Surgery exact sequence

A degree one normal map ${\displaystyle (f,b)\colon M\to X}$ consists of the following data: an ${\displaystyle n}$-dimensional oriented closed manifold ${\displaystyle M}$, a map ${\displaystyle f}$ which is of degree one (that means ${\displaystyle f_{*}([M])=[X]}$), and a bundle map ${\displaystyle b\colon TM\oplus \varepsilon ^{k}\to \xi }$ from the stable tangent bundle of ${\displaystyle M}$ to some bundle ${\displaystyle \xi }$ over ${\displaystyle X}$. Two such maps are equivalent if there exists a normal bordism between them (that means a bordism of the sources covered by suitable bundle data). The equivalence classes of degree one normal maps are called normal invariants.

When defined like this the normal invariants ${\displaystyle {\mathcal {N}}(X)}$ are just a pointed set, with the base point given by ${\displaystyle (id,id)}$. However the Pontrjagin-Thom construction gives ${\displaystyle {\mathcal {N}}(X)}$ a structure of an abelian group. In fact we have a non-natural bijection

${\displaystyle {\mathcal {N}}(X)\cong [X,G/O]}$

where ${\displaystyle G/O}$ denotes the homotopy fiber of the map ${\displaystyle J\colon BO\to BG}$, which is an infinite loop space and hence maps into it define a generalized cohomology theory. There are corresponding identifications of the normal invariants with ${\displaystyle [X,G/PL]}$ when working with PL-manifolds and with ${\displaystyle [X,G/TOP]}$ when working with topological manifolds.

The ${\displaystyle L}$-groups are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. See the main article for more details. Here only the properties of the L-groups described below will be important.

The map ${\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))}$ is in the first instance a set-theoretic map (that means not necessarily a homomorphism) with the following property (when ${\displaystyle n\geq 5}$:

A degree one normal map ${\displaystyle (f,b)\colon M\to X}$ is normally cobordant to a homotopy equivalence if and only if the image ${\displaystyle \theta (f,b)=0}$ in ${\displaystyle L_{n}(\mathbb {Z} [\pi _{1}(X)])}$.

Any homotopy equivalence ${\displaystyle f\colon M\to X}$ defines a degree one normal map.

This arrow describes in fact an action of the group ${\displaystyle L_{n+1}(\pi _{1}(X))}$ on the set ${\displaystyle {\mathcal {S}}(X)}$ rather than just a map. The definition is based on the realization theorem for the elements of the ${\displaystyle L}$-groups which reads as follows:

Let ${\displaystyle M}$ be an ${\displaystyle n}$-dimensional manifold with ${\displaystyle \pi _{1}(M)\cong \pi _{1}(X)}$ and let ${\displaystyle x\in L_{n+1}(\pi _{1}(X))}$. Then there exists a degree one normal map of manifolds with boundary

${\displaystyle (F,B)\colon (W,M,M')\to (M\times I,M\times 0,M\times 1)}$

with the following properties:

1. ${\displaystyle \theta (F,B)=x\in L_{n+1}(\pi _{1}(X))}$

2. ${\displaystyle F_{0}\colon M\to M\times 0}$ is a diffeomorphism

3. ${\displaystyle F_{1}\colon M'\to M\times 1}$ is a homotopy equivalence of closed manifolds

Let ${\displaystyle f\colon M\to X}$ represent an element in ${\displaystyle {\mathcal {S}}(X)}$ and let ${\displaystyle x\in L_{n+1}(\pi _{1}(X))}$. Then ${\displaystyle \partial (f,x)}$ is defined as ${\displaystyle f\circ F_{1}\colon M'\to X}$.

This is an example in the smooth category, ${\displaystyle n\geq 5}$.

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

${\displaystyle {\mathcal {S}}^{DIFF}(S^{n})=\Theta ^{n}}$

${\displaystyle {\mathcal {N}}^{DIFF}(S^{n})=\Omega _{n}^{alm}}$ the cobordism group of almost framed ${\displaystyle n}$ manifolds, ${\displaystyle {\mathcal {N}}_{\partial }^{DIFF}(S^{n}\times I)=\Omega _{n+1}^{alm}}$

${\displaystyle L_{n}(1)=\mathbb {Z} ,0,\mathbb {Z} _{2},0}$ where ${\displaystyle n\equiv 0,1,2,3}$ mod ${\displaystyle 4}$ (recall the ${\displaystyle 4}$-periodicity of the L-groups)

The surgery exact sequence in this case is an exact sequence of abelian groups. In addition to the above identifications we have

${\displaystyle bP^{n+1}=\mathrm {ker} (\eta \colon {\mathcal {S}}^{DIFF}(S^{n})\to {\mathcal {N}}^{DIFF}(S^{n}))=\mathrm {coker} (\theta \colon {\mathcal {N}}_{\partial }^{DIFF}(S^{n}\times I)\to L_{n+1}(1))}$

Because the odd-dimensional L-groups are trivial one obtains these exact sequences:

${\displaystyle 0\to \Theta ^{4i}\to \Omega _{4i}^{alm}\to \mathbb {Z} \to bP^{4i}\to 0}$

${\displaystyle 0\to \Theta ^{4i-2}\to \Omega _{4i-2}^{alm}\to \mathbb {Z} /2\to bP^{4i-2}\to 0}$

${\displaystyle 0\to bP^{2j}\to \Theta ^{2j-1}\to \Omega _{2j-1}^{alm}\to 0}$

The results of Kervaire and Milnor are obtained by studying the middle map in the first two sequences and by relating the groups ${\displaystyle \Omega _{i}^{alm}}$ to stable homotopy theory.

The generalized Poincaré conjecture in dimension ${\displaystyle n}$ can be phrased as saying that ${\displaystyle {\mathcal {S}}^{TOP}(S^{n})=0}$. It has been proved for any ${\displaystyle n}$ by the work of Smale, Freedman and Perelman. From the surgery exact sequence for ${\displaystyle S^{n}}$ for ${\displaystyle n\geq 5}$ in the topological category we see that

${\displaystyle \theta \colon {\mathcal {N}}^{TOP}(S^{n})\to L_{n}(1)}$

is an isomorphism. (In fact this can be extended to ${\displaystyle n\geq 1}$ by some ad-hoc methods.)

The complex projective space ${\displaystyle \mathbb {C} P^{n}}$ is a ${\displaystyle (2n)}$-dimensional topological manifold with ${\displaystyle \pi _{1}(\mathbb {C} P^{n})=1}$. In addition it is known that in the case ${\displaystyle \pi _{1}(X)=1}$ in the topological category the surgery obstruction map ${\displaystyle \theta }$ is always surjective. Hence we have

${\displaystyle 0\to {\mathcal {S}}^{TOP}(\mathbb {C} P^{n})\to {\mathcal {N}}^{TOP}(\mathbb {C} P^{n})\to L_{2n}(1)\to 0}$

From the work of Sullivan one can calculate

${\displaystyle {\mathcal {N}}(\mathbb {C} P^{n})\cong \oplus _{i=1}^{\lfloor n/2\rfloor }\mathbb {Z} \oplus \oplus _{i=1}^{\lfloor (n+1)/2\rfloor }\mathbb {Z} _{2}}$ and hence ${\displaystyle {\mathcal {S}}(\mathbb {C} P^{n})\cong \oplus _{i=1}^{\lfloor (n-1)/2\rfloor }\mathbb {Z} \oplus \oplus _{i=1}^{\lfloor n/2\rfloor }\mathbb {Z} _{2}}$

An aspherical ${\displaystyle n}$-dimensional manifold ${\displaystyle X}$ is an ${\displaystyle n}$-manifold such that ${\displaystyle \pi _{i}(X)=0}$ for ${\displaystyle i\geq 2}$. Hence the only non-trivial homotopy group is ${\displaystyle \pi _{1}(X)}$

One way to state the Borel conjecture is to say that for such ${\displaystyle X}$ we have that the Whitehead group ${\displaystyle Wh(\pi _{1}(X))}$ is trivial and that

${\displaystyle {\mathcal {S}}(X)=0}$

This conjecture was proven in many special cases - for example when ${\displaystyle \pi _{1}(X)}$ is ${\displaystyle \mathbb {Z} ^{n}}$, when it is the fundamental group of a negatively curved manifold or when it is a word-hyperbolic group or a CAT(0)-group.

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.