Homotopy groups of the space of homeomorphisms on a $2$-manifold
Homotopy groups of the space of homeomorphisms on a $2$-manifold
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DOI：
10.1215/ijm/1256054895
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发表时间：
1966-12
影响因子：
0.6
通讯作者：
M. Hamstrom
中科院分区：
文献类型：
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作者：
M. Hamstrom
这是关于2-多重上同胚空间的同伦群的pps系列的最后一个pper。设M是有界紧2-多重集M '，nd K是闭子集，记H（M，K）为M到自身上的同胚空间，使K点态不动点nd被H_0（M，K）的单位分支所固定. Kneser [14]证明了S上刚体运动的空间是Ho（S）的形变收缩。因此，对于ech n，m_n H_0（S ~ 2）_n（P）;对于n > 1，m_n H_0（S）_n（.3）;对于n > 2，m_n H_0（S）_n（S）_n（.3）。具体来说，H 0（S）Z nd r，H 0（S）0。若M是带孔圆盘或Moebius带，则Ho（M，M '）是同伦平凡的（[6]，[8] nd [12]）.在fct亚历山大的经典结果[1]中，n个n-胞腔到其自身且边界逐点固定的同胚空间是可收缩的且局部可收缩的是研究这些问题的最重要工具。若M是环面，则rHo（M）Try（M）对ech i，且若M是除去有限个不相交圆内部的环面，则Ho（M，M '）是同伦平凡的[11].对于rel射影空间，rH_0（P）r（P），其中i > 2，rH_0（P）0，rH_0（P）Z，rH_0（P，x）Z，其中xeP和Ho（P，x）0，其中i > 1（见[12]）.对于Klein瓶K，'Ho（K）0（i > 1），r Ho（K）Z nd 'Ho（K，x）0（ech i）[12].本文证明了：对于亏格大于1的11个紧2-多重（无边界），若可定向，若不可定向，则H_0（M）是同伦平凡的;若M是有非空边界的紧2-多重，则H_0（M，M '）是同伦平凡的。进一步的相关结果可以在McCrty的pper [16]中找到，其中他证明了，
This is the final pper in series of ppers concerning the homotopy groups of the spce of homeomorphisms on 2-mnifold. If M is compact 2-mnifold with boundary M’, nd K is closed subset, denote by H(M, K) the spce of homeomorphisms of M onto itself leaving K pointwise fixed nd by H0(M, K) its identity component. Kneser proved [14] tht the spce of rigid motions on S is deformation retract of Ho(S). Thus 7rn H0(S2) "n’n(P) for ech n, rn Ho(S) 7rn(.3) for n > 1, nd mHo(S) -,(S) for n > 2. In prticulr r H0(S) Z nd r, H0(S) 0. If M is disc with holes or Moebius strip, Ho(M, M’) is homotopiclly trivial ([6], [8] nd [12]). In fct Alexander’s classic result [1] that the spce of homeomorphisms of n n-cell onto itself leaving the boundary pointwise fixed is contractible nd locally contractible is most important tool in the study of these problems. If M is torus, rHo(M) Try(M) for ech i, nd if M is torus with the interiors of finite number of disjoint discs removed, Ho(M, M’) is homotopically trivial [11]. For rel projective spce, r H0(P) r(P) for i > 2, rH0(P) 0, rH0(P) Z, rH0(P, x) Z, where xeP nd Ho(P, x) 0 for i > 1 (see [12]). For the Klein bottle K, ’Ho(K) 0 for i > 1, r Ho(K) Z nd ’Ho(K, x) 0 for ech i [12]. In this present pper, it is shown that H0(M) is homotopiclly trivial for ll compact 2-mnifolds (without boundary) of genus greater thn 1, if orientble, and greter than 2, if non-orientble; nd that, if M is compact 2-mnifold with nonempty boundary, H0(M, M’) is homotopiclly trivial. Further related results my be found in McCrty’s pper [16], where he proves among other things, that