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
is a simple homotopy equivalence (as in an s-cobordism), with no further requirement on the inclusion
The original creator of this topic, Jean-Claude Hausmann, used the notation M− for the right-hand boundary of the cobordism.
A consequence of (W, M, M−) being a semi-s-cobordism is that the kernel of the derived homomorphism on fundamental groups
solves the group extension problem
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. 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.