THE TOTAL SURGERY OBSTRUCTION II
PHILIPP KÜHL
Abstract. The total surgery obstruction s(X) of a finite n-dimensional Poincaré complex X is an element of a certain abelian group Sn (X) with the
property that s(X) = 0 if and only if X is homotopy equivalent to a closed ndimensional topological manifold for n ≥ 5. These are the notes of the second
of three talks which want to give a brief overview of the techniques used from
algebraic surgery theory to define and proove this statement. Following the
first part of [Ran92] particularly chapters 3-5 and 11-14 we concentrate in
this part on the question of how algebraic bordism categories capture local
Poincaré duality, which leads us to a braid of exact sequences using L-theory
spectra.
1. Algebraic bordism categories
We denote the algebraic mapping cone of a chain map f by C(f ).
Definition 1.1. An algebraic bordism category Λ = (A, B, C, (T, e)) consists of the
following data:
A is an additive category.
B is a subcategory of the additive category of bounded chain complexes over
A.
C is a full subcategory of B closed under taking cones, i.e. C(f ) ∈ C for all
morphisms f ∈ C. This category will be used to define different types of
(Poincaré) duality.
T is a contravariant functor T : A → B which builds together with e the chain
duality of Λ.
e is a natural transformation e : T 2 → (id : A → B) such that
– eM : T 2 (M ) → M is a chain equivalence.
– eT (M ) ◦ T (eM ) = id.
(by id : A → B we mean an object A ∈ A goes to the 0-dimensional chain
complex C ∈ B with C0 = A)
and finally for any object B ∈ B
• C(id : B → B) ∈ C and
'
• C(e(B) : T 2 (B) −
→ B) ∈ C
has to be satisfied.
Remark 1.2. It will be important that the dual T (M ) of a chain module is a chain
complex. We can extend T : A → B in a natural way to a duality TB : B → B on
chain complexes by using the total chain complex as follows. Let
C : . . . → Mr → Mr−1 → Mr−2 → . . . ∈ C.
1
2
PHILIPP KÜHL
T is contravariant, hence we get the following picture
T (Mr )s o
T (Mr−1 )s o
T (Mr−2 )s
T (Mr )s−1 o
T (Mr−1 )s−1 o
T (Mr−2 )s−1
T (Mr )s−2 o
T (Mr−1 )s−2 o
T (Mr−2 )s−2
TB (C)p =
P
(T(Mr )s )
r−s=p
Example 1.3. Let R be a ring with involution. We denote by
Λ(R) = (A(R), B(R), C(R), (T, e))
the algebraic bordism category with
A(R) the additive category of finitely generated free R-modules,
B(R) the bounded chain complexes in A(R),
C(R) the contractible chain complexes of B(R) and
T is defined by T (M ) := HomR (M, R), so we get
e as the inverse of the isomorphism M → HomR (HomR (M, R), R),
x 7→ (f 7→ f (x)).
Before we introduce the main example of an algebraic bordism category which
we will use for the total surgery obstruction we want to transfer the concept of
L-groups of the previous talk to this setting.
1.1. L-groups of algebraic bordism categories. For objects M and N in A
define the chain complex of abelian groups
M ⊗A N := HomA (T (M ), N )
and the duality isomorphism
TM,N : M ⊗A N → N ⊗A M, (f : T M → N ) 7→ (eM ◦ T (f ) : T N → T 2 M → M ).
The properties of e yield
TM,N ◦ TN,M (f ) = eM ◦ T (eN ◦ T (f )) = eM ◦ T 2 (f )T (eN ) = f
which gives us (TM,N )−1 = TN,M . We can extend this to finite chain complexes C
and D in A:
C ⊗A D := HomA (T (C), D)
and
X
TC,D =
(−1)pq TCp ,Dq : C ⊗A D → D ⊗A C,
X
X
(Cp ⊗A Dq )r →
(Dq ⊗A Cp )r
p+q+r=n
p+q+r=n
2
Using TC,C
= 1 we can consider C ⊗A C as a Z[Z2 ]-module chain complex and we
are able to use the constructions of the previous talk to get symmetric, quadratic
and normal complexes in this setting of additive categories.
W% C := W ⊗Z[Z2 ] (C ⊗A C)
W % C := HomZ[Z2 ] (W, C ⊗A C)
In the first talk Q-groups were introduced as homology groups of W % C and W% C.
But in this setting it is enough to work with cycles.
THE TOTAL SURGERY OBSTRUCTION II
3
Definition 1.4. An n-dim. symmetric algebraic complex (C, ϕ) in Λ = (A, B, C, (T, e))
is a chain complex C ∈ B together with an n-cycle ϕ ∈ (W % C)n such that
∂C = Σ−1 C(ϕ0 : Σn T C → C) ∈ C
Definition 1.5. An (n + 1)-dimensional symmetric pair (f : C → D, (δϕ, ϕ)) in
Λ is a chain map f : C → D with C, D ∈ B, (δϕ, ϕ) ∈ C(f % ) (n + 1)-cycle and
C(δϕ0 , ϕ0 f ∗ ) ∈ C.
Definition 1.6. A cobordism between two n-dimensional symmetric complexes
(C, ϕ) and (C 0 , ϕ0 ) is an (n + 1)-dim. symmetric pair (C ⊕ C 0 → D, (δψ, ϕ ⊕ −ϕ0 )
in Λ.
Analogously to the first talk we can also state all these definitions in quadratic
and normal flavour. We define the symmetric/quadratic/normal L-groups
Ln (Λ) / Ln (Λ) / N Ln (Λ)
as the cobordism groups of n-dimensional symmetric/quadratic/normal algebraic
complexes in Λ.
We use QAC as an abbreviation for ‘quadratic algebraic complex respectively a
SAC for symmetric one.
Main example 1.7. Let Λ = (A, B, C, (T, e)) be an algebraic bordism category and
K a locally finite simplicial complex. By ∆∗ (K) we denote the simplicial chain
complex and by ∆∗ (K) the simplicial cochain complex of K. Local Poincaré duality
means for each simplex there is a duality. This local duality comes in two flavours.
(1) We can think of each simplex as a manifold with boundary and so there is
a duality between ∆∗ (σ, ∂σ) and ∆∗ (σ)
D(σ)
Instead of the simplex itself we can consider
its dual cell and in the case that K is a ho(2)
mology manifold we get a duality between
∆∗ (D(σ, K), ∂D(σ, K)) and ∆∗ (D(σ, K))
K
σ
We want to encode these two phenomena in two so called K-based algebraic bordism
categories which differ only in their morphism sets.
Definition 1.8. The addivitive categories of K-based objects A∗ (K) and A∗ (K)
are defined by
P
Obj(A∗ (K)) = Obj(A∗ (K)) = { σ∈K Mσ | Mσ ∈ A},
∗
(1) Mor(A
P (K)) =P
P
{
fτ,σ :
Mσ →
Nτ | (fτ,σ : Mσ → Nτ ) ∈ Mor(A)}
σ≤τ
σ∈K
τ ∈K
(2) Mor(A
P∗ (K)) =P
P
{
fτ,σ :
Mσ →
Nτ | (fτ,σ : Mσ → Nτ ) ∈ Mor(A)}
σ≥τ
∗
σ∈K
τ ∈K
So in A the morphisms only go from bigger to smaller simplices and in A∗ the
other way round. We call a map fτ,σ a skew map if σ 6= τ and a straight map
if σ = τ . Let’s have a look at an example. We can consider the simplicial chain
4
PHILIPP KÜHL
complex of the standard 1-simplex ∆∗ (∆1 ) as a chain complex in A(Z)∗ (∆1 ) using
the following construction:
σ0
C2 :
C1 :
C0 :
σ1
•
τ
C(σ0 ) = ∆∗ (σ0 , ∂σ0 )
C(τ ) = ∆∗ (τ, ∂τ )
•
C(σ1 ) = ∆∗ (σ1 , ∂σ1 )
0
j 0 TTTTT
TTTT
jjjj
j
j
j
T
j
T
j
TTTT
jj
TTTT
jjjj
TTT* tjjjjjj
TTTT
0
0
Z
j
j
j
T
TTTT
jjj
∂0 jjjj
∂
T
1
TTTT
j
TTTT
jjjj
TTTT tjjjjjjj
*
0
Z
Z
0
Let C be a chain complex in A∗ (K). For each simplex σ we get a chain complex
C(σ) by restriction to the corresponding column.
We want to encode in the chain duality functor T ∗ of A∗ (K) the local Poincaré
duality of all simplices of K such that we recover the local Poincaré duality in each
column chain complex. But the dimension of these local Poincaré dualities varies
with the dimension of the simplices and we have to deal with the boundaries. We
need three steps for the construction of T ∗ :
(1) We assemble in each column chain complex C(σ) the column chain complexes of the boundary of σ:
C(σ)r := C(σ)r ⊕
M
C(τ )r .
τ ∈∂σ
We integrate the the skew maps of C into the straight maps of C. The skew
maps of C are all defined to be 0 but we introduce horizontal inclusions
C r (τ ) → C r (σ) for τ ∈ ∂σ which in the next step will become the skew
maps in the dual chain complex.
C1 :
C0 :
/Zo
0
Z
0
(∂∂01 )
i0
/ Z⊕Z o
i1
Z
(2) To each column chain complex C(σ) we apply the chain duality T of A. T
gives us a chain complex for each chain module so we have to use the total
chain complex to get the dual chain complex
C r (σ) =
X
p+q=r
(TA(Z) (C(σ)−p ))q .
THE TOTAL SURGERY OBSTRUCTION II
5
In the example it is rather simple because the dual of each module is a
0-dimensional chain complex.
C1 :
0O o
/0
O
ZO ∗
∗
(∂∂01∗ )
C0 :
Z∗ o
(Z ⊕ Z)∗
i∗
0
i∗
1
/ Z∗
(3) In the whole dual complex we shift each column C(σ) by the dimension of
σ:
T ∗ (C)(σ)r = C r−|σ| (σ).
C 2 = (T ∗ C)−2 :
0O cGG
w; 0O
GG
ww
GG
w
w
GG
ww
GG
ww
G
w
w
Z∗ cG
ZO ∗
Z∗
GG
w;
w
GG ∂0∗
w
G ( ∗)
ww∗
w
∗ GG∂1
w
i0
G
ww i1
(Z ⊕ Z)∗
C 1 = (T ∗ C)−1 :
C 0 = (T ∗ C)0 :
So we end up with
T ∗ : A∗ (K) → B∗ (K), T ∗ (
X
Mσ ))r (τ ) = (T (
M
Mτ̃ ))r+|τ | .
τ ≥τ̃
σ∈K
for which Hn−k (C(σ)) ∼
= H k (C(σ)) holds where n is independent of σ. C in the
example is supposed to be an 1-dimensional Poincaré chain complex and in fact a
1-dimensional structure map ϕ0 : Σ1 T ∗ C → C induces isomorphisms on homology
H 1−n (T ∗ C(σ)) ∼
= Hn (C(σ))
∆∗ (σ0 , ∂σ0 )
∆∗ (τ, ∂τ )
∆∗ (σ1 , ∂σ1 )
0
Z OOO
OOO
ooo
∂
∂0 oooo
OOO1
o
OOO
ooo
OOO o
o
woo
'
C0 : Z
0
Z
C1 : 0
_o _ϕ0_ _
∆∗ (σ0 )
∆∗ (τ )
(Z ⊕ Z)∗
: (Σ1 T ∗ C)1
LL ∗
r
r
∗ LL i1
rr
(∂∂01∗ ) LLLL
rr
r
LL
r
yrr
% ∗
∗
Z∗
Z : (Σ1 T ∗ C)0
Z
M
M
q
M
q
M
q
MMM
qq
MMM
qqq
q
MM& q
xqq
0
0 : (Σ1 T ∗ C)−1
i∗
0
_o _ϕ0_ _
∆∗ (σ1 )
In A∗ (K) we want to encode in each column chain complex C(σ) a chain complex
coming from the dual cell of σ. There are two crucial observations for the definition
of T∗ :
(1) The higher the dimension of a simplex σ the lower the dimension of the
corresponding dual cell, so we have to shift the dual complex in the other
direction as for T ∗ to encode the local Poincaré dualities in the right way.
(2) The boundary of a dual cell D(σ, K) consists of the dual cells of bigger
simplices by which we mean the simplices which contain σ. So we have to
assemble in each column chain complex all column chain complexes coming
from bigger simplices. But that’s fine because in A∗ the skew maps go only
to bigger simplices.
6
PHILIPP KÜHL
This leads us to the definition
T∗ : A∗ (K) → B∗ (K), T∗ (
X
Mσ ))r (τ ) = (T (
σ∈K
M
Mτ̃ ))r−|τ | .
τ ≤τ̃
An algebraic bordism category induces a K-based algebraic bordism category by
the following construction
Definition 1.9. Let Λ = (A, B, C, (T, e)) be an algebraic bordism category and K
a locally finite simplicial complex. Then Λ∗ (K) is defined by
A∗
B∗
C∗
∗ ∗
(T , e )
= A∗ (K) the additive category of K-based objects in A.
the bounded chain complexes in A∗
= {C ∈ B∗ | C(σ) ∈ C for all σ ∈ K}.
the K-based chain duality as constructed above.
Definition 1.10. Let K be a simplicial complex, L ⊂ K a subcomplex of K and
Λ an algebraic bordism category. The induced K-based algebraic bordism category
relative L Λ∗ (K, L) is defined as subcategory of Λ∗ (K) with C(σ) = 0 for all σ ∈ L
and C ∈ B∗ chain complex in Λ∗ (K, L).
1.2. Constructions. We have seen in the previous talk how to get out of a “geometric situation” symmetric and quadratic algebraic complexes in Λ(Z). There is
a way to extend these constructions to Λ(Z)∗ (X). We start with a triangulated
Poincaré complex X for the symmetric construction. For each simplex σ ∈ X we
get a map
φσ : (X, ∅) → (X, X\D(νσ )) ' (D(νσ ), ∂D(νσ )) ' S |σ| ∧ (D(σ, ∂D(σ))
using excision. This gives us for each σ ∈ X a class
φσ ([X]) = [D(σ), ∂D(σ)] ∈ Cn−|σ| (D(σ), ∂D(σ))
which fit together to create an n-dimensional symmetric algebraic complex in
Λ(Z)∗ (X).
The input for the quadratic construction was a normal degree one map
f : M → X. We used Spanier-Whitehead duality to get a map Σk X → Σk M and
ended up with a quadratic complex in Λ(Z). Now let X be a triangulated manifold.
Making f transverse to D(σ) we get a map
fσ : f −1 (D(σ), ∂D(σ)) → (D(σ), ∂D(σ))
for each σ ∈ X. We apply to these maps the quadratic construction to get a
quadratic complex in Λ(Z)∗ (X).
2. Surgery sequences
Definition 2.1. A functor of algebraic bordism categories
F : Λ = (A, B, C) → Λ0 = (A0 , B0 , C0 )
is a covariant functor of additive categories, such that
• F (B) ∈ B0 for all B ∈ B,
• F (C) ∈ C0 for all C ∈ C,
• for every A ∈ A there is a chain map
GA : T 0 F (A) → F T (A)
THE TOTAL SURGERY OBSTRUCTION II
7
with C(GA ) ∈ C0 and a commutative diagram
T 0 F T (A)
GT (A)
T 0 GA
T 02 F (A)
/ F T 2 (A) .
F eA
e0F (A)
/ F (A)
Proposition 2.2 ( [Ran92, Prop. 3.8]). For a functor F : Λ → Λ0 of algebraic
bordism categories there are relative L-groups Ln (F ), Ln (F ) and N Ln (F ) which fit
into the long exact sequences
. . . → Ln (Λ) → Ln (Λ0 ) → Ln (F ) → Ln−1 (Λ) → . . . ,
. . . → Ln (Λ) → Ln (Λ0 ) → Ln (F ) → Ln−1 (Λ) → . . . ,
. . . → N Ln (Λ) → N Ln (Λ0 ) → N Ln (F ) → N Ln−1 (Λ) → . . . .
There will be no change of A in the following so it will be omitted from the
notation of algebraic bordism categories and we will write Ln (B, C) for the L-groups
of an algebraic bordism category (A, B, C). We need the statement of proposition 2.2
only in the special case where F is an inclusion of algebraic bordism categories. In
this situation we get a non-relative description for the relative L-groups.
Proposition 2.3 ( [Ran92, Prop. 3.9]). Let (B, C, D) be a triple of categories of
chain complexes over A with D ⊂ C ⊂ B. The relative symmetric L-groups for the
inclusion F : (B, D) → (B, C) are given by
(i) Ln (F ) = Ln−1 (C, D)
and in the quadratic and normal case by
(ii) Ln (F ) = Ln−1 (C, D) = N Ln (F ).
We are particularly interested in case (ii) where we get by Proposition 2.2 the
long exact sequences
(1) . . . → Ln (C, D) → Ln (B, D) → Ln (B, C) → Ln−1 (C, D) → . . .
and
(2) . . . → Ln (C, D) → N Ln (B, D) → N Ln (B, C) → Ln−1 (C, D) → . . . .
We can complete these two sequences to a whole braid of sequences by considering
the inclusions (B, C) → (B, B) and (B, D) → (B, B). They induce the long exact
sequences
(3) . . . → Ln (B, C) → N Ln (B, C) → N Ln (B, B) → Ln−1 (B, C) → . . .
and
(4) . . . → Ln (B, D) → N Ln (B, D) → N Ln (B, B) → Ln−1 (B, D) → . . . .
8
PHILIPP KÜHL
We get the following braid.
(4)
$
%
N Ln (B, D)
N Ln (B, B)
Ln−1 (B, C)
NNN
OOO
o7
nn7
NNN(2)
OOO
ooo
nnn
o
n
NNN
O
o
n
O
OOO
oo
nn
NN'
'
ooo
nnn
n
N L (B, C)
Ln−1 (B, D)
PPP
OOO
o7
p7
o
p
PPP
O
o
p
OOO
o
p
o
p
PPP
o
p
O
o
p
O
o
p
PP'
OO'
ppp (3)
ooo
Ln−1 (C, D)
Ln (B, C)
N Ln−1 (B, D)
:
9
(1)
Comments on the proofs of 2.2 and 2.3. Let F : Λ → Λ0 be a functor of algebraic
bordism categories. An element in Ln (F ) is an (n − 1)-dimensional QAC (C, ψ) in
Ln (Λ) together with a quadratic pair (F (C) → D, (δψ, F (ψ))). The maps in the
sequence are given by
/ Ln (Λ)
(C, ψ) F
/ Ln (Λ0 )
/ Ln (F )
/ Ln−1 (Λ)
/
/ (F (C), F (ψ))
(C 0 , ψ 0 )
/((0, 0), F (0) → C 0 , (ψ 0 , F (0)))
((C, ψ), F (C) → D, (δψ, F (ψ))) / (C, ψ)
Essentially this exact sequence is an analogue of the long exact sequence of cobordism groups Ωn (X). For the non-trivial parts of the proof of exactness a definition
of cobordism of pairs is necessary which is given in [Ran81].
The isomorphism Ln (F ) ∼
= Ln−1 (C, D) is given by
((C, ψ), C → D, (δψ, ψ)) 7→ (C 0 , ψ 0 )
where (C 0 , ψ 0 ) is the effect of algebraic surgery on (C, ψ) by the pair (C → D, (δψ, ψ)).
The inverse is given by
(C, ψ) 7→ ((C, ψ), C → 0, (0, ψ)) .
∼
Similar for N Ln (F ) = Ln−1 (C, D). We perform algebraic surgery on an (n −
1)-dimensional algebraic complex (C, ψ) using the normal pair (C → D, (δψ, ψ))
where ((C, ψ), C → D, (δψ, ψ)) ∈ N Ln (F ) and get an (n−1)-dimensional quadratic
complex (see [Ran92] 2.8(ii)).
Now we define for a locally finite simplicial complex K a special algebraic bordism
category that we put into the braid to get the exact sequences we need for the total
surgery obstruction.
(1) A = A(R)∗ (K) is the additive category of finitely generated free R-modules
based over K.
(2) B = B(A(R)∗ (K)) as usual the bounded chain complexes in A
(3) C = {C ∈ B | A(C) ' ∗} are the globally contractible chain complexes
where A : A∗ (K) → A[π1 (K)] is the assembly functor defined by
X
M 7→
M (p(σ̃))
σ̃∈K̃
THE TOTAL SURGERY OBSTRUCTION II
9
(4) D = {C ∈ B | C(σ) simeq ∗ ∀σ ∈ K} is the category of locally contractible
chain complexes.
We use the equations
(2.1)
Ln−1 (C, D)
n
=
Sn (R, K)
(2.2)
N L (B, C)
=
V Ln (R, K)
(2.3)
Ln (B, C)
=
Ln (R[π1 (K)])
(2.4)
Ln (B, D)
=
Hn (K, L• (R))
(2.5)
N L (B, D)
=
Hn (K, L• (R))
(2.6)
N Ln (B, B)
=
Hn (K, NL• (R))
n
to get the following version of the braid.
$
$
Hn (K, L• (R))
Ln−1 (R[π1 (K)]
Hn (K, NL• (R))
MMM
8
8
MMM
MMM
qqq
MMM
qqq
q
q
q
q
MMM
MMM
qq
qq
M&
M&
qqq
qqq
V Ln (R, K)
Hn−1 (K, L• (R))
8
MMM
MMM
q8
q
M
MMM
q
qqq
M
q
q
M
q
q
MMM
MMM
q
q
q
q
q
q
M&
M&
qq
qq
Sn (R, K)
Hn−1 (K, L• (R))
Ln (R[π1 (K)])
:
:
(2.1) is just a definition. The structure group Sn (R, K) is where the total surgery
obstruction lives. Strictly speaking not exactly in this group but a connective
version. See chapter 15 and 17 of [Ran92] for details or the notes of the third talk
for a brief overview.
(2.2) is also just a definition. The groups V Ln (R, K) are called visible L-groups.
They coincide with the visible L-groups of [Wei92] in some special cases. See
Remark 9.8 in [Ran92].
(2.3) is the result of the algebraic π-π-theorem ([Ran92, chapter 10]).
(2.4)-(2.6) The algebraic bordism categories (A, B, D) and (A, B, B) are induced
by (A(R), B(R), C(R)) and (A(R), B(R), B(R)) where C(R) is the category of contractible chain complexes of finitely generated free R-modules. That is the formal
reason why we can describe these groups in terms of homology groups with coefficients in L-theory spectra. For (2.5) we also use that every symmetric algebraic
complex in (B, D) has a normal structure ( [Ran92, Prop. 3.5]).
3. L-Spectra
Let’s have a closer look at L-theory spectra to get an idea why we get a description for some terms in the braid as homology groups. We restrict in the following
to the quadratic case. Everything works in exactly the same way for the symmetric
and normal cases just by replacing the word “quadratic” in the definitions. The
standard n-simplex is denoted by ∆n .
We briefly recall the definitions and most important facts we will need about
(Kan) ∆-sets. We consider ∆-sets only without degeneracy maps so a ∆-set K
comes with a sequence of sets K n for each dimension n ≥ 0 and with face maps
n
∂in : K n → K n−1 such that ∂in ∂jn−1 = ∂j−1
∂in−1 for j > i. A ∆-map f : K → L
(k)
sends k-simplices to k-simplices: f (K ) ⊂ L(k) . For ∆-sets K, L we define the
10
PHILIPP KÜHL
function ∆-set LK to be the ∆-set whose n-simplices are the ∆-maps K × ∆n → L
and face maps induced by the inclusions ∂i : ∆n−1 → ∆n .
Let Λni = ∂∆n \ ∂in ∆n be the boundary of the standard n-simplex with the i-th
face removed. We say a ∆-set K has the Kan extension property if any ∆-map
f : Λni → K extends to a ∆-map f˜ : ∆n → K not necessarily in an unique way.
For instance in the simplicial world the standard simplices are Kan, in the singular
world everything is Kan. The Kan property gives us that homotopy is a equivalence
relation where a homotopy of ∆-maps f0 , f1 : K → L is defined to be a ∆-map
h : K ⊕ ∆1 → L with the expected conditions. A ∆-set K is pointed if there is
a base simplex ∅ ∈ K (n) for each n ≥ 0. We denote by K+ the ∆-set K with an
added base point in each dimension. For a pointed Kan ∆-set K the homotopy
groups πn (K) are given by
πn (K) := [∂∆n+1 , K] = {x ∈ K (n) | ∂i x = ∅ ∈ K (n−1) , 0 ≤ i ≤ n}/∼
with x ∼ y if there is a z ∈ K (n+1) such

 x
y
∂i z =

∅
that
if i = 0
if i = 1
otherwise
The loop ∆-set ΩK is defined by
1 (n)
(ΩK)(n) = K S
= {x ∈ K (n+1) | ∂0 ∂1 . . . ∂n x = ∅ ∈ K (0) , ∂n+1 x = ∅ ∈ K n }
where S 1 is the pointed ∆-set defined by
{s, ∅} if n = 1
1 (n)
(S ) =
{∅}
if n 6= 1.
Let Λ be an algebraic bordism category. We assume K to be a finite ∆-set. Everything works fine with even locally finite ∆-sets using certain colimit constructions.
Proposition 3.1 (Shift principle). Let (C, ϕ) be an n-dimensional quadratic complex in Λ∗ (K) and σ ∈ K. Then (C, ϕ) restricted to the column (C(σ), ϕ(σ)) in
Λ∗ (K) is an n + |σ|-dimensional quadratic complex in Λ.
Proof. ϕ is an n-dimensional quadratic structure on the 0-simplex column chain
complexes C(τ ), |τ | = 0. Because of the shift in the chain duality
X
M
T ∗ : A∗ (K) → B∗ (K), T ∗ (
Mσ ))r(τ ) = (TA(Z) (
Mτ̃ ))r+|τ | .
σ∈K
τ ≥τ̃
ϕ is an n + |σ|-dimensional quadratic structure restricted to C(σ).
Definition 3.2. Let Ln (Λ) be the pointed ∆-set with k-simplices
Ln (Λ)(k) = {n-dim. quadratic complexes in Λ∗ (∆k )}.
The face maps
Ln (Λ)(k) → Ln (Λ)(k−1) ,
(C, ϕ) ∈ Λ∗ (∆k ) 7→ (C 0 , ϕ0 ) ∈ Λ∗ (∆k−1 )
are induced by the face inclusions ∂i : ∆k−1 → ∆k and the base point is the 0-chain
complex.
Proposition 3.3. L• (Λ) := {Ln (Λ) | n ∈ Z} is an Ω-spectrum of pointed Kan
∆-sets with πn (L• (Λ)) ∼
= Ln (Λ).
Proof. For the proof of the Kan extension condition we refer to [Ran92, p. 137].
For the Ω-spectrum property we prove that in fact (Ln+1 (Λ))(k) and (ΩLn (Λ))(k)
are different descriptions of the same ∆-set. Be aware that the index goes in the
opposite direction than usual. Let
THE TOTAL SURGERY OBSTRUCTION II
11
(C, ϕ) be an (n + 1)-dimensional QAC in Λ∗ (∆k ) which is a k-simplex in Ln+1 (Λ)
and
(C 0 , ϕ0 ) an n-dimensional QAC in Λ∗ (∆k+1 , {k + 1} ∪ ∂0 ∆k+1 ) = Λ∗ (Ω∆(k)) which
is a k-simplex in ΩLn (Λ)
There is a one-to-one correspondence between n-dimensional simplices in ∆k and
(n + 1)-dimensional simplices in Ω∆(k).
∅
∆1 :
•
01
0
?
• ???
/
12 
??02

 012 ???

•
•
•
1
∅
∅
: Ω∆1
∅
0
On the chain complex level we have C ({k + 1}) = 0 and C 0 (σ) = 0 for every
σ ∈ ∂0 ∆k+1 . So the shift principle yields a one-to-one correspondence between
(n + 1)-dimensional QACs in Λ∗ (∆k ) and n-dimensional QACs in Λ∗ (∆k+1 , {k +
1} ∪ ∂0 ∆k ) The following picture visualizes what happens for k = 1.
C ∗ (0) C ∗ (01) C ∗ (1)
•
:C n+1
C(0)
C(01)
C1 :
•
•
C(1)
ϕ0
• o_ _ _ _
C0 :
•
•
ϕ0
• o_ _ _ _ •
(C, ϕ) ∈ Λ∗ (∆1 )
•
C ∗ (01) C ∗ (012) C ∗ (02)
C(01) C(012) C(02)
ϕ00
•
•
• }{
{
{
ϕ00
C0 :
•
•
:C n
•
•
(C 0 , ϕ0 ) ∈ Λ∗ (∆2 , {2} ∪ ∂0 ∆2 )
•
{
• }{
{
{
{
{
{
{
•
{ •
•
•
•
:C n+1
•
:C n
The proof of πn (L• (Λ)) = Ln (Λ) is essentially the same story. A k-simplex in
πn (Lm (Λ)) is an m-dimensional QAC (C, ϕ) in Λ∗ (K) with C(σ) = 0 for all σ ∈ K
with |σ| < n. There is only one non-trivial column chain complex in C namely
the biggest simplex τ with |τ | = n. Using the shift principle we see that (C, ϕ)
is an (n + m)-dimensional QAC in Λ. The homotopy relation corresponds to the
cobordism relation hence πn (L• (Λ)) = πn+m (L−m (Λ)) = Ln (Λ).
Proposition 3.4. Let K be a finite simplicial complex and Λ an algebraic bordism
category. Then
∼ L• (Λ∗ (K))
(i) L• (Λ)K+ =
(ii) K+ ∧ L• (Λ) ' L• (Λ∗ (K))
Remark 3.5. In particular we get
Ln (Λ) = πn (L• (Λ∗ (K))) = πn (K+ ∧ L• (Λ)) = Hn (K, L• (Λ))
and so for our special choice we made for the braid
Ln (B, D) = Ln (Λ(R)∗ (K)) = Hn (K, L• (R)).
Proof of (i). We have to show L• (Λ∗ (K)) = L• (Λ)K+ . In the upper star category
the skew maps go only from bigger simplices to smaller simplices. Hence we can
split an n-dimensional QAC (C, ϕ) ∈ Λ∗ (K) into a collection of n-dimensional QAC
12
PHILIPP KÜHL
{(Cσ , ϕσ ) ∈ Λ∗ (∆|σ| )} such that the (Cσ , ϕσ ) are related to each oter in the same
way the corresponding simplices are related to each other. the chain complexes
(Cσ , ϕσ ) fit together. (Cσ , ϕσ ) is an |σ|-simplex in Ln (Λ) and the compatibility
conditions are contained in the notion of ∆-maps. Hence we get
(C, ϕ)
= {n-dim. QAC (Cσ , ϕσ ) ∈ Λ∗ (∆|σ| ) |
σ ∈ K and Cσ (∂i σ) = C∂i σ (∂i σ)}
=
∆-map fC : K+ → Ln (Λ) with f (σ) = Cσ
Thus
Ln (Λ∗ (K))(k)
=
{n-dim. QAC (C, ϕ) ∈ Λ∗ (K)∗ (∆k ) = Λ∗ (K × ∆k )}
=
{f : (K × ∆k )+ → Ln (Λ) | f is a pointed ∆-map}
=
(Ln (Λ)K+ )(k)
Proof of (ii). For m ∈ N large enough there is an embedding i : K → ∂∆m+1 .
Let Σm be the simplicial complex with one k-simplex σ ∗ for each (m − k)-simplex
σ ∈ ∂∆m+1 and σ ∗ is a face of τ ∗ in Σm if and only if τ is a face of σ in ∂∆m+1 .
The supplement of K is the the subcomplex
K := {σ ∗ ∈ Σm | σ ∈ ∂∆m+1 \ i(K)} ⊂ Σm .
There are two ways in which we map between Σm and ∂∆m+1 . First there is a map
∗ : ∂∆m+1 → Σm , σ 7→ σ ∗
which is a bijection between (∂∆m+1 )(m−k) and (Σm )(k) and sends face maps
∂i : (∂∆m+1 )(m−k) → (∂∆m+1 )(m−k−1) to inclusions ∂i∗ : (Σm )(k) ) → (Σm )(k+1) .
∂∆3
Σ2
K∗
K
This map yield
(3.1)
Ln (Λ∗ (Σ, K)) = Ln (Λ∗ (K)).
On the other hand we get for each σ ∗ ∈ Σm a homotopy equivalence between
σ and the dual cell D(σ, ∂∆m+1 ). We apply this homotopy equivalence to the
supplement K and we get
∗
Σm /K ' N (K)/∂N (K)
where N (K) is a neighbourhood of K in ∂∆m+1 . For m = 2 it looks like this:
D(∗−1 (K))
K
'
K
THE TOTAL SURGERY OBSTRUCTION II
13
Using Spanier-Whitehead duality (see [Whi62], 7.5) we get
K+ ∧ Ln (Λ)
SW
m
=
Ln (Λ)(Σ
(i)
Ln (Λ∗ (Σm , K))
=
(3.1)
=
,K)
Ln (Λ∗ (K))
Remark 3.6. There is also a proof without these simplex constructions by Weiss
in [Wei92] where he shows that the functor K → L• (Λ∗ (K)) from ∆-sets to spectra
is homotopy invariant and excisive.
References
[Ran81] A. Ranicki. Exact sequences in the algebraic theory of surgery, volume 26 of Mathematical
Notes, 1981.
[Ran92] A. Ranicki. Algebraic L-theory and topological manifolds. Cambridge Univ Pr, 1992.
[Wei92] M. Weiss. Visible L-theory. In Forum Math, volume 4, pages 465–498, 1992.
[Whi62] G.W. Whitehead. Generalized homology theories. Transactions of the American Mathematical Society, pages 227–283, 1962.