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 [math]\displaystyle{ M \hookrightarrow W }[/math] is a simple homotopy equivalence (as in an s-cobordism), with no further requirement on the inclusion [math]\displaystyle{ M^- \hookrightarrow W }[/math] (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 [math]\displaystyle{ K = \ker(\pi_1(M^{-}) \twoheadrightarrow \pi_1(W)) }[/math] is perfect. A corollary of this is that [math]\displaystyle{ \pi_1(M^{-}) }[/math] solves the group extension problem [math]\displaystyle{ 1 \rightarrow K \rightarrow \pi_1(M^{-}) \rightarrow \pi_1(M) \rightarrow 1 }[/math]. The solutions to the group extension problem for prescribed quotient group [math]\displaystyle{ \pi_1(M) }[/math] 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.
- Hausmann, Jean-Claude; Vogel, Pierre (1978), "The Plus Construction and Lifting Maps from Manifolds", Proceedings of Symposia in Pure Mathematics 32: 67–76, http://www.ams.org/bookstore-getitem/item=PSPUM-32.
- Hausmann, Jean-Claude (1978), "Manifolds with a Given Homology and Fundamental Group", Commentarii Mathematici Helvetici 53 (1): 113–134, doi:10.1007/BF02566068, https://doi.org/10.1007%2FBF02566068.
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