Semi-s-cobordism

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" (n.b.: the original creator of this topic, Jean-Claude Hausmann, used the notation "M" for the right-hand boundary of the cobordism), 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) but the inclusion M^- hookrightarrow W is not a homotopy equivalence at all.

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^{-}) woheadrightarrow pi_1(W)) is perfect; it is a non-obvious fact that the kernel must always be superperfect. A corollary of this is that pi_1(M^{-}) solves the group extension problem 1 ightarrow K ightarrow pi_1(M^{-}) ightarrow pi_1(M) ightarrow 1. The solutions to the group extension problem for proscribed quotient group 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.

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

*citation | last=MacLane | year=1963 | title=Homology | pages=124-129 | isbn=0387586628
*citation|url=http://links.jstor.org/sici?sici=0003-486X(197611)2%3A104%3A3%3C573%3AHS%3E2.0.CO%3B2-B | first=Jean-Claude | last=Hausmann | title=Homological Surgery | journal=The Annals of Mathematics, 2nd Ser. | volume=104 | issue=3 | year=1976 | pages=573-584.
*citation|url=http://www.springerlink.com/content/d0740ht7537775h5/ | first=Jean-Claude | last=Hausmann | | first2=Pierre | last2=Vogel | title=The Plus Construction and Lifting Maps from Manifolds | journal=Proceedings of Symposia in Pure Mathematics | volume=32 | year=1978 | pages=67-76.
*citation|url=http://www.ams.org/bookstore-getitem/item=PSPUM-32 | first=Jean-Claude | last=Hausmann | title=Manifolds with a Given Homology and Fundamental Group | journal=Commentarii Mathematici Helvetici | volume=53 | issue=1 | year=1978 | pages=113-134.


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