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).