zbMATH — the first resource for mathematics

Fixed point free equivariant homotopy classes. (English) Zbl 0548.55002
Let f:\(M\to M\) be an equivariant self-map of a compact smooth G-manifold M, G a compact Lie group. The author defines an equivariant Lefschetz number L(f) which is a family of integers indexed by ((H),C) where (H) is an isotropy type on M such that the Weyl group W(H) is finite and C is a connected component of \(M_{(H)}/G\), where \(M_{(H)}=\{x\in M|\) the isotropy subgroup at x is conjugate to \(H\}\). He then proves the Lefschetz fixed point theorem and its converse. In this context, he also discusses the question of the existence of a nonsingular equivariant vector field.
Reviewer: K.Komiya

55M20 Fixed points and coincidences in algebraic topology
57S15 Compact Lie groups of differentiable transformations
57R25 Vector fields, frame fields in differential topology
57R91 Equivariant algebraic topology of manifolds
Full Text: DOI EuDML