Surgery Structure Set - Remarks


Notice that in the definition of resp. is an h-cobordism resp. an s-cobordism. Using the s-cobordism theorem we obtain another description for the simple structure set, provided that n>4: The simple structure set is the set of equivalence classes of homotopy equivalences from closed manifolds of dimension n to X with respect to the following equivalence relation. Two homotopy equivalences (i=0,1) are equivalent if there exists a diffeomorphism (or PL-homeomorphism or homeomorphism) such that is homotopic to .

As long as we are dealing with differential manifolds, there is in general no canonical group structure on . If we deal with topological manifolds, it is possible to endow 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 whose equivalence class is the base point in . Some care is necessary because it may be possible that a given simple homotopy equivalence 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 .

The basic tool to compute the simple structure set is the surgery exact sequence.

Read more about this topic:  Surgery Structure Set

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