arXiv:1302.6014v2 [math.AT] 3 Mar 2013
EQUIVARIANT HOMOTOPY DIAGRAMS
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Abstract. We discuss obstructions to extending equivariant homotopy commutative diagrams to more highly commutative ones and rectifications of these equivariant homotopy commutative diagrams. As an application we show that in some
cases these obstructions vanish and the rectifications can be used to construct group
actions on products of spheres.
1. Introduction
Let A and M be two categories and U : A → M be a forgetful functor. We will say
an object Y of A is a free object if there exists an object X of M and a morphism
η : X → U(Y ) in M such that (Y, η) is an initial object in the comma category
X ↓ U. For example when you consider the forgetful functor from the category of
groups to the category of sets, free objects are free groups. Similarly one can define
free modules. However in some categories it is notoriously hard to construct free
objects. In this paper, we discuss one such category and a method for constructing
free objects in that category.
Let G be a finite group, M be a category, and GG denote the groupoid which has
one object ∗G with morphism set equals to the group G. A functor from GG to M
will be called a G-object of M. Given X a G-object of M we can obtain a functor
X from the opposite category of the poset of subgroups of G to M by sending H to
limit of X over GH if all these limits exist. From now on, we assume that M is a
category for which all the limits lim X exist.
GH
Let A denote the category with G-objects of M as objects and natural transformations from X to Y as morphisms between X, Y two G-objects of M. A free G-object
is a free object for the forgetful functor U : A → M which send X to X(∗G ). For
example if M = Top the category of topological spaces and continuous functions
then free G-objects are spaces with a free G-action. In this case we know that there
is no free Z/3-object Y with U(Y ) = S2 .
Let M, A be as above. If we further assume that M is a homotopical category
then we have a localization functor l from M to its homotopy category Ho(M).
For an object X of M, we will still write X instead of l(X) which is an object of
Ho(M). Considering the weak equivalences on M = Top defined by isomorphisms of
homotopy groups, we have Y = EG×S2 as a free G-object with U(Y ) ∼
= S2 in Ho(M).
The authors are supported by TÜBİTAK-TBAG/110T712.
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ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Considering diffeomorphisms as weak equivalences on M = Man the category of
smooth manifolds and smooth maps, the existence of free G-objects Y with U(Y ) ∼
=
S3 in Ho(M) are studied as a part of the generalized Poincare conjecture.
Here the case we are interested in is the case when M = CW the category of
finite CW-complexes as objects and admissible cellular maps as morphisms. Admissible means open cells are send homeomorphically onto open cells and such homeomorphisms between open cells commute with the identity map on the interior of the
disk after possible compositions with their characteristic maps. Here free G-objects
of CW are free G-CW-complexes. Consider weak equivalences in CW as the weak
equivalences of the underlying topological spaces. Given a group G and a product of
k spheres Sn1 × · · · × Snk , we will try to find if there exist a free G-object X of CW
such that X ∼
= Sn1 × · · · × Snk in Ho(CW).
We say a G-object X : G → M has isotropy in F if for all H subgroupoid of G not
in F the limit of X over H is an initial object. Notice that X is a free G-object if
it has isotropy in the family which only contains the trivial subgroup of G. So after
fixing a homotopy action the construction of free G-objects in Top is the same as
constructing objects with certain fixed point systems which is discussed in [8],[19],
[13], and [4]. However there are obstructions to these constructions. Here we discuss
a recursive method for handling these obstructions.
Swan [21] proved that there exists a free G-object of CW if G does not contain
a subgroup isomorphic to Z/p × Z/p for any prime p. The converse of this result
is proved by Smith [20]. Define the rank of a group G as the number k such that
G contains a subgroup isomorphic to an elementary abelian group of rank k but no
subgroup isomorphic to an elementary abelian group of rank k + 1. It is conjectured
(see [5]) that if the finite group G has rank less than or equal to k, there exist a finite
G-CW-complex homotopy equivalent to a product of k spheres. So this conjecture is
known to be true in case k = 1. The conjecture is also proved for k = 2 when Qd(p)
is not involved in the groups we consider (see [2], [15]).
Madsen-Thomas-Wall [18] proved that a finite group G acts freely on a sphere if
(i) G has no subgroup isomorphic to the elementary abelian group Z/p × Z/p for
any prime number p and (ii) has no subgroup isomorphic to the dihedral group D2p
of order 2p for any odd prime number p. It is also known (see [14], [1], [25]) that a
finite p-group can act freely and smoothly on a product of two spheres if and only if
the rank of the group is less than or equal to 3. Methods developed in these results
were also used in [26] to show that for every finite group G there exists a finite free
G-CW-complex homotopy equivalent to a product of spheres where the action is
homologically trivial.
In these later results, the idea is to construct Xn → . . . X1 → X0 = ∗ a sequence
of G-objects and {{e}} = Fn ⊆ · · · ⊆ F0 a sequence of subgroup families of G so
that Xi has isotropy in Fi and Xi ∼
= Xi−1 × Sni in the chosen homotopy category for
some ni ≥ 1. In the construction of Xi from Xi−1 , the starting point is a diagram
in the homotopy category. However gluing these over Xi−1 has some obstructions.
EQUIVARIANT HOMOTOPY DIAGRAMS
3
Here we discuss these obstructions. These obstructions are similar to the obstructions discussed in [7] and [12]. More precisely we replace homotopy diagrams with
cubically enriched functors called homotopy commutative diagrams and discuss their
rectifications which can be used to construct the cubically enriched functors in the
next step of the recursive process.
In Sections 2, 3 and 4, we remind some basic definitions and facts about simplicial sets and cubical sets and set the notations we will be using. In Section 5, we
introduce a bar construction which will be used to rectify homotopy commutative
diagrams. In Section 6, we discuss properties of the homotopy coherent nerve that
we obtain by using the bar construction of the previous section. In Section 7, we
discuss homotopy commutative diagrams and maps between homotopy commutative
diagrams. In Section 8, we discuss the obstructions to extending homotopy commutative diagrams to ones that are more highly commutative. In the last section, we
give some applications.
2. Some closed symmetric monoidal categories
Let Set denote the category of sets, Top denote the category of compactly generated topological spaces, and Cat denote the category of small categories. These are
all examples of closed symmetric monoidal categories with cartesian product.
2.1. Simplicial sets. Let [n] denote the category with objects {0, 1, 2, . . . , n} and
exactly one morphism from i to j when i < j. Let ∆ be the finite ordinal number
category, i.e., the full subcategory of Cat with objects [n] for n a non-negative integer.
Let C be a small category. Then simplicial objects of C are functors from ∆op to C.
Simplicial objects of C form a category denoted by sC with morphisms given by
natural transformations. If D is a small category and C is an arbitrary category, the
op
category of functors from D to C is denoted by C D . Hence we have sC = C ∆ . For
K a simplicial object of a category, we write Kn instead of K([n]) and call it the
n-simplices of K. For i in {0, 1, 2, . . . n}, we define a functor di from [n − 1] to [n] by
(
j
if j < i
di (j) =
j + 1 if j ≥ i
and define a functor si from [n + 1] to [n] by
(
j
if j ≤ i
si (j) =
j − 1 if j > i.
It is know that all morphisms of ∆ can be written as compositions of the above
morphisms. For a simplicial object K, the map di : Kn → Kn−1 induced by di is
called an i-th face map of K, and the map si : Kn → Kn+1 induced by si is called an
i-th degeneracy map.
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ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Objects of sSet are called simplicial sets. There is a standard embedding of ∆ in
sSet which takes [n] to the standard n-simplex ∆n =: ∆(−, [n]) : ∆op → Set. Let
∆≤n be the full subcategory of the finite ordinal number category on the objects
{[0], [1], · · · , [n]}. The functor induced from the inclusion ∆≤n ֒→ ∆ is called the
op
n-truncation functor τn : Set∆≤n → sSet. This functor has a left adjoint and composing two gives the n-th skeleton functor skn : sSet → sSet. Therefore, given a
simplicial set K, skn K is the subsimplicial set of K generated by K0 , . . . , Kn under
the degeneracies. If K and L are two simplicial sets, their simplicial tensor K ⊗ L in
sSet is defined:
(K ⊗ L)n := Kn × Ln .
This defines a symmetric monoidal structure on simplicial sets.
2.2. Cubical sets. Let n be a non-negative integer. We will write I n to denote
the set of functions from {1, 2, . . . , n} to {0, 1} for n ≥ 0. Here I 0 contains a single
function namely the empty function. For i in {1, 2, . . . n} and ε in {0, 1}, define a
function d(i,ε) from I n−1 to I n by


if j < i
f (j)
(i,ε)
d (f )(j) = ε
if j = i

f (j − 1) if j > i
for f in I n−1 and define a function si from I n to I n−1 by
(
f (j)
if j < i
si (f )(j) =
f (j + 1) if j ≥ i
for f in I n and define a function ci from I n+1 to I n by


if j < i
f (j)
i
c (f )(j) = max(f (j), f (j + 1)) if j = i

f (j + 1)
if j > i
for f in I n+1 . Let c denote the Box category with connections, whose objects are
(i,ε)
{I n }∞
, si and ci
n=0 and the morphisms of c are generated by the functions d
op
where n ≥ 0 and 1 ≤ i ≤ n. Let C be a category, a functor K : c → C is called
a cubical object of C. We write Kn instead of K(I n ) and call it the n-cubes of K.
The two maps d(i,ε) : Kn−1 → Kn induced by d(i,ε) is called an i-th face map of K,
the map si : Kn → Kn−1 induced by si is called an i-th degeneracy map, and the
map ci : Kn+1 → Kn induced by ci is called an i-th connection map. The collection
of all cubical objects of a category C form a category which we denote by cC, in
other words, cC denotes the functor category from op
c to C. Objects of cSet are
called cubical sets. There is a standard embedding of c in cSet called the standard
n-cube I n which is the cubical set c (−, I n ) : op
c → Set. For a cubical set K, the
subcubical set of K generated by K0 , . . . , Kn under the degeneracies is called the
EQUIVARIANT HOMOTOPY DIAGRAMS
5
n-th cubical skeleton of K, written skcn K. As in the simplicial sets, one could also
see the n-skeleton functor skcn : cSet → cSet as a composition of the n-truncation
functor τn on cubical sets with its left adjoint. If K and L are two cubical sets, their
cubical tensor K ⊗ L in cSet is defined:
K ⊗ L := colimI j →K,
I k →L
I j+k .
This defines a symmetric monoidal structure on cubical sets.
There are adjoint functors
cSet ⇋TS sSet
relating the cubical sets and the simplicial sets. The singular functor S associates to
every simplicial set X, a singular cubical set S(X) with n-cells S(X)n = sSet((∆1 )n , X).
The triangulation functor T is defined by T K = colimI n →K (∆1 )n . These functors
induce an equivalence of homotopy categories.
3. Some enriched categories
Let V be a bicomplete closed symmetric monoidal category with unit object E and
product ⊗. Let [B, C] denote the internal hom-object from B to C. The product
and the internal hom-objects satisfy an adjunction
V(A ⊗ B, C) ∼
= V(A, [B, C]).
A category M enriched over V means that M(M, M ′ ) is an object of V for every
M, M ′ object of M and the composition of M is given by a morphism of V
M(M ′ , M ′′ ) ⊗ M(M, M ′ ) → M(M, M ′′ ),
that satisfies the associativity and unit axioms.
Note that each V-category M gives rise to an ordinary category M0 which has the
same objects and M0 (M, M ′ ) = V(E, M(M, M ′ )).
A V-functor F : M → N between V-categories M and N assign an object F (M)
of N to each object M of M and a morphism
M(M, M ′ ) → N (F (M), F (M ′ ))
in V that satisfy the functoriality diagrams. One can similarly define a V-adjoint
pair and a V-natural transformation. In particular, these notions give rise to an
underlying ordinary functor F0 : M → N , underlying ordinary adjoint pairs and
natural transformation.
3.1. Cofibrant replacement of simplicially enriched categories. One way of
assigning a simplicial category to a small category is the standard resolution construction of [9] and [10] (see also [22]). Given a small category C, the free category on
C is the free category F C, where obj(F C) = obj(C), and a morphism in F C from c to
d is a word of composable non-identity morphisms (fn , . . . , f1 ) such that the domain
of f1 is c and the range of fn is d. Here the composition is given by the concatenation
6
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
of words and the identity morphism on c is the empty word (). For simplicity, we
write F f := (f ) ∈ F C(c, d) for a morphism f ∈ C(c, d).
For every functor F : C → D, there is an induced functor F F : F C → F D which
sends F f to F (F (f )). This makes F a functor from small categories to small categories. The functor F has a co-monad structure with the co-unit functor UC : F C → C
which is the identity on objects and sends F f to f . The co-multiplication functor
∆C : F C → F 2 C is defined by ∆C (F f ) = F 2 f . Clearly these functors satisfy the
comonad identities. Therefore one can construct a simplicial category (with discrete
object set) F• C by letting Fn C = F n+1C. Here the face and the degeneracy maps are
identity on objects and on morphisms they are given by
di = F i (UF n+1−i C ) : F n+1 C(c, d) → F n C(c, d)
si = F i (∆F n−i C ) : F n+1 C(c, d) → F n+2 C(c, d).
The category F• C is also a simplicially enriched category.
3.2. Cofibrant replacement of cubically enriched categories. The W-construction
on a small category C is the cSet-category W C where Obj(W C) = Obj(C) and for
every a, b objects of C, the cubical mapping complex W C(a, b) constructed as follows:
.
a
∼
W C(a, b) =
Iσn
σ : [n+1]→C,
σ(0)=a, σ(n+1)=b
where the equivalence relation is given by
n−1
i,0 n
n+1
Iσd
(Iσ ) and Iσs
n+1−i ∼ d
n+1−i
(
si (Iσn )
∼
ci (Iσn )
if i = 1, n + 1
otherwise.
The cubical composition
W C(a0 , ai ) ⊗ W C(ai , an+1 ) → W C(a0 , an+1 ) = W C(a, b)
n
where (β · σ)(j → j + 1) = σ(j → j + 1)
sends the (n − 1)-cube Iβi−1 ⊗ Iσn−i to d(i,1) Iβ·σ
if j ≤ n − i and (β · σ)(j → j + 1) = β((j + i − n − 1) → (j + i − n)), otherwise.
Moreover for every functor f : C → D, the W-construction induces a cSet-functor
W f : W C → W D which is given on objects by f and on morphisms by sending an
n-cube Iσn to Ifn◦σ . This makes W a functor from the category of small categories to
the category of cSet-categories. For every small category C, the simplicially enriched
categories T W C and F• C are naturally equivalent (see Lemma 3.6 in [3]).
4. Some tensored categories
We say that a V-category M is tensored if for every object K of V and M of M,
there is an object K ⊙ M of M and a V-natural isomorphism:
[K, M(M, M ′ )] ∼
= M(K ⊙ M, M ′ ).
EQUIVARIANT HOMOTOPY DIAGRAMS
7
Definition 4.1. Let M be a tensored V-category, D be a small V-category, and
G : D op → V and F : D → M be V-functors. Then their tensor product is defined to
be the following coequalizer:
!
a
a
(1)
G ⊙D F = coeq
Gd′ ⊗ D(d, d′) ⊙ F d ⇒
Gd ⊙ F d .
d,d′
d
4.1. Geometric realization. Let M be a tensored V-category, D be a small Vcategory, and R : D → M be a V-functor. We can then define the geometric realization with respect to R of any functor X : D op → V to be
|X|R = X ⊙D R.
Proposition 4.2. Let f : D1 → D2 be a fibration and R : D2 → M. Then for every
e R where X
e
functor X : D1op → V, there is a homeomorphism between |X|f ∗ (R) and |X|
op
is the left Kan extension of X along f .
e
Proof. Since X(c)
= colimz∈f −1 (c) X for every c ∈ D2 , the following diagram commutes
`
`
x,y∈D1 X(y) ⊗ MorD1 (x, y) ⊙ R(f (x))
z∈D1 X(z) ⊙ R(f (z))
`
a,b∈D2
e ⊗ MorD1 (a, b) ⊙ R(a)
X(b)
`
c∈D2
e ⊙ R(c).
X(c)
e R is a co-cone for the first row. In particular, we have
Therefore |X|
e ⊙ R(c) = colimz∈f −1 (c) X ⊙ R(c)
X(c)
e R the co-equalizer.
which makes |X|
4.2. Topological spaces. The category Top can be given the structure of a sSetcategory by applying the total singular complex functor to each hom-space. We will
denote the resulting sSet-category by TopsSet . The category TopsSet is tensored
with K ⊙sSet X = |K| × X.
Similarly we denote by TopcSet the cSet-category obtained by setting
TopcSet (X, Y )n = Top(X × I n , Y ).
The category TopcSet is tensored with K ⊙cSet X := |K| × X. Therefore for every
cubical set A and a space X ∈ Top, we have the following equivalence
T A ⊙sSet X ≃ A ⊙cSet X.
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ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
5. Equivariant double bar construction
5.1. Grothendieck constructions. Let Cat denote the category of small categories and functors. Given a functor γ : C → Cat we define two categories called
Grothendieck constructions
of γ.
R
The category C γ is a category with objects
{(c, x) | c is an object of C and x is an object of γ(c)} ,
with morphisms from (c1 , x1 ) to (c2 , x2 ) is
{(g, f ) | g : c1 → c2 is a morphism in C and f : γ(g)(x1 ) → x2 is a morphism in γ(c2 )} ,
with composition
The category
R op
C
(g2 , f2 ) ◦ (g1 , f1 ) = (g2 ◦ g1 , f2 ◦ γ(g2 )(f1 )).
γ is a category with objects
{(c, x) | c is an object of C and x is an object of γ(c)} ,
with morphisms from (c1 , x1 ) to (c2 , x2 ) is
{(g, f ) | g : c1 → c2 is a morphism in C and f : x2 → γ(g)(x1 ) is a morphism in γ(c2 )} ,
with composition
(g2 , f2 ) ◦ (g1 , f1 ) = (g2 ◦ g1 , γ(g2 )(f1 ) ◦ f2 ).
Let V-Cat denote the category of small V-categories and V-functors. From now on
we assume that V is closed under
R opcoproducts.
R Hence for any functor γ : C → V-Cat
the Grothendieck constructions C γ and C γ can be considered as a V-category.
5.2. Definition of double bar construction. We now introduce the enriched twosided bar construction. If MRis a tensored V-category,
C is a small category, γ : C →
R
op
V-Cat is a functor and G : C γ → V and F : C γ → M are V-functors then the
two-sided simplicial bar construction is a functor B• (G, γ, F ) : C → sM is defined as
follows. For c an object of C, the n-simplices Bn (G, γ, F )(c) of the simplicial object
B• (G, γ, F )(c) is given by
a
G(c, αn ) ⊗ γ(c)(αn−1 , αn ) ⊗ · · · ⊗ γ(c)(α0 , α1 ) ⊙ F (c, α0)
α0 , α1 , . . . αn
objects of γ(c)
and the face and the degeneracy maps are defined by using composition in γ(c), the
evaluation maps γ(c)(α0 , α1 ) ⊙ F (c, α0) → F (c, α1 ) and γ(c)(αn−1 , αn ) ⊗ G(c, αn ) →
G(c, αn−1), and the insertion of identities E → γ(c)(α, α′ ). For a morphism f : c → d
in the category C a morphism from Bn (G, γ, F )(c) to Bn (G, γ, F )(d) is defined by considering the morphisms G(f, 1γ(f )(αn ) ) : G(c, αn ) → G(d, γ(f )(αn )), γ(f ) : γ(c)(α, α′) →
γ(d)(γ(f )(α), γ(f )(α′)), and F (f, 1γ(f )(α0 ) ) : F (c, α0 ) → F (d, γ(f )(α0)).
EQUIVARIANT HOMOTOPY DIAGRAMS
9
The notion of double bar construction for enriched categories was first introduced
by Shulman [23] and this construction can be considered as an equivariant version of
Shulman’s double bar construction.
5.3. Properties of double bar construction. Let M be a tensored
R op V-category,
C beR a small category, γ0 , γ1 : C → V-Cat be functors, and G : C γ1 → V and
F : C γ1 → M be V-functors.
R Then every natural transformation from N∗ : γ0 → γ1
∗
induces a V-functor N F : C γ0 → M which is defined on objects by N F (c, x) =
F (c, Nc (x))R for every c ∈ C and x ∈ γ0 (c). For a morphism (g, f ) : (c1 , x1 ) →
(c2 , x2 ) in C γ0 , we
define N ∗ F (g, f ) = F (g, Nc2 (f )). Similarly, one can define a
R
op
V-functor N ∗ G : C γ0 → V. Hence N induces a natural transformation N ∗ from
B• (N ∗ G, γ0 , N ∗ F ) to B• (G, γ1 , F ).
R op
R
Let γ : C → V-Cat be a functor, and G : C γ → V and F : C γ → M be Vfunctors. Then every functor p : D → C from a small category D induces
R op a natural
∗
∗
∗
∗
transformation
p from B• (p G, γ ◦ p, p F ) to B• (G, γ, F ) where p G : D γ ◦ p → V
R
and p∗ F : D γ ◦ p → M are induced V-functors.
5.4. Opfibrations. Let G be a groupoid and γ : G → Cat be a functor. We define
a functor
dγ : G → Cat
on each object a of G by letting dγ(a) be the full subcategory of the simplex category
∆ ↓ γ(a) with object set consisting of σ : [n] → γ(a) where n = 1 or for each i the
morphism σ(i → i + 1) is not identity. Assume that Iγ denotes the Grothendieck
construction
Z
Iγ =
dγ.
G
Let pγ denote the natural projection
pγ : Iγ → G.
Note that pγ is an opfibration. Now define
f γ : Iγ → Cat
by f γ(a, σ) = [n] when σ : [n] → γ(a). Note that we can consider γ as the left Kan
extension Lanpγ (f γ) with the universal natural transformation ǫ : f γ → γ ◦ pγ given
by ǫ(a,σ) = σ.
R op
R
Theorem 5.1. Let G : G W γ → cSet and F : G W γ → M be morphisms in
cSet-Cat. Then we have
B(G, W γ, F ) = Lanp B(ǫ∗ p∗γ G, W f γ, ǫ∗ p∗γ F ).
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ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Proof. For every a ∈ G and σ : [n] → γ(a), define
m
ηa,σ
: (B(ǫ∗ p∗γ G, W f γ, ǫ∗ p∗γ F )(a, σ))m → (B(G, W γ, F )(a))m
by sending the summand G(a, W (σ)(αm ))⊗W ([n])(αm−1 , αm ) · · ·⊗W ([n])(α0 , α1 ) ⊙
F (a, W (σ)(α0)) corresponding to α0 , · · · , αm in W ([n]) to a summand corresponding to W (σ)(α0 ), · · · , W (σ)(αm) in W (γ(a)) by a map induced from the functor
W (σ) : W ([n]) → W (γ(a)). For a map (g, f ) : (a, σ : [n] → γ(a)) → (b, β : [m] →
γ(b)), we have βf = γ(g)σ and hence W (β)W (f ) = W (γ(g))W (σ). This makes
η = (ηa,σ ) a natural transformation.
For a given functor M : G → cSet-Cat and a natural transformation
α : B(ǫ∗ p∗γ G, W f γ, ǫ∗ p∗γ F ) → M ◦ p,
we define µ : B(G, W γ, F ) → M as follows. For every a ∈ G, define
µ1a : B(G, W γ, F )(a)1 → M(a)1
by sending (Iaj ⊗Iβk )⊙Ial in (G(a, x1 )⊗W (γ(a)(x0 , x1 ))⊙F (a, x0 ) to αa,β (Iaj ⊗Iιkk+1 )⊙Ial
where ιk+1 : [k + 1] → [k + 1] is the identity functor. For s > 1, define µsa by using the
cubical decomposition defined in Section 3.2. Since α is a natural transformation, we
have M(f )α(a,β) = α(b,γ(f )(β)) B(f, idγ(β) ) for every map f : a → b and β : [k + 1] →
γ(a). Therefore M(g)(µ1a ((Iaj ⊗ Iβk ) ⊙ Ial ) = µ1b B(f, idγ(β) )((Iaj ⊗ Iβk ) ⊙ Ial ) and hence
µ is a natural transformation.
Note that for every (a, σ : [n] → γ(a)) ∈ f γ and g : [k + 1] → [n], the following
diagram commutes
B(ǫ∗ p∗γ G,W fγ,ǫ∗ p∗γ F )(ida ,f )
B(ǫ∗ p∗γ G, W f γ, ǫ∗ p∗γ F )(a, σ ◦ g) −−−−−−−−−−−−−−−−→ B(ǫ∗ p∗γ G, W f γ, ǫ∗ p∗γ F )(a, σ)


α(a,σ) 
α(a,σ◦g) 
y
y
M(a)
M(a).
Therefore we have η(µp) = α as desired. Clearly µ is unique.
Note that the above theorem still holds if we replace the functor W with the
functor skcn W : Cat → cSet-Cat or the functor T skcn W : Cat → sSet − Cat. Here
the skeleton skcn C of a cSet-category C is generated by the tensor products of the
cubes of dimension less than equal to n.
6. Classifying Spaces
6.1. Category [n]. The morphisms of the free simplicial category of [n] is given as
follows (see [9])
∆[1]j−i−1 , i ≤ j;
F• [n](i, j) =
∅,
otherwise.
EQUIVARIANT HOMOTOPY DIAGRAMS
Define
a
Bin =
11
|F• [n](ri−1 , ri )| × · · · × |F• [n](r0 , r1 )|
r=(r0 ,··· ,ri )∈Rn
i
where Rin = {r = (r0 , · · · , ri )| 0 ≤ r0 < r1 < · · · < ri ≤ n}. Then the standard
geometric realization of the double bar construction B(∗, F•[n], ∗) is given by
a
.
|B(∗, F•[n], ∗)| =
Bin × |∆i |
∼
i≥0
where the equivalence relation has (di x, t) = (x, di t) for every (x, t) ∈ Bin × |∆i−1 |.
From now on, we write |C n (r)| := |F•[n](ri−1 , ri )| × · · · × |F• [n](r0 , r1 )| for short. We
(i)
(i)
(1)
(1)
also denote an element (t1 , · · · , tri −ri−1 −1 ; · · · ; t1 , · · · , tr1 −r0 −1 ) of |C n (r)| by t(r)
and an element (t0 , · · · , ti ) ∈ |∆i | by t.
e to sSet where ∆
e is the subcategory
Now consider B(∗, F• , ∗) as a functor from ∆
e are generated by the
of ∆ which has the same object set as ∆ and morphisms in ∆
injections [k] ֒→ [m] and the surjection [1] → [0].
Theorem 6.1. There is a natural homeomorphism h : |B(∗, F•, ∗)| → ∆• where the
e → Top is given by ∆• ([n]) = |∆n |.
functor ∆• : ∆
In order to define the map hn := h([n]), we first introduce some notation. Let
be the subset of |∆n | consisting of all a = (a0 , · · · , an ) satisfying the following
conditions
Anr
(1) am = 0 when m > ri or m < r0 ,
(2) au ≤ min(ark , ark+1 ) when rk < u < rk+1 ,
(3) min(arp , ars ) ≤ arp+1 when p < s,
for each r = (r0 , · · · , ri ) ∈ Rin and 1 ≤ i ≤ n. Now define hn (r) : |Cn (r)| × |∆i | → Anr
by
1
hn (r)(t(r), t) = Pn
(a0 , · · · , an )
j=0 aj
where
 k+1
 tu−rk maxp≤k<s{min(tp , ts )} , rk < u < rk+1 ,
au =
0,
u < r0 or u > ri ,

maxp<k<s{tk , min(tp , ts )},
u = rk ,
for every (t(r), t) ∈ |Cn (r)| × |∆i |. It easily follows that (a0 , · · · , an ) ∈ Anr . Note that
(i−1)
dj (t(r)) = (s1
(i−1)
(1)
(1)
, · · · , sdj ri −dj ri−1 −1 ; · · · ; s1 , · · · , sdj r1 −dj r0 −1 )
12
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
where dj rk = rk if k < j, dj rk = rk+1 otherwise, and
 k
tu ,
k < j;



j

,
k = j and u < rj − rj−1 − 1;
t
 u
k
1,
k = j and u = rj − rj−1 − 1;
su =

j+1

t j +rj−1 , k = j and u > rj − rj−1 − 1;


 u−r
tk+1
k > j.
u ,
When tj = 0, min(tp , ts ) = 0 whenever j ∈ {p, s}. So it does not contribute to au
and hence we have hn (r)(t(r), t) = hn (dj r)(dj t(r), t′ ) where dj t′ = t. Therefore the
maps hn (r) induce a map hn : |B(∗, F•[n], ∗)| → |∆n |.
Proof. To show that hn is onto, take (b0 , · · · , bn ) ∈ ∆n . Let r = (r0 , · · · , ri ) where r0
is the smallest integer for which bu 6= 0 and rk ’s are defined inductively as follows.
If for all u ≥ rk , brk < bu , define rk+1 = rk + 1; otherwise let rk+1 be the largest
integer such that brk ≥ bu for all rk < u < rk+1 . This procedure terminates at i
where bu = 0 for all u > ri . Let tk =
brk
b
where b = br0 + · · · + bri and tkj =
b(rk−1 +j)
brk−1
.
Then hn ([t(r), t]) = (b0 , · · · , bn ). Note that we have br0 < br+1 < · · · < bri by
construction and hence r = (r0 , r1 , · · · , ri ) is the smallest subset of {1, · · · , n} for
which (b0 , · · · , bn ) ∈ Anr .
It remains to show that hn ([t(r), t]) = hn ([s(r ′ ), s]) implies (t(r), t) ∼ (s(r ′ ), s). For
this assume that t(r), t, and r = (r0 , · · · , ri ) is defined as above. Since the intersection
of Anr and Anr′ is Anr′′ where r ′′ = r ∩ r ′ , we have r ′ ⊇ r as a set. So r = dik · · · di1 r ′
for some i1 > · · · > ik . Without loss of generality, we can let r = dj r ′ , that is,
r ′ = (r0 , · · · , rj−1 , rj′ , rj , · · · , ri ). Once we show that s is (t0 , · · · , tj−1 , 0, tj , · · · , ti ), it
immediately follows that dj s(r ′ ) = t(r). Therefore we obtain (t(r), t) ∼ (s(r ′ ), s) as
desired.
By definition, we have a0 = t0 = s0 and ari = ti = si+1 . By inductive hypothesis
on k < j, tl = sl for all l ≤ m < k. Since min(sm , sz ) < sm , we have
′
sm = tm < tm+1 = arm+1
= max{sm+1 , maxm+1<z {min(sm , sz )}} = sm+1 .
Similarly, one can show that sm = tm−1 for all m > j. Since the sum of sm ’s is 1, this
forces sj to be zero. This proves that the map hn is a homeomorphism for each n.
e is generated by the face maps dj : [n − 1] → [n] and the
Since the morphisms in ∆
0
degeneracy map s : [1] → [0], to show that h is natural, it suffices to show that the
following diagrams commute
dj
∗
|B(∗, F•[n − 1], ∗)| −−−
→ |B(∗, F•[n], ∗)|




hn−1 y
hn y
∆• ([n − 1])
dj
−−−→
ƥ ([n])
s0
∗
→ |B(∗, F•[0], ∗)|
|B(∗, F•[1], ∗)| −−−




h1 y
h0 y
ƥ ([1])
s0
−−−→
ƥ ([0])
EQUIVARIANT HOMOTOPY DIAGRAMS
13
for every 0 ≤ j ≤ n. Since ∆• ([0]) ∼
= ∗, the second diagram com= |B(∗, F•[0], ∗)| ∼
mutes. On the other hand, the induced map dj∗ : |B(∗, F•[n−1], ∗)| → |B(∗, F• [n], ∗)|
is explicitly given by

[t(r), t],
j > ri ;



l
[s(r+ ), t], j = rl , 0 < l ≤ i;
dj∗ [t(r), t] =
l+1
[p(r+
), t], rl < j = u < rl+1 , 0 ≤ l ≤ i;



0
[t(r+ ), t],
j ≤ r0 ,
m
where r+
= (r0 , · · · , rm−1 , rm + 1, · · · , ri + 1) and

m = l + 1, v = u − rl ,
 0,
0, m = l, v = rl − rl−1 ,
m
m
t
,
m
= l + 1, v > u − rl ,
p
=
sm
=
v−1
v
v
tm
 m
v , otherwise,
tv , otherwise.
By using this formula, one can easily check that the map hn respects the face maps
in the sense that it makes the first diagram above commute.
6.2. In general. Let B : Cat → Top be the classifying space functor. Then we have
the following result.
Theorem 6.2. There is a natural homeomorphism between the functors Bγ and
|B• (∗, F•γ, ∗)|.
Proof. First by Section 5.4, the bisimplicial set B(∗, F• γ, ∗) is the left Kan extension
of B(∗, F•f γ, ∗) over pγ . Now by the previous theorem there is a natural homeomorphism between |B(∗, F•f γ, ∗)| and Bf γ. We also know Bγ is the left Kan extension
of Bf γ over pγ . Hence we have a natural homeomorphism between the functors Bγ
and |B• (∗, F•γ, ∗)|.
7. Homotopy diagrams
Let M be a homotopical category (see [11]), in other words, M is a category
with a class of morphisms called weak equivalences that contains all the identities
and satisfies the 2-out-of-6 property: if hg and gf are weak equivalences, then f ,
g, h, and hgf are also weak equivalences. Any homotopical category M has a
homotopy category Ho(M), obtained by formally inverting the weak equivalences.
There is a localization functor l : M → Ho(M) which is universal among the functors
inverting the weak equivalences. Here we have enriched homotopical categories and
their homotopy categories have small hom-sets.
7.1. Homotopy diagrams. For the following assume that G is a small category,
γ : G → Cat is a functor, M is a cSet-enriched
homotopical category with localizaR
tion functor l : M →RHo(M), andR f : G γ → Ho(M) is a functor. Also let pn denote
the projection from G skcn W γ to G γ.
14
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Definition 7.1. An n-commutative homotopy γ-diagram over f is a cSet-functor
Z
F:
skcn W γ → M
G
which satisfies f ◦ pn = l ◦ F .
Remark 7.2. Our definiton of a homotopy commutative diagram is in agreement
with Vogt’s definition [27] when M = TopcSet . For c an object of G and σ : [k + 1] →
γ(c) a functor, let fi denote the map σ(i → (i + 1)) for 0 ≤ i ≤ k, A = F (c, σ(0)),
and B = F (c, σ(k + 1)). Then F (idc , −) gives a cubical map from the cube Iσ to
TopcSet (A, B). If we consider this map as a continuous function D from the k-cube
[0, 1]k to TopcSet (A, B) and use the notation D(fk , tk , · · · , t0 , f0 ) for the image of
(tk , . . . , t1 ) in [0, 1]k then we have the following equations
D(fn , tn , · · · , f0 ) ◦ D(gm , um , · · · , g0 ) = D(fn , tn , · · · , f0 , 1, gm, um , · · · , g0 ).
and

D(fn , tn , · · · , f1 ),



D(fn , tn , · · · , fi+1 , max{ti+1 , ti }, fi−1 , · · · , f0 ),
D(fn , tn , · · · , f0 ) =
 D(fn−1 , tn−1 , · · · , f0 ),

 D(f , t , · · · , t , f f , t , · · · , f ),
n n
i+1 i i−1 i−1
0
f0 = id,
fi = id, 0 < i < n,
fn = id,
ti = 0.
7.2. Maps between homotopy diagrams. For a non-negative integer k let γ × [k]
denote the functor from G to Cat given by the cartesian product of the functor γ
and the constant functor [k].
Definition 7.3. Let n, k be two non-negative integers. Then a (k − 1)-map of ncommutative homotopy γ-diagrams over M is a cSet-functor
Z
F:
skcn+k W (γ × [k]) → M.
R
G
Definition 7.4. Let F1 , F2 : G skcn W γ → M be two n-commutative homotopy γdiagram over f . Then we say that F1 is homotopicRto F2 if there exists two 1-maps
of n-commutative homotopy
γ-diagrams H1 , H2 : G skcn+2 W (γ × [2]) → M such
R
that Hi restricted to G skcn W (γ × {j}) is the same as Fk when i + j ≡ k mod 2
R
c
and
H
1 restricted to G skn+2 W (γ × {j → j + 1}) is the same as H2 restricted to
R
skc W (γ × {j ′ → j ′ + 1}) when j + j ′ = 1, and both H1 and H2 restricted to
RG cn+2
skn+2 W (γ × {0 → 2}) are induced by the identity.
G
The following two theorems will explain why such cSet-functors are called maps
of homotopy diagrams. First we introduce some notations. If k is a non-negative
integer and A is a subset of {0, 1, 2, . . . , k} then A will denote the full subcategory
of [k] with the object set equal to A and for a positive integer r, sr,A will denote the
cSet-category
sr,A = skcr W (γ × A)
15
EQUIVARIANT HOMOTOPY DIAGRAMS
and Fr,A will denote the restriction of F to sr,A . Given F a (k − 1)-map of ncommutative homotopy γ-diagram over TopcSet , we will define a map
fk (F ) : |B• (∗, T sn,{0}, Fn,{0} )| × |I k−1| → |B• (∗, T sn,{k}, Fn,{k})|
which satisfies the following two theorems
Theorem 7.5. Let F be a n-commutative homotopy γ-diagram over TopcSet . Then
f1 (F ×[1]) is homotopy equivalent to the identity map on |B• (∗, T skcn W γ, T F )| where
F × [1] denotes the functor obtained by the projection from γ × [1] to γ.
The above theorem says up to homotopy identity goes to identity.
Theorem 7.6. The map fk also satisfies the following property
fk (F )(−, iB ) = f1 (Fn+1,{bs−1 ,bs } ) ◦ · · · ◦ f1 (Fn+1,{b0 ,b1 } )
where B = {0 = b0 < b1 < · · · < bs = k} and iB = (a1 , . . . ak−1 ) in |I k−1 | with ai = 1
if i = bj for some 1 ≤ j ≤ s − 1 and ai = 0 otherwise.
Now to define fk (F ), we first give a double bar construction B ′ (∗, e
γ , F ) for γe =
c
′
γ , F ) is a functor from G to the functor category from
T skn+k W (γ × [k]) where B (∗, e
∆ ↓ [k] to Top defined for c an object of G and σ : [m] → [k] in ∆ ↓ [k] by
a
B ′ (∗, e
γ , F )(c)(σ) =
γe(c)(αm−1 , αm )⊗· · ·⊗e
γ (c)(α0 , α1 ) ⊙F (c, α0 ).
For 0 ≤ i ≤ m,
αi is an object of
γ(c) × {σ(i)}
Let i be an integer with 0 ≤ i ≤ k. Then we define functors Ri , Ri , and Ri from
∆ ↓ [k] to Top as follows:
Ri (σ) = {t0 e0 + . . . tm em ∈ |∆m | | t′0 = t′1 = t′i−1 = 0},
j−1
X
1
Ri (σ) = {t0 e0 + . . . tm em ∈ Ri (σ) | ∀j ≥ i, j 6= k ⇒ t′j ≤ (1 −
ts }, and
2
s=i
1
Ri,t (σ) = {t0 e0 + . . . tm em ∈ Ri (σ) | t′i = t} where t ∈ [0, ]
2
X
′
tl for 0 ≤ j ≤ k. Notice that for
where σ : [m] → [k] is a functor and tj =
l∈σ−1 (j)
0 ≤ i ≤ k − 1 and u in [0,1], there is a natural transformation ri,u from Ri to Ri
1−ut′
which sends (t0 e0 + · · · + tm em ) to u(t0 e0 . . . td−1 ed−1 ) + 1−t′i (td ed . . . tm em ) where d
i
is the minimum element of the set {σ −1 (j)| i + 1 ≤ j ≤ k}. Note that Rk = Rk and
ri,0 (Ri ) = Ri+1 . Hence the composition rk−1,0 ◦ · · · ◦ r1,0 ◦ r0,0 induces a map from
|B ′ (∗, T skcn+k W (γ × [k]), F )|R0 to |B ′ (∗, T skcn W (γ × [k]), F )|Rk . To define fk (F ), we
need the following observations.
16
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Lemma 7.7. |B ′ (∗, T W [k], ∗)|R
0, 1
2
∼
= I k−1
Proof. Since B ′ (∗, T W [k], ∗)(σ) = B ′ (∗, T W [k], ∗)(si∗(σ)), we can suppress the degenerate objects in ∆ ↓ [k]. In addition, if σ −1 (0) or σ −1 (k) is empty then R0, 1 (σ) = ∅.
2
Therefore, we have
.
a
′
m
|B (∗, T W [k], ∗)|R =
∼
|C (σ)| × R0, 1 (σ)
0, 1
2
2
σ : [m]֒→[k],
σ(0)=0, σ(m)=k
where |C m (σ)| = |F• [k](σ(m − 1), σ(m))| × · · · × |F• [k](σ(0), σ(1))|. Note that for
every σ : [m] ֒→ [k] with σ(0) = 0 and σ(m) = k, we have R0, 1 (σ) ∼
= I m−1 and
2
|C m (σ)| ∼
= I k−m . Therefore |B ′ (∗, T W [k], ∗)|R 1 is obtained by gluing (k − 1)-cubes
0, 2
k−1
Iσk−1. On the other hand, for each m ≤ k, there are m−1
-many inclusions σ : [m] ֒→
[k] for which σ(0) = 0 and σ(m) = k and for each such σ, the size of the set
{(di )∗ (σ)| 0 < i < m} is (m − 1). Moreover there are exactly (k − m + 1)-many
γ : [m + 1] ֒→ [k] such that (di )∗ (γ) = σ for some 0 < i < m + 1. This means that
Iσk−1 is glued to (k − 1)-many different cubes. Indeed it is glued to each cube along a
different face containing the vertex ((fk−1) · · · (f0 ), ( 12 , 0, · · · , 0, 21 )). Therefore, these
2k−1-many (k − 1)-cubes are glued together to form another (k − 1)-cube with center
((fk−1) · · · (f0 ), ( 21 , 0, · · · , 0, 12 )).
((f3 )(f2 )(f1 )(f0 ), ( 12 , 0, 12 ))
Lemma 7.8. There is a natural inclusion of |B• (∗, T sn,{0} , Fn,{0} )|×|B ′ (∗, T W [k], ∗)|R
′
in |B (∗, T
skcn+k
0, 1
2
W (γ × [k]), F )|R0 .
Proof. Set U = |B ′ (∗, T skcn+k W (γ × [k]), ∗)|R0 . We can consider U ⊆ B(γ × [k])
because |B ′ (∗, T W (γ × [k]), ∗)| ∼
= B(γ × [k]). Consider V ⊆ Bγ. The projection
p1 : γ × [k] → γ induces Bp1 : B(γ × [k]) → Bγ. We have Bp1 (U) ⊆ V . Define f as
the restriction of Bp1 to U. So f : U → V . The projection p2 : γ × [k] → [k] induces
W p2 : W (γ × [k]) → W [k] and this induces g : U → |B ′ (∗, T W [k], ∗)|R ∼
= I k−1 . We
have (f × g) : U → V × I
k−1
0, 1
2
an injective map. Define X = |B (∗, T skcn+k W (γ ×
′
17
EQUIVARIANT HOMOTOPY DIAGRAMS
[k]), F )|R
0, 1
2
. Then X is a subspace of |B ′ (∗, T skcn+k W (γ × [k]), F )|R0 . Define Fe =
Fn+k,{0} . The space X is same as the pull back of |B(∗, T skcn+k W γ, Fe)| × I k−1 →
V × I k−1 by the map f × g. Define Z = |B(∗, T skcn W γ, ∗)| and consider Z ⊆
Bγ. We have Z × I k−1 ⊆ (f × g)(U). Now we are done because the pull back
of |B(∗, T skcn+k W γ, Fe)| × I k−1 → V × I k−1 by the map Z × I k−1 → V × I k−1 is
|B• (∗, T sn,{0}, Fn,{0} )| × |B ′ (∗, T W [k], ∗)|R .
0, 1
2
Lemma 7.9. There is a natural homeomorphism between |B ′ (∗, skcn W (γ ×[k]), F )|Rk
and |B• (∗, T sn,{k}, Fn,{k})|.
Proof. For each [n] ∈ ∆, let σ[n] : [n] → [k] be defined by Im(σ[n] ) = {k}. Then
p : ∆ → ∆ ↓ [k] defined by p([n]) = σn is a fibration with Rk (p([n])) = |∆n |. Since
we have B ′ (∗, skcn W (γ × [k]), F )(c)(σ[n] ) = B• (∗, T sn,{k}, Fn,{k} )(c)([n]), the result
follows from Proposition 4.2.
Now we can define the map fk (F ) as the composition of the homeomorphism
and
|B• (∗, T sn,{0}, Fn,{0} )| × |I k−1 | ∼
= |B• (∗, T sn,{0}, Fn,{0} )| × |B ′ (∗, T W [k], ∗)|R
′
the inclusion of |B• (∗, T sn,{0} , Fn,{0} )(c)| × |B (∗, T W [k], ∗)|R
0, 1
2
0, 1
2
in the realization
|B ′ (∗, T skcn+k W (γ × [k]), F )|R0 and the above map to |B ′ (∗, skcn W (γ × [k]), F )|Rk
which is homeomorphic to the space |B• (∗, T sn,{k}, Fn,{k} )|.
Note that Theorem 7.5 follows directly from the construction of the map f1 and
the fact that r0,0 is a deformation retract. On the other hand, the cubes in the proof
of Lemma 7.7 are glued together to form the (k − 1)-cube I k−1 as follow. For each
σ : [m] ֒→ [k] with σ(0) = 0 and σ(m) = k, there is one cube Iσk−1 and each such
cube is glued to others except the three faces passing through a vertex which forms
a vertex v = {v1 , · · · , vk−1 } of I k−1 with vj = 1 if j is in the image of σ. Therefore,
Theorem 7.6 directly follows.
8. Obstructions
8.1. Bredon cohomology. In this section, assume that G is a groupoid and Ab
denotes the category of abelian groups. Let γ : G → Cat be a functor. We write dγ
for the composition of γ with (∆ ↓ ·) : Cat → Cat. We define
Z
Iγ = dγ.
G
Definition 8.1. A local coefficient system on γ is a functor M : Iγ → Ab.
Let D denote the projection from Iγ to ∆. Given a local coefficient system M on
γ we obtain a cosimplicial abelian group M : ∆ → Ab as the right Kan extension
of M over π. Let CG∗ (γ; M) denote the associated cochain complex to M. In other
18
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
words
CG∗ (γ; M) = lim C ∗ (γ; M)
G
∗
where C (γ; M) is considered as a functor from G to the category of cochain complexes defined as follows:
Y
C n (γ; M)(c) =
M(c, σ)
σ : [n]→γ(c)
for c an object of G and
δc (f )(τ ) =
X
(−1)i M (idc , di )(f (τ ◦ di ))
i
n
for f in C (γ; M)(c) and τ : [n + 1] → γ(c). For g : c → c′ a morphism in G, a map
between C n (γ; M)(c) and C n (γ; M)(c′ ) is defined by (gf )(σ) = M ((g, idσ ))(f (g −1)).
Definition 8.2. The Bredon cohomology of γ with coefficients in M is defined as
the cohomology of the cochain complex CG∗ (γ; M).
To define relative Bredon cohomology take α = (αi )i∈I where αi : G → Cat is a
functor such that the category α(c) is a subcategory of γ(c) for every object c of G
and the inclusions give a natural transformation
from αi to γ for each i in I. Let
R
Mαi denote the restriction of M to G αi for i in I. Notice that CG∗ (αi ; Mαi ) can
be considered as a subcochain complex of CG∗ (γ; M). Let CG∗ (α; Mα) denote cochain
complex which is generated by the cochain complexes CG∗ (αi ; Mαi ) as i ranges over
I. Hence we obtain a relative cochain complex
CGn (γ, α; M) = CGn (γ; M)/CGn(α; Mα )
for n an integer.
Definition 8.3. The relative Bredon cohomology of the pair (γ, α) with coefficients
in M is defined as the cohomology of the cochain complex CG∗ (γ, α; M).
R
Let dn γ = π −1 ([n]) be the subcategory of G dγ with objects which are sent to
[n] by π and morphisms which are sent to identity by π. We can write dn γ as a
disjoint union of two full subcategories Ndn γ and Ddn γ where an object (c, σ) of dn γ
is in Ndn γ if σ is not in the image of a degeneracy map in the category dγ(c) and
the image of σ is not in αi (c) for any i in I and an object (c, σ) of dn γ is in Ddn γ
otherwise. Now we can define a subcochain complex of CG∗ (γ; M) as follows:
DCGn (γ, α; M) =
lim
(c,σ)∈Ddn γ
M(c, σ).
This cochain complex contains CG∗ (α; Mα ). Hence we can define a new cochain
complex
n
C G (γ, α; M) = CGn (γ; M)/DCGn(γ, α; Mα ).
EQUIVARIANT HOMOTOPY DIAGRAMS
19
The relative Bredon cohomology of the pair (γ, α) with coefficients in M is isomor∗
phic to the cohomology of the cochain complex C G (γ, α; M). We can also define a
subgroup of CGn (γ; M) as follows:
NCGn (γ, α; M) =
lim
(c,σ)∈N dn γ
M(c, σ).
n
Now notice that as an abelian group C G (γ, α; M) is isomorphic to NCGn (γ, α; M)
because we have
CG∗ (γ, α; M) = NCGn (γ, α; M) ⊕ DCGn (γ, α; M).
8.2. Obstructions to extending homotopy diagrams. Assume that we have a
cSet-functor
Z
F:
skck (W γ, skck+r W α) → M
G
where k and r are positive integers. For c an object of G and σ : [n] → γ(c) an object of
dγ(c), let MF (c, σ) denote the path component of morM (F (c, σ(0)), F (c, σ(n))) which
contains the morphism F (idc , σ(0 → n)) and Aγ (c, σ) denotes the automorphism
group of the object (c, σ) in the category π −1 ([n]). Then we define a functor
Z
mF :
dγ → cSet
such that for (c, σ) an object of
R
G
G
dγ
mF (c, σ) = lim MF (c, σ)
Aγ (c,σ)
R
and for a morphism (g, 1) in G dγ, the morphism mF (g, 1) : mF (c, σ) → mF (c′ , γ(g)σ)
is induced by the morphisms
F (g −1, 1) : F (c′ , γ(g)σ(0)) → F (c, σ(0)) and F (g, 1) : F (c, σ(n)) → F (c, γ(g)σ(n))
and for σ : [n] → γ(c) and τ : [m] → γ(c) two objects of dγ(c) and f : [n] → [m] a
morphism from σ to τ in dγ(c), the morphism mF (1, f ) : mF (c, σ) → mF (c, τ ) is
induced by the morphisms
F (1, τ (0 → 1)) ◦ · · · ◦ F (1, τ ((f (0) − 1) → f (0))) : F (c, τ (0)) → F (c, σ(0))
and
F (1, τ (f (n) → (f (n) + 1))) ◦ · · · ◦ F (1, τ ((m − 1) → m)) : F (c, σ(n)) → F (c, τ (m)).
Now we will discuss obstructions to extending a relative k-commutative homotopy
diagram F , in Rother words, we want to define obstructions to the existence of a cSetfunctor from G skck+1 (W γ, skck+r W α) to M which is equal to F when restricted
R
c
c
k+1
to
and ∂I k+1 denote the constant functors from
R G skk (W γ, skk+r W α). Let I
d γ to cSet constantly equal to the standard (k + 1)-cube and its boundary
G k+2
20
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
respectively. We define a natural transformation ok+1
from ∂I k+1 to mF restricted
F
to dk+2γ as follows:
k+1 ∼
ok+1
= ∂ morW [k+2] (0, k + 2)
F (c, σ) : ∂I
F (idc ,·)◦W σ
−→ mF (c, σ)
for c an object of G and σ : [k + 2] → γ(c) an object
of dk+2 γ(c). Assuming mF (c, σ)
R
is fibrant and simple for every object (c, σ) of G dγ, we can define a local coefficient
system on γ
πk F :
Z
dγ → Ab
G
k+2
by composing mF with πk . Now we can consider ok+1
as an element in C G (γ, α; πk F ).
F
Now the following result shows that ok+1
can
be
considered
as an obstruction to exF
tension.
R
Proposition 8.4. A cSet-functor F : G skck (W γ, skck+r W α) → M lifts to a cSetR
= 0.
functor F ′ : G skck+1 (W γ, skck+r W α) → M if and only if ok+1
F
Proof. We have
ok+1
F (c, σ) = 0 ∈ πk F (c, σ)
for all (c, σ) in Ddk+2γ. Moreover we have
πk (
lim
(c,σ)∈N dk+2 γ
mF (c, σ)) ∼
=
Hence the result follows.
lim
(c,σ)∈N dk+2 γ
πk F (c, σ).
Now we show that ok+1
is a cocycle.
F
Lemma 8.5. δok+1
= 0.
F
For k = 1 and τ : [4] → γ(c) an object of d4 γ(c), (δo2F )(c, τ ) is represented by the
following cube:
EQUIVARIANT HOMOTOPY DIAGRAMS
21
F014
F024
F24 F012
F
4
23
F
4
13
23
4F
12
F
01
2
F0
F
F0
F34 F123 F01
F34 F023
F34 F013
F124 F01
1
34
F0
F34 F23 F012
F24 F012
F 23
F34 F23 F012
1
F0
F 02
4
F
2
F1
4
3
2
where Fi1 i2 ···is denotes the map F (idc , τ (i → i2 · · · → is )). As shown in the picture,
(δo2F )(idc , τ ) is also represented by the boundary of the below square. Since the
square can be filled by the map
F234 F012 : I 2 → mF (c, τ ),
it vanishes.
Proof. First notice that for any c an object of G and τ : [k + 3] → γ(c) an object of
dk+3γ(c), we have
1
k+2 X
X
(−1)i+ǫ F (idc , ·) ◦ W τ ◦ (d(i,ǫ) |∂I k+1 ) = 0
i=1 ǫ=0
in the abelian group [∂I k+1 , mF (c, τ )] where we consider the face map d(i,ǫ) from I k+1
to I k+2 ∼
= morW [k+3](0, k + 3) and let d(i,ǫ)|∂I k+1 denote the restriction of the face map
(i,ǫ)
d
to ∂I k+1 . Assume that A(i, ǫ) denote F (idc , ·) ◦ W τ ◦ (d(i,ǫ)|∂I k+1 ). Then
δok+1
F (c, τ )
=
k+2
X
i=1
i
(−1) A(i, 0) − A(0, 1) + A(k + 2, 1) =
k+1
X
(−1)i A(i, 1) = 0.
i=2
To show that [ok+1
is a cohomological obstruction for lifting the cSet-functor F
F ] R
to a cSet-functor F ′ : G skck+1 (W γ, skck+r W α) → M, we first note that there is an
action of πk (mF (c, σ)) on homotopy classes of maps from I k to mF (c,
R σ) relative to
k
∂I since mF (c, σ) is fibrant and simple for every object (c, σ) of G dγ. Let f · g
22
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
denote the action of g in πk (mF (c, σ)) on f a homotopy class of a map from I k to
mF (c, σ) relative to ∂I k .
R
Proposition 8.6. Let F : G skck (W γ, skck+r W α) → M be a cSet-functor. Then the
R
restriction of F to a (k−1)-th skeleton lifts to a cSet-functor on G skck+1 (W γ, skck+r W α)
k+2
if and only if [ok+1
(γ, α; πk F ).
F ] = 0 in H
k+1
k+2
Proof. We have [ok+1
(γ, α; πk F ). Hence there exists g in C G (γ, α; πk F )
F ] = 0 in H
k+1
such that δg = oF . The restriction of F to (k − 1)-skeleton can be extended
R
to a cSet-functor on G skck (W γ, skck+r W α) denoted by Fe so that Fe|Iσn is equal to
(F |Iσn )·g(c, σ)−1 as a map from Iσn to mF (c, σ) for (c, σ) in Ndk+2 γ(c). Hence ok+1
=0
Fe
R
c
c
and Fe lifts to a cSet-functor on sk (W γ, sk W α).
G
k+1
k+r
9. Applications
9.1. Equivariant homotopy. Let M be a category and G be a groupoid. A Gobject of M is a functor from G to M. Let G be a group and GG be the associated
groupoid. Then a G-object of Top can be also called a G-space, a G-object of sSet
can be considered as a simplicial set with a G action, and a G-object of CW can be
called a G-CW-complex. Let F be a family of subgroupoids of G. We say G-object
X : G → M has isotropy in F if for all H subgroupoid of G not in F the limit lim X
H
is an initial object.
Given A, B two G-objects of M, a G-map from A to B is a natural transformation
from A to B. Further assume that M is a model category. Let f, g be two G-maps
from A to B we say f is homotopic to g if there exists a cylinder object I with two
trivial cofibrations i0 : ∗ → I and i1 : ∗ → I and a natural transformation H from
A × I to B such that f = H ◦ i0 and g = H ◦ i1 . So one can also define homotopy
equivalence between G-objects. We call such G-maps G-homotopy equivalences.
9.2. Equivariant classifying spaces. Let G be a groupoid and X : G → sSet be
a functor. We define Sdγ : G → Cat to be the functor which send the object c to
Sdγ(c), the opposite of the full subcategory of ∆ ↓ Z with object set equal to nondegenerate objects. Assume N denotes the nerve functor from Cat to sSet. Then
there is a natural homeomorphism between the realizations |X| and |NSd(X)|.
Let F be a family of subgroups of a group G which is closed under conjugation.
Define EF as the functor from GG to the category Cat which sends the object ∗G in GG
to the category EF (∗G ) whose objects are pairs (G/H, xH) where H ∈ F and x ∈ G
and morphisms from (G/H, xH) to (G/K, yK) are the G-maps from G/H to G/K
which sends xH to yK. Moreover g ∈ G sends the object (G/H, xH) to the object
(G/H, gxH) and sends the morphism f : G/H → G/K to the morphism gf : G/H →
G/K which satisfies (gf )(gxH) = gyK. The functor EF has the following universal
property
EQUIVARIANT HOMOTOPY DIAGRAMS
23
Proposition 9.1. For any functor X : GG → sSet with isotropy in F there exists a
natural transformation fX from SdX to EF .
Proof. Let X be a functor from GG to sSet. We want to define a natural transformation from Sd(X) to EF . This means that we want to define a functor from Sd(X)(∗G )
to EF (∗G ). Choose I a set of representatives from each orbit of the action of G on
the set of objects in Sd(X)(∗G ). Send x in I to (G/Gx , Gx ) in EF (∗G ) and extend
the functor equivariantly on the rest of the objects in Sd(X)(∗G ). Then there is a
unique way to send the morphisms.
9.3. Pullbacks. Let G be a group and F be a family of subgroups of G, and let
X : GG → sSet be a G-object of sSet with isotropy in F . By the above section we
know that there exists a natural transformation f from SdX to EF . This induces a
natural transformation from T skcn W SdX to T skcn W EF . Therefore given
Z
F:
skcn W EF → M
GG
we can define the pullback of F over X as follows:
Pf (X, F ) = B(∗, T skcn W SdX, f ∗ (T F )).
Since we could also pull back homotopies we obtain the following result.
Theorem 9.2. If F1 and F2 are homotopic as n-commutative diagrams over TopcSet
then |Pf (X, F1 )| and |Pf (X, F2 )| are homotopic as G-objects of Top.
Proof. By definition F1 and FR2 are homotopic as n-commutative diagrams over TopcSet
means there exists H1 , H2 : G skcn+2 W (γ × [2])R → M which satisfies certain conditions which guarantees that H1 restricted to G skcn+1 W (γ × {0 → 1}) induces a
R
map from |Pf (X, F1 )| to |Pf (X, F2 )| and H1 restricted to G skcn+1 W (γ × {1 → 2})
induces a map from |Pf (X, F2 )| to |Pf (X, F1 )| and the compositions of these maps
are G-homotopic to identity by Theorem 7.5 and Theorem 7.6.
9.4. Finiteness. Assume
F:
Z
GG
skcn W EF → TopcSet
with isotropy in F . Let U : CW → Top denote the functor that sends a CW-complex
to its underlying topological space. Here CW can be considered as a homotopical
category with weak equivalences as cellular maps which induce isomorphisms on
homotopy groups and U can be considered as a homotopical functor. We will say a
simplicial set is finite if it has finitely many non-degenerate simplices.
R
Theorem 9.3. If X is finite simplicial set and F restricted to GG skc−1 W EF factors
through CW then |Pf (X, F )| has isotropy in F and it is G-homotopy equivalent to
a finite G-CW-complex.
24
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Proof. Let g be a map from A to |X|. Define Bg as the pull back of the diagram
A → |X| ← |Pf (X, F )| where the map on the left is g and the map on the right
is the composition |Pf (X, F )| → |Pf (X, ∗)| ֒→ |X|. For i ≥ 0,R define Ai = | ski X|.
We have an inclusion gi : Ai → |X|. Since F restricted to GG skc−1 W EF factors
through CW we can say that Bg0 = |Pf (sk0 X, F )| is G-homotopy equivalent to a
finite G-CW-complex. Assume that we proved Bgi is G-homotopy equivalent to a
finite G-CW-complex. Define Ai+1 as | ski+1 X| with a smaller open disc removed
from every open (i + 1)-cell. There is an inclusion g i+1 : Ai+1 → |X|. Define Ai+1
as disjoint union of the closures of the smaller open discs removed above. There is
an inclusion g i+1 : Ai → |X|. Now Bgi+1 is equal to the union of Bgi+1 and Bg i+1 .
The space Bgi+1 is homotopy equivalent to Bgi hence, it is G-homotopy equivalent to
a finite G-CW-complex. The space is a disjoint union of products of finite G-CWcomplexes. Moreover the inclusion of the intersection of Bgi+1 and Bgi+1 in Bgi+1
is a cofibration. Hence Bgi+1 is G-homotopy equivalent to a finite G-CW-complex.
Hence by induction |Pf (X, F )| is G-homotopy equivalent to a finite G-CW-complex.
Moreover |Pf (X, F )| has isotropy in F as it is obtained as a pullback of a space which
has isotropy in F .
9.5. Group actions on products of spheres. Let G be a finite group and let X
be a G-object of sSet with all isotropies in F where F is a family of subgroups of
G which is closed under conjugation and taking subgroups. Suppose also that X is
a finite G-CW -complex with dimX = k (that is the dimension of X(∗G )). Then the
standard geometric realization |X| of X is a finite G-CW -complex.
A family of representations {αH : H → U(m)}H∈F is called compatible if for every
map cg : H → K induced by conjugation cg (h) = ghg −1, there is a γ ∈ U(m) such
that the following diagram commutes
αH
U(m)
H
cg
cγ
U(m).
K α
K
Given a family of representations
{αH }H∈F , we will construct a k-commutative homoR
c
topy EF -diagram F1 : GG skk W EF → TopcSet such that F1 |skck W EF (∗G ) is homotopic
to the k-commutative homotopy diagram F2 : skck W EF → TopcSet which sends every
object to Sq and every morphism to the identity map on Sq for some q >> 0. Since
|Pf (X, F2 )| = |X| × Sq , the following result will follow by Theorem 9.2 and Theorem
9.3.
Theorem 9.4. Let {αH }H∈F be a compatible family of representations where F is
a family of subgroups of G which is closed under conjugation and taking subgroups.
EQUIVARIANT HOMOTOPY DIAGRAMS
25
Then for every finite G-object X of sSet with all isotropies in F , there is a finite GCW -complex Y ∼
= |X| × Sq whose isotropy subgroups are in the family V = {Hx |x ∈
S(αH ), H ∈ F } for some q >> 0.
Let X, F and {αH }H∈F be defined as in the above theorem with dim(X) = k.
For every H ∈ F and u ∈ G/H, fix x(H,u) ∈ u. We write x = xH,u when it is
clear from the context. Let XH = (G × S2m−1 )/ ∼ where (gh, s) = (g, αH (h)s) and
XH,u = {[xH,u , s] ∈ XH }. The group G acts on XH by g ′ [g, s] = [g ′g, s] and the
restriction of this action to x H induces a x H-action on XG/H,u . Define
Z
D:
EF → Ho(TopcSet )
GG
on objects by D(∗G , (G/H, u)) = XH,u .
For H, K ∈ F with a H ∈ K for some a ∈ G, let NG (H, K) = {g ∈ G | g H ≤ K}.
Write NG (H, K) as a disjoint union of double cosets:
[
NG (H, K) =
Kgi H.
i∈IG (H,K)
For i in IG (H, K), fix an element γi in U(n) such that cγi ◦ αH = αK ◦ cgi . For i in
IG (H, K), write Kgi H as a disjoint union of cosets
[
Kgi H =
tij gi H
j∈JG (H,K,i)
where tij gi = x(G/H,tij gi H) . For g in NG (H, K), define
γg = γg (H, K) = αK (tij )γi αH (h)
where g = tij gi h, i in IG (H, K), j in JG (H, K, i), and h in H. We denote the unique
map from (G/H, u) to (G/K, v) in EF (∗G ) by fu,v . Now for the map (g, fgu,v ) from
(∗G , (G/H, u)) to (∗G , (G/H, v)), let D(g, fgu,v ) = [Dfu,v ] where Dfu,v : XH,u → XK,v
is defined by
Dfu,v ([x, s]) = [y, γy−1x (H, K)s]
where x = x(H,u) and y = y(K,v) .
R
Lemma 9.5. D : GG EF → Ho(TopcSet ) is a functor.
Proof. Note that
Didu,u ([x, s]) = [x, αH (gi−1)γi s]
when the identity element e of the group is in the coset Hgi H. Since U(m) is pathconnected, there is a path p : I → U(m) with p(0) = id and p(1) = αH (gi−1 )γi . Then
the map H : XH,u × I → XH,u defined by H([x, s], t) = [x, p(t)s] gives a homotopy
between idXH,u and Didu,u .
26
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
Let fu,w : (G/H, u) → (G/L, w) and fw,v : (G/L, w) → (G/K, v) be composable
maps for some H, L and K in H with a H ≤ L and b L ≤ K. Let
[
[
NG (H, K) =
Kgi H and Kgi H =
tim gi H,
[
[
NG (H, L) =
Laj H and Laj H =
rjn aj H,
[
[
NG (L, K) =
Kbk L and Kbk L =
skl bk L.
Then we have
Dfw,v ◦ Dfu,w ([x, s]) = [y, αK (skl )γk αL (lrjn )γj αH (h1 )s]
Dfw,v fu,w ([x, s]) = [y, αK (tij )γi α(h)s]
where z −1 x = rjn aj h1 , y −1 z = skl bk l and y −1x = tij gi h. Here, y = x(G/K,v) and
z = x(G/L,w) . Since y −1x = skl bk lrjn aj h1 , we have
−1
αK (tij )γi αH (h) = αK (skl )γK αK (lrjn )γK
γi αH (gi−1 bk aj )αH (h1 ).
Therefore Hu,w,v : XH,u × I → XK,v defined by
Hu,w,v ([x, s], t) = [y, αK (skl )γK αK (lrjn )θgi ,aj ,bk αH (h1 )s]
gives a homotopy between Dfw,v fu,w and Dfw,v ◦ Dfu,w where θgi ,aj ,bk is a path from
−1
γj to γK
γi αH (gi−1bk aj ).
Since skc1 W EF is defined by including the cubical compositions
of 1-cubes in W EF ,
R
1
we can lift D to a 1-commutative homotopy EF -diagram D : GG skc1 W EF → TopcSet
by letting D 1 (g, Iσ ) = Hu,w,v : XH,u × I → XH,v for every σ : [2] → EF with σ(0 →
1) = fgu,w and σ(1 → 2) = fw,v . We use the obstruction theory from Section 8 to lift
it to a k-commutative homotopy EF -diagram for given k. The following observation
makes it easier to deal with the obstruction classes.
R
Lemma 9.6. For every lift D n : GG skcn W EF → TopcSet of a 1-commutative homotopy EF -diagram D 1 and for every σ : [k] → EF with σ(0) = (G/H, u), we have the
following homeomorphism
mD n (∗G , σ) ≃ AutIx H (XH,u )
(2)
where x = x(G/H,u) and AutIx H (XH,u ) denotes the identity component of Autx H (XH,u ).
T
Proof. Let σ(i) = (G/Hi , ui ), xi = x(G/Hi ,ui) , un = v, and xn = y. Since x H ⊆ xi Hi ,
AEF (∗G , σ) = {(g, idEF (g)σ )| σ = EF (g) ◦ σ} = {(g, idEF (g)σ )| g ∈ x H}.
Therefore we have
mD n (∗G , σ) =
lim
AEF (∗G ,σ)
(MorTopcSet (XH,u , XK,v ), Dfu,v ) = (Mapx H (XH,u , XK,v ), Dfu,v ).
EQUIVARIANT HOMOTOPY DIAGRAMS
27
For every φ : XH,u → XK,v , define φ′ : S 2m−1 → S 2m−1 by φ([x, s]) = [y, φ′(s)]. Then
φ is in Mapx H (XH,u , XK,v ) if and only if for every h ∈ H, φ′ satisfies the following
equality
φ′ (s) = γαH (h−1 )γ −1 φ′ (αH (h)s)
−1
where γ = αK (k)γi αH (h) and k = y x h.
Now we can define a homeomorphism Ψ : Mapx H (XH,u , XK,v ) → AutIx H (XH,u ) by
Ψ(φ)([x, s]) = [x, γ −1 φ′ (s)]. The map Ψ(φ) is an x H-map since
Ψ[φ]([xh x, s] = Ψ[φ]([x, αH (h)s] = [x, γ −1 φ′ αH (h)s] = [x, αH (h)γ −1 φ′ (s)]
= x h[x, γ −1 φ′ (s)] =x hΨ(φ)([x, s])
for every h ∈ H. The inverse of Ψ is given by Ψ−1 (f )[x, s] = [y, γf ′(s)] where f ′ is
defined by f ([x, s]) = [x, f ′ (s)]. Clearly, we have Ψ(D(fu,v )) = id|XH,u .
Since the spaces AutIx H (XH,u ) are both fibrant and simple, we can use the obstruction theory developed in Section 8. Since the space XH,u is x H-homeomorphic to the
representation sphere S(x αH ) where x αH : x H → U(m) is the representation defined
by x αH (x h) = αH (h), we can use the following theorem to kill the obstructions.
Theorem 9.7 (Libman [17]). Let α be a complex representation of a finite group H.
Then for any r ≥ 1, there exist integers Mr and tr such that |πr (AutIH (∗t S(α)))| ≤ Mr
for all t ≥ tr .
In order to use Rthe above result, we need to introduce the following terminology.
The join E ∗ F : G skcn W γ → TopcSet of an n-commutative homotopy γ-diagrams
E and F over TopcSet is an n-commutative homotopy γ-diagram defined on objects
by (E ∗ F )(c, x) = E(c, x) ∗ F (c, x). Here the join X ∗ Y of topological spaces
X and Y is defined as the quotients space X × Y × I under the identifications
(x, y1 , 0) ∼ (x, y2 , 0) and (x1 , y, 1) ∼ (x2 , y, 1). We write the points of the join
X ∗ Y as (xr1 , yr2) with r1 , r2 ∈ I and r1 + r2 = 1. On morphisms, we define
(E ∗ F )(Iσ ) : (E ∗ F )(c, σ(0)) × I r → (E ∗ F )(c, σ(r)) for every σ : [r] → γ(c) with
r ≤ n by letting (E ∗ F )(Iσ )((x1 r1 , x2 r2 ), t) = (E(Iσ )(x1 , t)r1 , F (Iσ )(x2 , t)r2 ).
Proposition 9.8.R For every integer n ≥ 1, there is an n-commutative homotopy
EF -diagram F n : GG skcn W EF → TopcSet over ∗t D : EF → Ho(TopcSet ) for some
t >> 0.
Proof. Let F ′ = {(H, x)| x = x(H,u) for some H ∈ F , u ∈ G/H}. By Theorem
(H,x)
(H,x)
9.7, for every (H, x) ∈ F ′ and r ≥ 1, there are integers Mr
and tr
such that
(H,x)
(H,x)
I
x
|πr (Autx H (∗t S( αH )))| ≤ Mr
for all t > tr . Let t1 = 1 and for n > 1, let
tn =
n−1
Y
Y
r=1 (H,x)∈F ′
t(H,x)
r
Y
(H,x)
i≤Mr
i .
28
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
We prove by induction on n that there is an n-commutative homotopy EF -diagram
F1n over ∗tn D : EF → Ho(TopcSet ). For n = 1, let F 1 = D 1 . By induction hypothesis,
there is an (n − 1)-commutative homotopy diagram F n−1 over ∗tn−1 D. Now, consider
the (n − 1)-commutative homotopy EF -diagram D n−1 = ∗t′n F n−1 over ∗tn D where
tn
t′n = tn−1
. Under the equivalence 2, the map D n−1 (∗, −) factors through
∂ morW [n+1] (0, n + 1)
F n−1 (∗G , −)
D n−1 (∗G , −)
AutIx H (∗tn S(x αH ))
φ
AutIx H (∗tn−1 S(x αH ))
where φ is defined by φ(a)(x1 r1 , · · · , xt′n rt′n ) = (a(x1 )r1 , · · · , a(xt′n )rt′n ). Therefore,
by Lemma 2.6 in [26], the obstruction class onDn−1 sends (∗G , σ) to t′n onF n−1 (∗G , σ) in
πn−1 (AutIx H (∗tn S(x αH ))) where σ(0) = (G/H, u) and x = x(H,u) . Since (H, x) ∈ F ′ ,
|πn−1 (AutIH (∗tn S(x αH )))| divides t′n and hence onDn−1 (∗G , σ) = 0. Therefore, we can
extend D n−1 to an n-commutative homotopy EF -diagram F n over ∗tn D.
For every k-commutative homotopy diagram F , the restriction
F |skck W EF is the
R
c
c
composition F ◦ j where the functor j : skk W EF (∗G ) → GG skk W EF is given by
j(G/H, u) = (∗G , (G/H, u)) and j(Iσ ) = (e, Iσ ) where e is the identity element of the
group G. Now let F2 (p, r) : skcr W EF (∗G ) → TopcSet be an r-commutative homotopy
diagram which sends every object to Sq and every morphism to the identity map on
Sq where q = 2mp − 1. Note that for every p ≤ s, we have F2 (s, r) = ∗ ps F2 (p, r).
R
Proposition 9.9. There is a k-commutative homotopy EF -diagram F1 : GG skck W EF →
TopcSet over ∗t D such that F1 |skck W EF (∗G ) and F2 (t, r) are homotopic for some t >> 0.
R
Proof. Let F : GG skck W EF → TopcSet be a k-commutative homotopy EF -diagram
over ∗t′ D with t′ > 1 given by the above proposition. Then F |skc1 W EF (∗G ) = ∗t′ D 1 .
For every (G/H, u) ∈ EF (∗), let iH,u : Sq → XH,u and jH,u : X(H,u) → S q be defined
by iH,u (s) = [x(H,u),s ] and jH,u ([x(H,u),s ]) = s. For every (G/H, u) and (G/K, v) in
EF (∗), choose a path pH,K : I → U(m) from the identity to γx−1 xH,u (H, K). Now
K,v
define E 1 : skc1 W (EF × [2]) → TopcSet on objects by E 1 ((G/H, u), 1) = XH,u and
29
EQUIVARIANT HOMOTOPY DIAGRAMS
E1 ((G/H, u), 0) = E1 ((G/H, u), 2) = S2m−1 . We define E 1 on morphisms as follows

idS2m−1 ◦ p1 , σ(0 → 1) = (fu,w , 0) and σ(1 → 2) = (fw,v , 0);




iK,v ◦ p1 ,
σ(0 → 1) = (fu,w , 0) and σ(1 → 2) = (fw,v , 0 → 1);




idS2m−1 ◦ p1 , σ(0 → 1) = (fu,w , 0) and σ(1 → 2) = (fw,v , 0 → 2);




P
σ(0 → 1) = (fu,w , 0 → 1) and σ(1 → 2) = (fw,v , 1);

L,K ,


idS2m−1 ◦ p1 , σ(0 → 1) = (fu,w , 0 → 1) and σ(1 → 2) = (fw,v , 1 → 2);
E 1 (Iσ ) =
idS2m−1 ◦ p1 , σ(0 → 2) = (fu,w , 0) and σ(1 → 2) = (fw,v , 2);




Hu,w,v ,
σ(0 → 1) = (fu,w , 1) and σ(1 → 2) = (fw,v , 1);




e
PH,L ,
σ(0 → 1) = (fu,w , 1) and σ(1 → 2) = (fw,v , 1 → 2);




j
◦
p
,
σ(0
→ 1) = (fu,w , 1 ≤ 2) and σ(1 → 2) = (fw,v , 2);

H,u
1

 id 2m−1 ◦ p , σ(0 → 1) = (f , 2) and σ(1 → 2) = (f , 2).
1
u,w
w,v
S
where P L,K : S2m−1 × I → XK,v is defined by P L,K ([s, t]) = [xK,v , pL,K (t)s] and
PeH,L : XH,u × I → S2m−1 is given by PeH,L ([xH,u , s], t) = pH,L (t)s. Then ∗t′ E 1 gives a
homotopy from F2 (t′ , 1) to F |skc1 W EF (∗) .
′
Define tn inductively by tn = |πn (S2mt qn −1 )| where q1 = 1 and qn = t1 · · · tn−1 for
n > 1. Then, let F2 = F2 (tt′ , k) and F1 = ∗t F where t = t1 t2 · · · tk−1 . To construct
the desired homotopy, we first need to construct k-commutative homotopy diagrams
T12 , T21 : skck W (EF × [1]) → TopcSet with Tij |sk,{0} = Fj and Tij |sk,{1} = Fi that lift
∗t (Es11 ,{0,1} ) and ∗t (Es11 ,{1,2} ), respectively. For this, consider the cSet-functor
1
T12
: skc1 (W (EF (∗) × [1]), skck W (EF (∗) × {0, 1})) → TopcSet
1
1
|skck W (EF (∗)×{0}) = F2 (t′ , 2),
defined by the relations T12
|skc1 W (EF (∗)×[1]) = ∗t′ Es11 ,{0,1} , T12
′
1
and T12
|skck W (EF (∗)×{1}) = F . Since o2(∗t T 1 ) = t1 (o2T 1 ) and it factors through π1 (S2mt −1 ),
12
1 12
1
extends to a cSet-functor, say
we have o2(∗t T 1 ) = 0. Therefore, the functor ∗t1 T12
1 12
c
c
2
T12 : (sk2 W (EF (∗) × [1]), skk W (EF (∗) × {0, 1}) → TopcSet . By applying this process
k
repeatedly, we obtain a k-commutative homotopy diagram T12 = T12
with the desired
properties. One can construct T21 similarly.
Now let
1
E : skc1 (W (EF (∗) × [2]), skck W (EF (∗) × ({0, 1}, {1, 2}, {0, 2}))) → TopcSet
1
1
be a cSet-functor defined by relations E |skc1 W (EF (∗)×[2]) = ∗t′ E 1 , E |skc W (EF (∗)×{0,1}) =
1
1
k
T12 , E |skc W (EF (∗)×{1,2}) = T21 , and E |skc W (EF (∗)×{0,2}) = F2 (tt′ , 2). Define sn induck
k
′
tively by sn = |πn (S2mtt rn −1 )| where r1 = 1 and rn = s1 · · · sn−1 for n > 1. As
1
above one can extend the functor ∗s1 E to a cSet-functor E 2 from the cSet-category
skc2 (W (EF (∗)×[2]), skck W (EF (∗)×({0, 1}, {1, 2}, {0, 2}))) to TopcSet and by repeating
the same argument, one can construct the homotopy E. Note that E is a homotopy
from F2 (u, k) to ∗ tu′ F where u = tt′ s1 s2 · · · sk−1
30
ASLI GÜÇLÜKAN İLHAN AND ÖZGÜN ÜNLÜ
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Asli Güçlükan İlhan, Department of Mathematics, Bilkent University, Bilkent,
Ankara, Turkey.
E-mail address: [email protected]
Özgün Ünlü, Department of Mathematics, Bilkent University, Bilkent, Ankara,
Turkey.
E-mail address: [email protected]