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 formula_1 is a simple homotopy equivalence (as in an "s"-cobordism) but the inclusion formula_2 is not a homotop
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 formula_3 is perfect. A corollary of this is that formula_4 solves the group extension problem formula_5. The solutions to the group extension problem for proscribed quotient group formula_6 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.
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".