Semi-s-cobordism

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 $1\rightarrow K\rightarrow \pi _{1}(M^{-})\rightarrow \pi _{1}(M)\rightarrow 1$. 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⁻) 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.

References

• MacLane (1963), Homology, pp. 124–129, ISBN 0-387-58662-8
• Hausmann, Jean-Claude (1976), "Homological Surgery", Annals of Mathematics, Second Series, 104 (3): 573–584, doi:10.2307/1970967, JSTOR 1970967.
• Hausmann, Jean-Claude; Vogel, Pierre (1978), "The Plus Construction and Lifting Maps from Manifolds", Proceedings of Symposia in Pure Mathematics, 32: 67–76.
• Hausmann, Jean-Claude (1978), "Manifolds with a Given Homology and Fundamental Group", Commentarii Mathematici Helvetici, 53 (1): 113–134, doi:10.1007/BF02566068.