# Surgery structure set

In mathematics, the surgery structure set ${\mathcal {S}}(X)$ is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or homeomorphic). There are different versions of the structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion is taken into account or not.

## Definition

Let X be a closed smooth (or PL- or topological) manifold of dimension n. We call two homotopy equivalences $f_{i}:M_{i}\to X$ from closed manifolds $M_{i}$ of dimension $n$ to $X$ ($i=0,1$) equivalent if there exists a cobordism ${\mathcal}(W;M_{0},M_{1})$ together with a map $(F;f_{0},f_{1}):(W;M_{0},M_{1})\to (X\times [0,1];X\times \{0\},X\times \{1\})$ such that $F$, $f_{0}$ and $f_{1}$ are homotopy equivalences. The structure set ${\mathcal {S}}^{h}(X)$ is the set of equivalence classes of homotopy equivalences $f:M\to X$ from closed manifolds of dimension n to X. This set has a preferred base point: $id:X\to X$.
There is also a version which takes Whitehead torsion into account. If we require in the definition above the homotopy equivalences F, $f_{0}$ and $f_{1}$ to be simple homotopy equivalences then we obtain the simple structure set ${\mathcal {S}}^{s}(X)$.

## Remarks

Notice that $(W;M_{0},M_{1})$ in the definition of ${\mathcal {S}}^{h}(X)$ resp. ${\mathcal {S}}^{s}(X)$ is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set ${\mathcal {S}}^{s}(X)$, provided that n>4: The simple structure set ${\mathcal {S}}^{s}(X)$ is the set of equivalence classes of homotopy equivalences $f:M\to X$ from closed manifolds $M$ of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences $f_{i}:M_{i}\to X$ (i=0,1) are equivalent if there exists a diffeomorphism (or PL-homeomorphism or homeomorphism) $g:M_{0}\to M_{1}$ such that $f_{1}\circ g$ is homotopic to $f_{0}$.
As long as we are dealing with differential manifolds, there is in general no canonical group structure on ${\mathcal {S}}^{s}(X)$. If we deal with topological manifolds, it is possible to endow ${\mathcal {S}}^{s}(X)$ with a preferred structure of an abelian group (see chapter 18 in the book of Ranicki).
Notice that a manifold M is diffeomorphic (or PL-homeomorphic or homeomorphic) to a closed manifold X if and only if there exists a simple homotopy equivalence $\phi :M\to X$ whose equivalence class is the base point in ${\mathcal {S}}^{s}(X)$. Some care is necessary because it may be possible that a given simple homotopy equivalence $\phi :M\to X$ is not homotopic to a diffeomorphism (or PL-homeomorphism or homeomorphism) although M and X are diffeomorphic (or PL-homeomorphic or homeomorphic). Therefore, it is also necessary to study the operation of the group of homotopy classes of simple self-equivalences of X on ${\mathcal {S}}^{s}(X)$.
The basic tool to compute the simple structure set is the surgery exact sequence.

## Examples

Topological Spheres: The generalized Poincaré conjecture in the topological category says that ${\mathcal {S}}^{s}(S^{n})$ only consists of the base point. This conjecture was proved by Smale (n > 4), Freedman (n = 4) and Perelman (n = 3).
Exotic Spheres: The classification of exotic spheres by Kervaire and Milnor gives ${\mathcal {S}}^{s}(S^{n})=\theta _{n}=\pi _{n}(PL/O)$ for n > 4 (smooth category).