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Semi-s-cobordism

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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 (not even being a homotopy equivalence).

Other notations

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The original creator of this topic, Jean-Claude Hausmann, used the notation M for the right-hand boundary of the cobordism.

Properties

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A consequence of (W, M, M) being a semi-s-cobordism is that the kernel of the derived homomorphism on fundamental groups is perfect. A corollary of this is that solves the group extension problem . The solutions to the group extension problem for prescribed quotient group 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

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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.

References

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  • 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.