Piecewise linear normal micro-bundles

C. T. C. Wall
1965 Bulletin of the American Mathematical Society
The object of this paper is to gain some information about the unstable piecewise linear groups. The tool that we use for this purpose is the s-cobordism theorem (which has been established for piecewise linear manifolds by J. Stallings  and D. Barden [l]). All manifolds and micro-bundles in this paper are piecewise linear, unless otherwise specified. THEOREM 1. Let M m be a compact manifold such that iri(dM)=wiM by inclusion, and let ƒ : K k -*M m be a simple homotopy equivalence of a
more » ... ivalence of a finite simplicial k-complex with M. Then if m^6, m^2k + l, there is a compact manifold L such that Tn(dL)Ç=wiL, and LXI==M. If m*z7, mèz2k+2, L is uniquely determined. PROOF. We first observe that the pair (M, dM) is (m -k -1)connected. Indeed, since inclusion induces an isomorphism of fundamental groups, we can use the relative Hurewicz theorem to compute the first nonvanishing relative homotopy group: TTi(M, dM) =TTi(M y dM) (where M denotes the universal cover), 7Ti(M, dM) Ç^Hi(M, dM)=H™~\M), by duality, where c denotes compact cohomology, and H?~\M)^H™-\K) vanishes for i<m-k. It follows that for m^2k + l, ƒ is homotopic to a map g: K-+dM. If, now m^2k+2 we can move g into general position (see e.g. [ll, Chapter 6, Theorem 18]) and so suppose it an imbedding. Take a regular neighbourhood L of g(K) in dM. Then L is a manifold, and the inclusion LQM is a simple homotopy equivalence. If m = 2k + l, g will in general have singularities, transverse selfintersections of fe-simplexes of K. For each such selfintersection Q = g(Pi) =g(P2), we join Pi to P 2 by a path a in K such that g (a) is a nullhomotopic loop (since g*: wi(K)-»7Ti(dilf) is onto, this is possible). As fe^3, we can now map a disc D 2 into 9Af, with its interior imbedded, and meeting g(K) only in its boundary, which is attached along g(a). Proceeding thus for each selfintersection Q, we obtain an imbedding of a complex K f simply homotopy-equivalent to K; we can then take a regular neighbourhood to obtain L, as above. Note in either case that as regular neighbourhood of a subcomplex K! of codimension ^ 3, L has the property 7TI(3L)=7TIL, for ÔL is a deformation retract of L-K'. 638