eop → εop .
For any category C there is an embedding εop : C → Dir(C) taking C ∈ ob C to C
γ
viewed as a functor [0] → C and taking C → C 0 to (1[0] , γ).
op op
op
op
op
The map φ : e → ε is given for [n] ∈ ob ∆ by (Inop → [0], φop
n : en → cn ) where:
[n]
op
• cop
n denotes the constant functor In → [0] → ∆.
op
op
• φop
n is given for 0 ≤ i ≤ j ≤ n by φn (i,j) : [j − i] → [n], where φn (i,j) (k) = k + i.
We remark that just as eop : ∆ → Dir(∆) is a special case of eop : CAT → Dir(CAT ), so is
φop : eop → εop . Briefly, for any small category A and any a, a0 ∈ ob A with A(a, a0 ) 6= ∅,
0
00
0
00
φop
A (a,a0 ) : (a/A/a ) → A is the functor taking a → a → a to a .
op
op
Using the isomorphism Dir(C) ≈ Inv(C ) (see [12] for the definition of Inv(C)) we
obtain functors e, ε: ∆op → Inv(∆op ) and a map φ: ε → e. Now let C be any simplicial
D-category, i. e. C: ∆op → CAT D . Thus Inv(C): Inv(∆op ) → Inv(CAT D ).
Recall from [12] the functor holim : Inv(CAT D ) → CAT D . Notice that the composition holim ◦ Inv(C) ◦ ε is (naturally isomorphic to) C. Define the simplicial D-category
C 0 as holim ◦ Inv(C) ◦ e and let κ = (holim ◦ Inv(C))φ. Thus κ: C → C 0 .
Proposition 11.1 For each [n] ∈ ob ∆ and each d ∈ ob D, κn d : Cn d → Cn0 d is a homotopy equivalence.
Proof: Let L: [0] → In be the functor determined by the initial object (0, n) of In . Thus
L has a right adjoint. Note that en ◦L = [n]. As in [12], we write Cˆ for the functor from D
to simplicial categories associated to C (given by “switching the independent variables”).
Thus holim (Cˆd ◦ en ) = Cn0 d and holim (Cˆd ◦ en ◦ L) = Cn d . Let α be the following map in
Inv(CAT ) from Cˆd ◦ en to Cˆd ◦ en ◦ L:
L
1
α = ([0] → In , Cˆd ◦ en ◦ L → Cˆd ◦ en ◦ L)
The map holim α is a homotopy equivalence by [12, Thm. 2] since L has a right adjoint.
(in the notation of [12], holim α = L∗ ). Thus it suffices to show that holim α ◦ κn d equals
the identity map of Cn d . This follows upon applying holim ◦ Inv(Cˆd ) to the composition
L
φn
1
([0] → In , en ◦ L → [n]) ◦ (In → [0], cn → en )
in Inv(∆op ), since that composition equals the identity of [n]: [0] → ∆op , QED.
Now let f Θ F = (ΘsF)0 , where ΘsF is viewed as a simplicial D-category whose value
at [n] is Θsn F (where D = Λop × EN Rop ).
Theorem 11.1 The map κ: ΘsF → f Θ F is a Ck -homotopy equivalence for all k ≥ 1.
Proof: This follows from the previous proposition since ((sdk ΘsF)Ck )0 is (naturally
isomorphic to) (sdk f Θ F)Ck (homotopy limits commute with limits, and, for G a group
and X a G-set, X G is a limit), QED.
30
We conclude this section by explicitly describing C 0 for any simplicial category C. The
op
following is a picture of eop
n : In → ∆:
...
- (0, n − 1) - (0, n)
(0, 0) - (0, 1) 6
6
(1, 1)
-
...
6
- (1, n − 1)
6
- (1, n)
6
..
.
..
.
6
6
(n − 1, n − 1)
- (n − 1, n)
6
(n, n)
eop
n
[0]
1
2
∂∂[1]
6
∂0
1
∂[0]
?
...
...
n−1
∂[n − 1]
n−2
6
∂0
∂[n − 2]
6
∂0
n
∂[n]
6
∂0
n−1
∂[n − 1]
6
∂0
..
.
..
.
6
∂0
[0]
1
6
∂0
∂[1]
6
∂0
[0]
It follows that a typical object of Cn0 = holim ∗ (C ◦ en ) consists of the following data:
• For 0 ≤ i ≤ j ≤ n an object Cij of Cj−i .
j
• For 1 ≤ i ≤ j ≤ n a map aji : Cij → ∂0 Ci−1
in Cj−i .
• For 0 ≤ i ≤ j ≤ n − 1 a map bji : Cij → ∂j+1−i Cij+1 in Cj−i .
such that for 1 ≤ i ≤ j ≤ n − 1 the following commutes in Cj−i :
Cij
aji
?j
∂0 Ci−1
bji
- ∂j+1−i C j+1
i
∂j+1−i aj+1
i
?
j+1
∂j+1−i ∂0 Ci−1
1
?
∂0 bji−1j+1
∂0 ∂j+2−i Ci−1
A map in Cn0 from {Cij , aji , bji } to {Cij 0 , aji 0 , bji 0 } is a family of maps {Cij → Cij 0 } that
“commutes with the a’s and the b’s”.
31
We now describe the face maps: ∂k {Cij , aji , bji } = {Cij 0 , aji 0 , bji 0 }, where:
(
k
∂k−i C∂∂k ij , i ≤ k − 1 < j
j0
Ci =
k
otherwise
C∂∂k ij ,
(
aji 0
and
ajk 0
=
k
∂k−i a∂∂ k ji ,
k
a∂∂ k ji ,
i≤k−1<j
i > k or j ≤ k − 1
is the composition
aj+1
k+1
j+1
→ ∂0 Ckj+1
Ckj 0 = Ck+1
∂0 aj+1
k
→
j+1
j+1
j
0
∂0 ∂0 Ck−1
= ∂0 ∂k−(k−1) Ck−1
= ∂0 Ck−1
.
Further,
(
bji 0
=
k
∂k−i b∂∂ k ji ,
k
b∂∂ k ji ,
i≤k−1<j
i ≥ k or j < k − 1
0
and finally bk−1
is the composition
i
bk−1
i
Cik−1 0 = Cik−1 →
∂k−i Cik
∂k−i bk
i
→
∂k−i ∂k−i+1 Cik+1 = ∂k−i ∂k−i Cik+1 = ∂k−i Cik 0 .
Thus we described the action of the face maps on objects. Their action on morphisms
can be read off from their action on the {Cij }.
We describe the degeneracy maps: sk {Cij , aji , bji } = {Cij 0 , aji 0 , bji 0 }, where:
(
k
sk−i Cssk ij , i ≤ k < j
j0
Ci =
k
Cssk ij ,
otherwise
(
aji 0
=
k
sk−i assk ji ,
i≤k<j
assk ji ,
i > k + 1 or j ≤ k
k
and ajk+1 0 is the identity map of Ckj−1 . Further,
(
bji 0
=
k
sk−i bssk ji ,
k
bssk ji ,
i≤k<j
i ≥ k + 1 or j < k
and finally bki 0 is the identity map of Cik . Thus we described the action of the degeneracy
maps on objects. As before, their action on morphisms can be read off from their action
on the {Cij }.
In this notation, κn : Cn → Cn0 takes an object C of Cn to {Cij , aji , bji }, where Cij =
∂j−i+1 . . . ∂n−1−i ∂n−i (∂0 )i (C) = (∂0 )i ∂j+1 . . . ∂n−1 ∂n (C) (same formula for morphisms)
and all the aji ’s and bji ’s are identities.
12
Separating the fixed points of the identity map of a filtered
space
Recall the embedding ∆ of §2.2. In this section we will write ∆[n] for the (compact EN R
which is the) image of [n] under this embedding, and we will write ∆[i, j] for the copy of
∆[j − i] in ∆[n] corresponding to the map i → j in [n].
Let C be a category and Ar[n] be the category [n][1] for [n] ∈ ob ∆. A functor
C: Ar[n] → C determines the following data:
• For each map i → j in [n], an object Cij of C.
32
• For each sequence i → j → k in [n], a sequence Cij → Cik → Cjk in C.
Recall the functor sE: ∆op ×EN Rop → SET of §4. A typical element of sn EB is a functor
E: Ar[n] → EB satisfying:
• For each object i of [n], Eii = B × R0 .
• For each sequence i → j → k in [n], the sequence Eij → Eik → Ejk is a cofibration
sequence in EB .
Ar[n]
Fix E as above. We say that a map h: ∆[n] × E → ∆[n] × E over ∆[n] in EB
separating homotopy provided that the following conditions are satisfied:
is a
SH1. For all maps i → j in [n], the restriction hji |∆[j, n] × Eij is the identity.
SH2. For all maps i → j in [n] with j − i ≥ 1, there exists a map
rij−1 : (Eij , sj−1
(Eij−1 )) → (Eij−1 , Eij−1 )
i
denotes the cofibration from Eij−1 to Eij associated to E,
in GB , where sj−1
i
such that:
• If we denote again by sj−1
the induced map from (Eij−1 , Eij−1 ) to
i
(Eij−1 )) in GB , then rij−1 sij−1 = 1(E j−1 ,E j−1 ) .
(Eij , sj−1
i
i
•
|∆[0, j
= (hj−1
− 1] ×
i
germs along sij−1 (Eij−1 ).
hji |∆[0, j
least as
Eij
− 1] ×
Eij−1 )
i
◦ (1∆[0,j−1] × rij−1 ), at
SH3. Away from the inverse image of sj−1
(Eij−1 ), hji does not depend on
i
the ∆[0, j − 1] direction. More precisely:
For all maps i → j in [n] with j − i ≥ 1, there exists a neighborhood V of
sj−1
(Eij−1 ) in Eij such that:
i
• ∆[0, j − 1] × V ⊂ (hji )−1 (∆[0, j − 1] × sj−1
(Eij−1 ))
i
• For d, d0 ∈ ∆[0, j − 1] and e ∈ Eij − V , hji (d, e) = hji (d0 , e).
Notice that the maps rij−1 are determined by h: They are obtained by taking the germ
along sj−1
(Eij−1 ) of hji |∆[j − 1, j − 1] × Eij . Also notice that h is determined by hn0
i
(either by restriction, or by passing to quotients).
Define hsE: ∆op × ∆op × EN Rop → SET by:
h
Ar[n]
hm sn EB = {∆[m] × E → ∆[m] × E in EB
|E ∈ sn EB and h is over ∆[m]}
A computation shows that the separating homotopies form a subobject of the following
composition:
∆op × EN Rop
diag.×1
→
hsE
∆op × ∆op × EN Rop → SET
Recall the monoid M of §2.4. View sE as a ∆op × EN Rop -(discrete) category, so that
M × sE is also a ∆op × EN Rop -category. Explicitly, there are precisely two self-maps of
E: Ar[n] → EB in M × sn EB , namely 1 (which is an identity) and 0, and there are no
other maps in sn EB . We need yet another ∆op × EN Rop -category hE defined as follows:
• ob hn EB = ob (M × sn EB ) = ob sn EB = sn EB
• The self-maps of E: Ar[n] → EB are pairs (m, h) such that m ∈ M , if m = 0 then
h is a separating homotopy of E, and if m = 1 then h is the identity of ∆[n] × E.
There are no other maps in hn EB .
Thus there is a forgetful map from hE to M × sE.
33
Lemma 12.1 The induced map N cy hE∆ → N cy (M × sE∆ ) is a homotopy equivalence.
Proof: We will show that for any object [p] of Λ and any object [n] of ∆, the map
Npcy hn E∆ → Npcy M × sn E∆ is a fibration and a homotopy equivalence. This will be
shown by induction on n, the case n = 0 being trivial.
Fix E: Ar[n] → E∆[q] in sn E∆[q] and m = (m0 , . . . , mp ) in Npcy M = M p+1 . We have
to prove that every circular diagram γ over ∂∆[q], say
γ = {∆[n] × E|∂∆[q]
h(l) |∂∆[q]
→
∆[n] × E|∂∆[q]}l∈Z/p+1
such that if ml = 1 then h(l) |∂∆[q] is the identity and if ml = 0 then h(l) |∂∆[q] is a
separating homotopy, extends to all of ∆[q]. It suffices to construct hn(l) 0 , and this is
trivial if ml = 1. Further, by the inductive hypothesis applied to ∂n γ (here the face
map ∂n is taken in the h-direction) we may assume that hn(l) 0 |∆[n − 1] × E0n−1 is already
constructed (we may assume that all the cofibrations of E are inclusions).
We first extend hn(l) 0 |(∆[n−1, n−1]×E0n−1 )∪(∆[n−1, n]×E0n |∂∆[q]) to ∆[n−1, n]×E0n
so that when restricted to (∆[n − 1, n] × E0n−1 ) ∪ (∆[n, n] × E0n ) it is the identity. This
can be done, because it is a lifting problem associated to the cofibration and homotopy
equivalence
incl.
(∆[n − 1, n] × E0n |∂∆[q]) ∪ (∆[n − 1, n] × E0n−1 ) ∪ (∆[n, n] × E0n ) → ∆[n − 1, n] × E0n
and to the fibration E0n → ∆[q].
Now find a neighborhood V∂ of E0n−1 |∂∆[q] in E0n |∂∆[q] as in the statement of SH 3,
let F∂ denote its closure and ∂F∂ denote its boundary (in E0n |∂∆[q]). Observe that over
∆[n − 1, n − 1] the restriction of hn(l) 0 to F∂ is a retraction onto E0n−1 |∂∆[q]. Use Cor. 3.2
to extend it to r: U → E0n−1 for some neighborhood U of F ∪ E0n−1 in E0n . Use Prop. 3.1
to extend hn(l) 0 to f : ∆[n−1]×F → E0n−1 for some compact neighborhood F of E0n−1 in U
with F ∩E0n |∂∆[q] = F∂ . Let d: I ×∆[n−1] → ∆[n−1] be a contraction rel. ∆[n−1, n−1]
with d0 the identity and d1 constant. Choose a continuous map λ: F − ∂F∂ → I which
vanishes on E0n−1 ∪ (F∂ − ∂F∂ ) and is equal to 1 on ∂F − ∂F∂ . Then we may define
g: ∆[n − 1] × F → E0n−1 extending hn(l) 0 |(∆[n − 1] × E0n−1 ∪ ∆[n − 1] × E0n |∂∆[q]) ∩ F
by g(δ, e) = f (d(λ(e), δ), e) on F − ∂F∂ , (because by SH 3 hn(l) 0 does not depend on
δ ∈ ∆[n − 1] when restricted to ∆[n − 1] × ∂F∂ ). Note that g does not depend on
δ ∈ ∆[n − 1] when restricted to ∆[n − 1] × ∂F . Thus we may extend it to all of
(∆[n − 1] ∪ ∆[n − 1, n]) × E0n by letting g(δ, e) = hn(l) 0 (δn−1 , e) for e ∈
/ F or δ ∈ ∆[n − 1, n]
where δn−1 denotes the point in ∆[n − 1, n − 1]. Observe that this extension does not
satisfy SH 2, and we correct this as follows:
Let U0 be a neighborhood of E0n−1 in F such that
hn(l) 0 |∆[n − 1] × (U0 ∩ E0n |∂∆[q]) = (hn(l) 0 |∆[n − 1] × E0n−1 |∂∆[q])(1∆[n−1] × r)
and let diag(E0n ) denote the diagonal copy of E0n in E0n ×E0n . Use the remark immediately
preceding Prop. 3.2 to find a homotopy d: I × W → E0n rel. diag(E0n ) between the two
projections of E0n × E0n onto E0n for some neighborhood W of diag(E0n ) in E0n × E0n .
Define G: I × (∆[n − 1] ∪ ∆[n − 1, n]) × F → E0n × E0n by
G(t, δ, e) = d(t, g(δ, e), g(δ, r(e)))
and observe that G restricted to I × (∆[n − 1] ∪ ∆[n − 1, n]) × E0n−1 takes values in
diag(E0n ). Thus there is a compact neighborhood F1 of (∆[n − 1] ∪ ∆[n − 1, n]) × E0n−1 in
(∆[n − 1] ∪ ∆[n − 1, n]) × U0 with G(I × F1 ) ⊂ W . Choose a (continuous) map µ: F1 → I,
equal to 1 near ∆[n − 1] × E0n−1 , and equal to 0 on ∂F1 . Let H: F1 → E0n be defined
by H(δ, e) = G(µ(δ, e), δ, e). Then H and g|((∆[n − 1] ∪ ∆[n − 1, n]) × E0n ) − int(F1 ) fit
together to define hn(l) 0 |(∆[n − 1] ∪ ∆[n − 1, n]) × E0n .
34
We now extend hn(l) 0 to all of ∆[n] × E0n so that it satisfies SH 1. Once again, this
can be done because it is a lifting problem associated to the cofibration and homotopy
equivalence
(∆[n] × E0n |∂∆[q]) ∪ (∆[n − 1] ∪ ∆[n − 1, n]) × E0n ∪
n
[
incl.
(∆[k, n] × E0k ) → ∆[n] × E0n
k=0
and to the fibration E0n → ∆[q], QED.
Let hΘ E be the pullback of Θ × sE ⊂ N cy (M × sE) ← N cy hE.
proj.
Theorem 12.1 In the diagram hΘ E∆ → Θ × sE∆ → sE∆ all maps are Ck -homotopy
equivalences for all positive integers k.
Proof: For the second map the conclusion follows since Θ has the Ck -homotopy type
of a point for all k ≥ 1. Note that N cy (M × sE) = Θ × sE q N cy {1} × sE and
also N cy hE = hΘ E q N cy {1} × sE. Thus the conclusion follows from the previous
lemma by observing that the (natural) isomorphism ψ: sdk (N cy hE)Ck → N cy hE, resp.
ψ: sdk (N cy (M × sE))Ck → N cy (M × sE), restricts to an isomorphism from sdk (hΘ E)Ck
to hΘ E, resp. from sdk (Θ × sE)Ck to Θ × sE, QED.
We are now able to define tr: hΘ E → f Θ F. The map will factor through the objects
of f Θ F. Fix σ ∈ hΘp
n EB with σ = (m, γ) where m ∈ Θp and
h(l)
γ = {∆[n] × E → ∆[n] × E}l∈Z/p+1
j
Let f(l) be the induced map E → E by the inclusion ∆[0] ⊂ ∆[n]. Let F(l)
i be the fixed
j
j
j
j
j
j
point set of f(l−1)
i . . . f(1) i f(0) i f(p) i . . . f(l+1) i f(l) i for 0 ≤ i ≤ j ≤ n and l ∈ Z/p + 1.
kj
j
Let Eik j be the image of Eik in Eij for 0 ≤ i ≤ k ≤ j ≤ n. Let F(l)
i = F(l) i ∩
(i+k) j
(i+k−1) j
(Ei
− Ei
) for 0 ≤ i ≤ j ≤ n, l ∈ Z/p + 1, and 1 ≤ k ≤ j − i. Define
tr(σ) = {Cij , aji , bji } ∈ ob fnΘp FB , where:
(j−i) j
1j
• Cij = {(Eij , F(l)
i , . . . , F(l) i
)
j
ml ,f(l)
i
→
(j−i) j
1j
(Eij , F(l+1)
i , . . . , F(l+1) i )}l∈Z/p+1
Thus Cij ∈ ob Θp sj−i FB .
j
• The map aji : Cij → ∂0 Ci+1
is given by {aj(l) i }l∈Z/p+1 where aj(l) i is (the section-
retraction pair determined by the appropriate germs of) ((qij )−1 , qij ) and qij is the
j
j
quotient map from Eij to Ei+1
, which is invertible away from the section of Ei+1
.
j
• The map bji : Cij → ∂j+1−i Cij+1 is given by {bj(l) i }l∈Z/p+1 where bj(l) i = (sji , r(l)
i)
j
and sji is (an appropriate germ of) the cofibration from Eij to Eij+1 . Above, r(l)
i
is (a restriction to the appropriate fixed point set of) the map determined by SH 2
j
j
0
and hj(l) i if ml = 0. If ml = 1 then r(l)
i = r(u) i where u is minimal among u ≥ l
0
0
with mu0 = 0 or if no such u exists then u is maximal among u ≤ l with mu0 = 0.
It is clear that tr is natural in the Λop and EN Rop directions, and the fact that it is
natural in the ∆op direction follows by a long but straightforward computation, using
Θp
the description of the face and degeneracy maps from fn±1
FB to fnΘp FB given at the end
of the previous section. However, the equation ∂0 tr = tr∂0 requires an extra argument
(because ∂ 0 is unique among cofaces and codegeneracies in that it does not preserve 0)
which we give below:
0
0
0
Let tr(σ) = {Cij , aji , bji } as above. Then ∂0 tr(σ) = {C∂∂0 ij , a∂∂ 0 ji , b∂∂ 0 ji }. Also ∂0 (σ) =
(m, γ 0 ) where:
∂0 h(l)
γ 0 = {∆[n − 1] × ∂0 E → ∆[n − 1] × ∂0 E}l∈Z/p+1
35
j
Let g(l) be the map induced from ∂0 h(l) by ∆[0] ⊂ ∆[n − 1]. Thus g(l)
i is the map
0
j+1
j
induced from h∂(l)j∂ 0 i by ∆[1, 1] ⊂ ∆[n − 1]. We must check that g(l)
i = f(l) i+1 as germs
j+1
1 j+1
j−i j+1
near F(l)
i+1 = F(l) i+1 q . . . q F(l) i+1 . Fix k with 1 ≤ k ≤ j − i. We will show that
j
j+1
k j+1
j+1
g(l)
i = f(l) i+1 near F(l) i+1 . This will follow if we show that h(l) i+1 does not depend on
k j+1
δ ∈ ∆[1] near F(l)
i+1 , and this is clear if ml = 1. Assume ml = 0.
k j+1
i+1+k j+1
Since F(l)
, condition SH 2 implies that hj+1
i+1 ⊂ Ei+1
(l) i+1 is determined near
k j+1
j
i+1+k
i+1+k
F(l)
i+1 by h(l) i+1 and a retraction r(l) i+1 . . . r(l) i+1 which does not depend on δ ∈ ∆[1].
k i+1+k
Thus it suffices to show that hi+1+k
(l) i+1 does not depend on δ ∈ ∆[1] near F(l) i+1 .
k i+1+k
But by the definition of F(l)
as a fixed point set of a composition of filtration
i+1
k i+1+k
i+k
preserving maps, hi+1+k
(l) i+1 (F(l) i+1 ) ∩ Ei+1 = ∅. Thus condition SH 3 implies that near
k i+1+k
F(l)
the map hi+1+k
i+1
(l) i+1 does not depend on δ ∈ ∆[i + k] which contains ∆[1] since k is
positive.
Appendix: Pushouts of fiberwise ENRs
p
We say that a space E → B over B has the Beginning Covering Homotopy Property
(BCHP ) if for any commutative diagram
X
i0
?
X ×I
φ E
p
?
d B
there exists a neighborhood U of X ×0 in X ×I and a map D: U → E satisfying pD = d|U
p
and Di0 = φ. Dold proves in [7] that if E is EN R then E → B is EN RB if and only if
it has the BCHP .
f
g
We now prove that the pushout of a diagram G ← F → E of EN RB s is EN RB
provided that F and E are compact and g is injective. We may assume that both
the cofibration F → E and the canonical map G → G ∪F E are inclusions and that
G ∪F E − G = E − F with the canonical map E → G ∪F E equal to the inclusion when
restricted to E − F . We then have the following commutative diagram:
F
f HH incl.
HH
incl.
j
H
- G ∪F E π
G
E
HH q
ρ
H
HH
p
j ?
B
Let W0 be a neighborhood of F in E deformable (over B) in E to F relative to F . Then
W = π(W0 ) ∪ G is a neighborhood of G in G ∪F E deformable in G ∪F E to G relative
to G, say via Ht : W → G ∪F E.
Lemma A.1 If E = B × C where C is a compact EN R and p is the projection B × C →
B then ρ is EN RB .
Proof: We show ρ has the BCHP . Fix a commutative diagram:
X
i0
?
X ×I
φG ∪F E
ρ
?
d
-B
36
From the proof in [7] of “BCHP implies EN RB ” we see that we may assume X is EN R.
Let X1 = φ−1 (G) and X2 = X − X1 . Let Y = X1 × 0 ∪ X2 × I. Since
φ|X2 : X2 → G ∪F E − G = E − F ⊂ E = B × C
if x ∈ X2 we may write φ(x) = (φ1 (x), φ2 (x)) where φ1 (x) ∈ B and φ2 (x) ∈ C. Define
D0 : Y → G ∪F E by:
(d(x, t), φ2 (x)), (x, t) ∈ X2 × I
0
D (x, t) =
φ(x),
(x, t) ∈ X1 × 0
Lemma A.2 D0 is continuous.
Proof: Suppose (xn , tn ) is a sequence in X2 × I converging to (x, 0) ∈ X1 × 0. Let
g = φ(x) and U0 be a neighborhood of g in G∪F E. Let b = q(g) ∈ B. Say f −1 (g) = b×C0
for some compact subset C0 of C. Let U1 = π −1 (U0 ), a neighborhood of b × C0 in E.
Find a neighborhood C1 of C0 in C so that b × C1 ⊂ b × C1 ⊂ U1 .
Since π is a closed map and B × C1 ∩ U1 is open and π −1 (g) ⊂ B × C1 ∩ U1 there is
a neighborhood U of g in G ∪F E with π −1 (U ) ⊂ B × C1 ∩ U1 (for example,
U = G ∪F E − π(E − (B × C1 ∩ U1 ))
is such a neighborhood). Find a neighborhood V of b in B with V × C1 ⊂ π −1 (U ).
Then π(V × C1 ) is open in the image of π. In fact, π −1 (π(V × C1 )) = V × C1 , i. e., if
(x, y) ∈ V × C1 and π(x, y) = π(x, z) then z ∈ C1 :
(x, y) ∈ V × C1 ⇒ (x, y) ∈ π −1 (U ) ⇒ (x, z) ∈ π −1 (U ) ⇒ (x, z) ∈ B × C1
Thus eventually d(xn , tn ) ∈ V and φ(xn ) ∈ V × C1 i. e.,
D0 (xn , tn ) = (d(xn , tn ), φ2 (xn )) ∈ V × C1 ⊂ U1 ⊂ U0
and D0 is continuous, QED.
Now let W1 = D0−1 (W ), a neighborhood of X1 × 0 in Y . Say W1 = W2 ∩ Y , W2 open
in X × I. Let W3 be open in X × I with X1 × 0 ⊂ W3 ⊂ W3 ⊂ W2 . Let U0 be open in
X × I with X × 0 − W3 ⊂ U0 ⊂ U0 ⊂ X2 × I. Let U1 = W3 ∪ U0 . Let Z = U1 − ∂X1 × 0.
We have two closed disjoint sets in Z: X1 × I ∩ Z and (X2 × 0 ∪ U0 ) ∩ Z. Find an open
set V with X1 × I ∩ Z ⊂ V ⊂ V0 ⊂ Z − (X2 × 0 ∪ U0 ), V0 denoting the closure of V
in Z. Choose a map λ: Z → I with λ(V0 ) = 1 and λ(X2 × 0 ∪ U0 ) = 0. Define a map
U1 ∩ Y → G ∪F E by

y ∈X ×0
 φ(y),
y 7→ D0 (y),
y ∈ U0

H(D0 (y), λ(y)), otherwise.
We continue denoting this map by D0 . We can easily check that D0 is continuous, that
D0 i0 = φ, and that ρD0 = d|U1 ∩ Y . Moreover, D0−1 (G) contains X1 × 0 ∪ V , V denoting
the closure of V in U1 (thus V = V0 ∪ ∂X1 × 0).
Using Proposition 3.1, extend D0 |(X1 × 0 ∪ ∂V ) to a map D1 : O → G for some
neighborhood O of X1 × 0 ∪ ∂V in U1 , where we consider U1 as a space over B via d.
Let U = (U1 − V ) ∪ O. This is a neighborhood of X × 0 in X × I. Further U
can be written as the union of two closed sets (in U ), namely U1 − V and O ∩ V , and
(U1 − V ) ∩ (O ∩ V ) = ∂V . We may therefore define a map D: U → G ∪F E by defining
D|O ∩ V = D1 |O ∩ V and D|U1 − V = D0 |U1 − V . Then pD = d|U and Di0 = φ, QED.
f
Proposition A.1 Let B be a compact EN R and G ← F ⊂ E be a diagram of EN RB s
with F and E compact. Then G ∪F E is EN RB .
37
Proof: For E is a neighborhood retract of some B × C for some disk C and thus G ∪F E
is a neighborhood retract of G ∪F (B × C), QED.
Acknowledgements— I would like to thank my advisor, Thomas Goodwillie, for suggesting this
project to me, as well as for his help and encouragement. Most of the main ideas of this paper
are present in a letter of his to Waldhausen (see [9], especially §5). I would also like to thank
the University of Bielefeld for its hospitality while I was preparing this version.
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Fakultät für Mathematik
Universität Bielefeld
33615 Bielefeld
Germany
39