Semi-s-cobordism

In mathematics, a cobordism (W, M, M&minus;) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M&minus;, 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&minus; for the right-hand boundary of the cobordism.

Properties
A consequence of (W, M, M&minus;) 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 $$1 \rightarrow K \rightarrow \pi_1(M^{-}) \rightarrow \pi_1(M) \rightarrow 1$$. 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&minus;) is a semi-s-cobordism, then (W, M&minus;, M) is a plus cobordism. (This justifies the use of M&minus; 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&minus;)+ must be diffeomorphic (respectively, piecewise-linearly (PL) homeomorphic) to M but there may be a variety of choices for (M+)&minus; for a given closed smooth (respectively, PL) manifold M.