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Piecewise linear normal micro-bundles

C. T. C. Wall

1965
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Bulletin of the American Mathematical Society
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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 [9] 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
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... 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

doi:10.1090/s0002-9904-1965-11373-8
fatcat:5ap37w3wuzcapenzujssx55km4