Piecewise Linear Normal Micro-bundles

نویسنده

  • C. T. C. WALL
چکیده

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(My dM) (where M denotes the universal cover), 7Ti(M, dM) Ç^Hi(M, dM)=HTM~\M), by duality, where c denotes compact cohomology, and H?~\M)^HTM-\K) vanishes for i<m-k. I t 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. [ l l , 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 P i 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 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 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'.

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تاریخ انتشار 2007