A consequence of (W, M, M−) being a semi-s-cobordism is that the kernel of the derived homomorphism on fundamental groups is perfect; it is a non-obvious fact that the kernel must always be superperfect. A corollary of this is that solves the group extension problem . The solutions to the group extension problem for proscribed quotient group 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.