Semi-s-cobordism

In mathematics, a cobordism (W, M, M) of an (n + 1)-dimensionsal 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 ${\displaystyle M\hookrightarrow W}$ is a simple homotopy equivalence (as in an s-cobordism) but the inclusion ${\displaystyle M^{-}\hookrightarrow W}$ is not a homotopy equivalence at all.

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 ${\displaystyle K=\ker(\pi _{1}(M^{-})\twoheadrightarrow \pi _{1}(W))}$ is perfect. A corollary of this is that ${\displaystyle \pi _{1}(M^{-})}$ solves the group extension problem ${\displaystyle 1\rightarrow K\rightarrow \pi _{1}(M^{-})\rightarrow \pi _{1}(M)\rightarrow 1}$. The solutions to the group extension problem for proscribed quotient group ${\displaystyle \pi _{1}(M)}$ and kernel group K are classified up to congruence (see Homology by MacLane, e.g.), so there are restrictions on which n-manifolds can be the right-hand boundary of a semi-s-cobordism with proscribed 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 (WMM) 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.