Fixed point sets and tangent bundles of actions on disks and Euclidean spaces
Fixed point sets and tangent bundles of actions on disks and Euclidean spaces
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磁盘和欧几里得空间上的不动点集和切线束动作
DOI：
10.1016/0040-9383(95)00043-7
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
B. Oliver
中科院分区：
文献类型：
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作者：
B. Oliver
本文的主要结果是确定，对于任何给定的有限群G不素数幂阶，究竟哪些光滑流形可以是光滑G-作用在圆盘或欧氏空间上的不动点集。在[O 1]中发展了在具有给定同伦类型的不动点集的圆盘上构造光滑作用的一般技术，在欧几里得空间上构造作用的过程是类似的（但更简单）。这里新的是一种在给定同伦类型的G-复形上构造G-向量丛的方法，它在不动点集上扩展了给定的G-丛。这样的G-丛可以用来控制G-复形的“加厚”过程，以得到一个具有光滑G-作用的流形;特别是控制不动点集的同构类型。这里的“G-复形”总是指G-CW复形：由胞腔的轨道G/H× Dn构成的复形（其中G平凡地作用在圆盘Dn上）。对于非素幂阶的有限群G，构造G-丛的主要技术结果在定理2.4中给出。设P（G）表示G的素数幂阶子群集.非常粗略地，给定一个有限G-复形X，一个XNP上的G-向量丛η def=<$H/∈ P（G）XH，以及对所有P∈ P（G）的P-向量丛<$P↓ X，定理2.4给出了能够将η和<$P（稳定化后）结合起来得到一个与X具有相同（非等变）同伦类型的G-复形X′上的G-丛的条件，并且（X′）NP= XNP。这个结果然后可以与Edmonds & Lee [EL]和Pawa lowski [Pa 2]的等变增厚定理相结合（见定理A. 12），构造具有光滑G-作用的流形，该流形具有给定的同伦类型和不动点集上的给定的切结构。注意，这个过程并不（直接）适用于构造具有G作用的闭流形，而仅适用于开（非紧）流形或具有边界的紧流形。
The main result of this paper is the determination, for any given finite group G not of prime power order, of exactly which smooth manifolds can be fixed point sets of smooth G-actions on disks or on euclidean spaces. General techniques for constructing smooth actions on disks with fixed point set of a given homotopy type were developed in [O1], and the procedure for constructing actions on euclidean spaces is similar (but simpler). What is new here is a way of constructing a G-vector bundle over a G-complex of given homotopy type which extends a given G-bundle over the fixed point set. Such a G-bundle can then be used to control the process of “thickening up” the G-complex to get a manifold with smooth G-action; and in particular to control the diffeomorphism type of the fixed point set. Here “G-complex” always means G-CW complex: a complex built up of orbits G/H× Dn of cells (where G acts trivially on the disk Dn).The main technical result for constructing G-bundles, for a finite group G not of prime power order, is given in Theorem 2.4. Let P (G) denote the set of subgroups of G of prime power order. Very roughly, given a finite G-complex X, a G-vector bundle η over XNP def=∪ H/∈ P (G) XH, and P-vector bundles ξP↓ X for all P∈ P (G), Theorem 2.4 gives conditions for being able to combine η and the ξP (after stabilization) to get a G-bundle over a G-complex X′ of the same (nonequivariant) homotopy type as X, and with (X′) NP= XNP. This result can then be combined with the equivariant thickening theorem of Edmonds & Lee [EL] and Pawa lowski [Pa2](see Theorem A. 12 below), to construct manifolds with smooth G-action having given homotopy type and given tangential structure on the fixed point sets. Note that this procedure does not (directly) apply to construct closed manifolds with G-action, but only open (noncompact) manifolds, or compact manifolds with boundary.