Semi s cobordism

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 K = ker ⁡ ( π 1 ( M − ) ↠ π 1 ( W ) ) is perfect. A corollary of this is that π 1 ( M − ) solves the group extension problem 1 → K → π 1 ( M − ) → π 1 ( M ) → 1 . The solutions to the group extension problem for proscribed quotient group π 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.

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.