Surgery structure set

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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[edit]

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).

References[edit]

External links[edit]