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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.## References

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; 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 $1\; rightarrow\; K\; rightarrow\; pi\_1(M^\{-\})\; rightarrow\; pi\_1(M)\; rightarrow\; 1$. 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.

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday May 20, 2008 at 15:41:32 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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