An Invariant for Almost-closed Manifolds

1. Let M be a compact, oriented, connected, ^-dimensional differential manifold with dM (boundary M) homeomorphic to the n—1 sphere 5~". Then dM represents an element [dM] of r~ , the group of differential structures (up to equivalence) on 5~. We consider the (much studied) problem of expressing [dM] in terms of "computable" invariants of M. Let 7Tn-i be the n — 1 stem, Jo: 7r»(BSO)—»7rw-i the classical J-homomorphism, and ir^x the cokernel of Jb. In [S], a map P: T""-***-* was defined (see below). We will define an invariant A(M) which is a subset of w'n„x (and often consists of a single element). The main theorem states: P[dM]EA(M). In a strong sense, the definition of A(M) involves only homotopy theory. Moreover, A(M) seems amenable to computation by standard techniques of algebraic topology. We illustrate this below and, as applications, give explicit examples (1) of a manifold M, n odd, with [dilf]^0, and (2) of Mf n even, with [dM] not only 5*0, but in fact with [dM] not even contained in T"^(dw)t the subgroup in r**""* 1 of elements which bound 7r-manifolds. (Examples of M, n even, with [dM]^0 are of course well known.) Other applications, and detailed proofs, will appear elsewhere. REMARK 1. By [5], kernel P^T^idr). If n is odd, T^dr)**!), so P is injective, while if n^2 (4), kernel PQZ2. If wsO (4), kernel P tends to be large (but see §5). Let BSO, BSPL, BSTop be the stable classifying spaces for orientable vector bundles» piecewise-linear ( = PL) bundles, topological bundles. There are maps Jo: 7rn(BSG)—>7r»-i (G = 0, PL, Top) and a commutative diagram with exact rows