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".

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