# On equivariant deformations of maps.

• Volume: 32, Issue: 1, page 115-121
• ISSN: 0214-1493

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## Abstract

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We work in the smooth category: manifolds and maps are meant to be smooth. Let G be a finite group acting on a connected closed manifold X and f an equivariant self-map on X with f|A fixpointfree, where A is a closed invariant submanifold of X with codim A ≥ 3. The purpose of this paper is to give a proof using obstruction theory of the following fact: If X is simply connected and the action of G on X - A is free, then f is equivariantly deformable rel. A to fixed point free map if and only if the usual Lefschetz number L(f|(X,A)) = 0. As a consequence we obtain a sepcial case of a theorem of Wilczynski (cf. [12, Theorem A]).Finally, motivated by Wilczynski's paper we present an interesting question concerning the equivalent version of the converse of the Lefschetz fixed point theorem.

## How to cite

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Vidal, Antonio. "On equivariant deformations of maps.." Publicacions Matemàtiques 32.1 (1988): 115-121. <http://eudml.org/doc/41027>.

@article{Vidal1988,
abstract = {We work in the smooth category: manifolds and maps are meant to be smooth. Let G be a finite group acting on a connected closed manifold X and f an equivariant self-map on X with f|A fixpointfree, where A is a closed invariant submanifold of X with codim A ≥ 3. The purpose of this paper is to give a proof using obstruction theory of the following fact: If X is simply connected and the action of G on X - A is free, then f is equivariantly deformable rel. A to fixed point free map if and only if the usual Lefschetz number L(f|(X,A)) = 0. As a consequence we obtain a sepcial case of a theorem of Wilczynski (cf. [12, Theorem A]).Finally, motivated by Wilczynski's paper we present an interesting question concerning the equivalent version of the converse of the Lefschetz fixed point theorem.},
author = {Vidal, Antonio},
journal = {Publicacions Matemàtiques},
keywords = {Grupos de transformación; Deformación equivariante; Número de Lefschetz; coefficient system; finite group acting on a simply-connected closed manifold; equivariantly deformable; fixed point free G-map; Lefschetz number},
language = {eng},
number = {1},
pages = {115-121},
title = {On equivariant deformations of maps.},
url = {http://eudml.org/doc/41027},
volume = {32},
year = {1988},
}

TY - JOUR
AU - Vidal, Antonio
TI - On equivariant deformations of maps.
JO - Publicacions Matemàtiques
PY - 1988
VL - 32
IS - 1
SP - 115
EP - 121
AB - We work in the smooth category: manifolds and maps are meant to be smooth. Let G be a finite group acting on a connected closed manifold X and f an equivariant self-map on X with f|A fixpointfree, where A is a closed invariant submanifold of X with codim A ≥ 3. The purpose of this paper is to give a proof using obstruction theory of the following fact: If X is simply connected and the action of G on X - A is free, then f is equivariantly deformable rel. A to fixed point free map if and only if the usual Lefschetz number L(f|(X,A)) = 0. As a consequence we obtain a sepcial case of a theorem of Wilczynski (cf. [12, Theorem A]).Finally, motivated by Wilczynski's paper we present an interesting question concerning the equivalent version of the converse of the Lefschetz fixed point theorem.
LA - eng
KW - Grupos de transformación; Deformación equivariante; Número de Lefschetz; coefficient system; finite group acting on a simply-connected closed manifold; equivariantly deformable; fixed point free G-map; Lefschetz number
UR - http://eudml.org/doc/41027
ER -

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