Semi-s-cobordism explained

In mathematics, a cobordism (W, M, M^-) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M^-, is called a semi-s-cobordism if (and only if) the inclusion

M \hookrightarrow W is a simple homotopy equivalence (as in an s-cobordism), with no further requirement on the inclusion M ^{-} \hookrightarrow W (not even being a homotopy equivalence).

Other notations

The original creator of this topic, Jean-Claude Hausmann, used the notation M_- for the right-hand boundary of the cobordism.

Properties

A consequence of (W, M, M^-) being a semi-s-cobordism is that the kernel of the derived homomorphism on fundamental groups

K = \ker(\pi ^{-} _{1 (M} ) \twoheadrightarrow \pi _{1(W))} is perfect. A corollary of this is that \pi ^{-} _{1 (M} ) solves the group extension problem 1 → K → \pi ^{-} _{1 (M} ) → \pi _{1(M)} → 1 . The solutions to the group extension problem for prescribed quotient group \pi _{1(M)} and kernel group K are classified up to congruence by group cohomology (see Mac Lane's Homology pp. 124-129), so there are restrictions on which n-manifolds can be the right-hand boundary of a semi-s-cobordism with prescribed left-hand boundary M and superperfect kernel group K.

Relationship with Plus cobordisms

Note that if (W, M, M^-) is a semi-s-cobordism, then (W, M^-, M) is a plus cobordism. (This justifies the use of M^- for the right-hand boundary of a semi-s-cobordism, a play on the traditional use of M⁺ for the right-hand boundary of a plus cobordism.) Thus, a semi-s-cobordism may be thought of as an inverse to Quillen's Plus construction in the manifold category. Note that (M^-)⁺ must be diffeomorphic (respectively, piecewise-linearly (PL) homeomorphic) to M but there may be a variety of choices for (M⁺)^- for a given closed smooth (respectively, PL) manifold M.

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