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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

##### MSC:
 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
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