On deforming $G$-maps to be fixed point free.
On deforming $G$-maps to be fixed point free.
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DOI：
10.2140/pjm.1988.132.277
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发表时间：
1988-04
影响因子：
0.6
通讯作者：
E. Fadell;P. Wong
中科院分区：
文献类型：
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作者：
E. Fadell;P. Wong
当/：M -· M是紧致流形的自映射，且dim M > 3，Wecken的经典定理指出，f同伦于不动点自由映射当且仅当f的Nielsen数n（f）为零。当M是单连通的，并且dim M > 3时，NASC变为L（f）= 0，其中L（f）是f的莱夫谢茨数。对于G-映射f：Λ i-· Λf，其中M是紧G-流形，后一结果的等变版本是由于D。Wilczyriski，在假设M H是维数> 3的单连通的条件下，对任意具有有限Weyl群WH的各向同性子群H.在这些假设下，f是G -同伦到一个不动点自由映射的当且仅当，L（f H）= 0，对于任何各向同性子群H（WH有限），其中f H = f\M H，M H表示M中被H固定的那些元素。这个结果的一个特例也由A.通过等变障碍理论。本文证明了类似的等变结果，而不假设M H是单连通的，假设n（fH）= 0，对所有H，WH有限。还有一个余维条件。这是主要的结果。
When /: M —• M is a self-map of a compact manifold and dim M > 3, a classical theorem of Wecken states that / is homotopic to a fixed point free map if, and only if, the Nielsen number n(f) of / is zero. When M is simply connected, and dim M > 3 the NASC becomes L(f) = 0, where L(f) is the Lefschetz number of/. An equivariant version of the latter result for G-maps /: Λί —• Λf, where M is a compact G-manifold, is due to D. Wilczyriski, under the assumption that M H is simply connected of dimension > 3 for any isotropy subgroup H with finite Weyl group WH. Under these assumptions, / is G -homotopic to a fixed point free map if, and only if, L(f H ) = 0 for any isotropy subgroup H {WH finite), where f H = f\M H and M H represents those elements of M fixed by H. A special case of this result was also obtained independently by A. Vidal via equivariant obstruction theory. In this note we prove the analogous equivariant result without assuming that the M H are simply connected, assuming that n(f H ) = 0, for all H with WH finite. There is also a codimension condition. Here is the main result.