Rétractes Absolus de Voisinage Algébriques

2000 Mathematics Subject Classification: 54C55, 54H25, 55M20.

We introduce the class of algebraic ANRs. It is defined by replacing continuous maps by chain mappings in Lefschetz’s characterization of ANRs. To a large extent, the theory of algebraic ANRs parallels the classical theory of ANRs. Every ANR is an algebraic ANR, but the class of algebraic ANRs is much larger; the most striking difference between these classes is that every locally equiconnected metrisable space is an algebraic ANR, whereas there exist metric linear spaces which are not ARs. This is important for applications of topological fixed point theory to functional analysis because all known results of fixed point for compact maps of ANRs extend to the algebraic ANRs. We prove here two such generalizations: the Lefschetz-Hopf fixed point theorem for compact maps of algebraic ANRs, and the fixed point theorem for compact upper semi-continuous multivalued maps with Q-acyclic compacts point images in a Q-acyclic algebraic ANR. We stress that these generalizations apply to all neighborhood retract of a metrisable linear space and, more generally, of a locally contractible metrisable group.