In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in dimension
>4
l{S}(X)
n
X
n
X
The basic idea is that in order to calculate
l{S}(X)
In practice one has to proceed case by case, for each manifold
l{}X
s
h
The original 1962 work of Browder and Novikov on the existence and uniqueness of manifolds within a simply-connected homotopy type was reformulated by Sullivan in 1966 as a surgery exact sequence.In 1970 Wall developed non-simply-connected surgery theory and the surgery exact sequence for manifolds with arbitrary fundamental group.
The surgery exact sequence is defined as
… \tol{N}\partial(X x I)\toLn+1(\pi1(X))\tol{S}(X)\tol{N}(X)\toLn(\pi1(X))
where:
the entries
l{N}\partial(X x I)
l{N}(X)
the entries
l{}Ln+1(\pi1(X))
l{}Ln(\pi1(X))
Z[\pi1(X)]
the maps
\theta\colonl{N}\partial(X x I)\toLn+1(\pi1(X))
\theta\colonl{N}(X)\toLn(\pi1(X))
the arrows
\partial\colonLn+1(\pi1(X))\tol{S}(X)
η\colonl{S}(X)\tol{N}(X)
There are various versions of the surgery exact sequence. One can work in either of the three categories of manifolds: differentiable (smooth), PL, topological. Another possibility is to work with the decorations
s
h
See main article: Normal invariants.
A degree one normal map
(f,b)\colonM\toX
n
M
f
f*([M])=[X]
b\colonTM ⊕ \varepsilonk\to\xi
M
\xi
X
When defined like this the normal invariants
l{N}(X)
(id,id)
l{N}(X)
l{N}(X)\cong[X,G/O]
where
G/O
J\colonBO\toBG
[X,G/PL]
[X,G/TOP]
See main article: L-theory.
The
L
See main article: Surgery obstruction.
The map
\theta\colonl{N}(X)\toLn(\pi1(X))
n\geq5
A degree one normal map
(f,b)\colonM\toX
\theta(f,b)=0
Ln(Z[\pi1(X)])
η\colonl{S}(X)\tol{N}(X)
Any homotopy equivalence
f\colonM\toX
\partial\colonLn+1(\pi1(X))\tol{S}(X)
This arrow describes in fact an action of the group
Ln+1(\pi1(X))
l{S}(X)
L
Let
M
n
\pi1(M)\cong\pi1(X)
x\inLn+1(\pi1(X))
(F,B)\colon(W,M,M')\to(M x I,M x 0,M x 1)
with the following properties:
1.
\theta(F,B)=x\inLn+1(\pi1(X))
2.
F0\colonM\toM x 0
3.
F1\colonM'\toM x 1
Let
f\colonM\toX
l{S}(X)
x\inLn+1(\pi1(X))
\partial(f,x)
f\circF1\colonM'\toX
Recall that the surgery structure set is only a pointed set and that the surgery obstruction map
\theta
For a normal invariant
z\inl{N}(X)
z\inIm(η)
\theta(z)=0
x1,x2\inl{S}(X)
η(x1)=η(x2)
u\inLn+1(\pi1(X))
\partial(u,x1)=x2
u\inLn+1(\pi1(X))
\partial(u,id)=id
u\inIm(\theta)
In the topological category the surgery obstruction map can be made into a homomorphism. This is achieved by putting an alternative abelian group structure on the normal invariants as described here. Moreover, the surgery exact sequence can be identified with the algebraic surgery exact sequence of Ranicki which is an exact sequence of abelian groups by definition. This gives the structure set
l{S}(X)
The answer to the organizing questions of the surgery theory can be formulated in terms of the surgery exact sequence. In both cases the answer is given in the form of a two-stage obstruction theory.
The existence question. Let
X
X
l{N}(X)
x\inl{N}(X)
\theta(x)=0
\theta\colonl{N}(X) → Ln(\pi1(X))
0\inLn(\pi1(X))
The uniqueness question. Let
f\colonM\toX
f'\colonM'\toX
l{S}(X)
l{}f
l{}f'
l{}η(f)=η(f')
l{N}(X)
(F,B)\colon(W,M,M')\to(X x I,X x 0,X x 1)
l{}\theta(F,B)
l{}Ln+1(\pi1(X))
l{}f
l{}f'
In his thesis written under the guidance of Browder, Frank Quinn introduced a fiber sequence so that the surgery long exact sequence is the induced sequence on homotopy groups.
This is an example in the smooth category,
n\geq5
The idea of the surgery exact sequence is implicitly present already in the original article of Kervaire and Milnor on the groups of homotopy spheres. In the present terminology we have
l{S}DIFF(Sn)=\Thetan
l{N}DIFF(Sn)=
alm | |
\Omega | |
n |
n
DIFF | |
l{N} | |
\partial |
(Sn x I)=
alm | |
\Omega | |
n+1 |
Ln(1)=Z,0,Z2,0
n\equiv0,1,2,3
4
4
The surgery exact sequence in this case is an exact sequence of abelian groups. In addition to the above identifications we have
bPn+1=ker(η\colonl{S}DIFF(Sn)\tol{N}DIFF(Sn))=coker(\theta\colon
DIFF | |
l{N} | |
\partial |
(Sn x I)\toLn+1(1))
Because the odd-dimensional L-groups are trivial one obtains these exact sequences:
0\to\Theta4i\to
alm | |
\Omega | |
4i |
\toZ\tobP4i\to0
0\to\Theta4i-2\to
alm | |
\Omega | |
4i-2 |
\toZ/2\tobP4i-2\to0
0\tobP2j\to\Theta2j-1\to
alm | |
\Omega | |
2j-1 |
\to0
The results of Kervaire and Milnor are obtained by studying the middle map in the first two sequences and by relating the groups
alm | |
\Omega | |
i |
The generalized Poincaré conjecture in dimension
n
l{S}TOP(Sn)=0
n
Sn
n\geq5
\theta\colonl{N}TOP(Sn)\toLn(1)
is an isomorphism. (In fact this can be extended to
n\geq1
The complex projective space
CPn
(2n)
\pi1(CPn)=1
\pi1(X)=1
\theta
0\tol{S}TOP(CPn)\tol{N}TOP(CPn)\toL2n(1)\to0
From the work of Sullivan one can calculate
l{N}(CPn)\cong
\lfloorn/2\rfloor | |
⊕ | |
i=1 |
Z ⊕
\lfloor(n+1)/2\rfloor | |
⊕ | |
i=1 |
Z2
l{S}(CPn)\cong
\lfloor(n-1)/2\rfloor | |
⊕ | |
i=1 |
Z ⊕
\lfloorn/2\rfloor | |
⊕ | |
i=1 |
Z2
An aspherical
n
X
n
\pii(X)=0
i\geq2
\pi1(X)
One way to state the Borel conjecture is to say that for such
X
Wh(\pi1(X))
l{S}(X)=0
This conjecture was proven in many special cases - for example when
\pi1(X)
Zn
The statement is equivalent to showing that the surgery obstruction map to the right of the surgery structure set is injective and the surgery obstruction map to the left of the surgery structure set is surjective. Most of the proofs of the above-mentioned results are done by studying these maps or by studying the assembly maps with which they can be identified. See more details in Borel conjecture, Farrell-Jones Conjecture.