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\(L_ 2\)-cohomology and group cohomology. (English) Zbl 0597.57020

Simplicial \(L_ 2\)-cohomology is based on the space of square-summable real-valued cochains on a simplicial complex. The authors show how to extend this \(L_ 2\)-cohomology to group equivariant singular \(L_ 2\)- cohomology on arbitrary topological spaces. Using the notion of von Neumann dimension, they then manage to define the appropriate equivariant notions of Betti number and Euler characteristic and prove the usual formal results therefore. They obtain thereby homotopy invariants of group actions with particularly nice properties. Indeed, the Euler characteristic is sometimes defined even though the orbit space of the action has infinite topological type; and the Betti numbers behave multiplicatively under finite coverings.
The authors give in some circumstances a local formula for their Euler characteristic and show that the local invariants are particularly easily calculable in the case of group actions by amenable groups. In the amenable case, the invariants can sometimes be expressed in terms of standard Euler characteristics; this fact allows the authors to recover and extend prior results of Gottlieb, Stallings, and Rosset. The authors also relate their results to discrete actions of groups on Riemannian manifolds having bounded covering geometry.
Reviewer: J.W.Cannon

MSC:

57S20 Noncompact Lie groups of transformations
20J05 Homological methods in group theory
55N91 Equivariant homology and cohomology in algebraic topology
55N10 Singular homology and cohomology theory
57R20 Characteristic classes and numbers in differential topology
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