The Homotopy Type of the Space of Diffeomorphisms . Ii
نویسنده
چکیده
The result (proved in Part I) that Diff(D", i) sa jj" (PL /O ) is used to compute some new homotopy of Diff(D", dD"). The relation between smooth and PL pseudo-isotopy is explored. Known and new results on the homotopy of PL are summarized. 5. Some remarks on PL and Top . We first summarize known results. In n n low dimensions we have the following are contractible: (1) PL,/Oj sa PL2/02 at *, Top/O, at Top2/02 ~ *. It is trivial to show that PL./Oj ~Topj/0j ~ *. Kneser [16] showed a long time ago that TopVO sa *. That PL /O ~ * has been shown recently by Akiba and P. Scott [1]. In the "stable" range, we recall the results of Haefliger and Wall [26a]: (2) rri(PLk+l, PLA)= Ofor z<*-l,and (2') Ker(^_1(PL/t) — »rjl_1(PLfe+1))= Image iJiKero^,,«^) -» ffA_j(0A+1))), ik: 0k —» PLfc the natural map. By standard smoothing theory [17], (3) ir ¡(PL k/0k)-ht.(PL/0) is bijective for i< k. By Theorem 4.6 (see also Hirsch [38]), (4) 77n+1(PLn/On)-» j7n+1(PL/0) is bijective, «> 5. (4') wn+2(PLn/On) -» n-n+2(PL/0) is surjective, n > 5. Combining these results we get Theorem 5.1. (a) ^.(PLfc+1/0Jt+1, PLk/Ok) = 0 for i < k + 2, k > 5. (b) 7Tj.(TopJt+1/0fe+1, Top^/O^) = 0 for i < k + 2, k>5. Part (b) follows from (a) and the results of Kirby and Siebenmann [l5l. Received by the editors June 13, 1973 and, in revised form, September 14, 1973. AMS (MOS) subject classifications (1970). Primary 57D50, 57D40, 57A35, 57E05. (1 ) During the preparation of this work the first named author was partially supported by Aarhus Universitet-Math. Inst. and by Sonderforschungsbereich Theoretische Mathematik an der Universität Bonn. (2 ) The second named author was partially supported by the National Science Foundation, USA. Copyright O 1974, American Mtthenuucal Society
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