By A. S. Smogorzhevski

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**L²-invariants: theory and applications to geometry and K-theory**

In algebraic topology a few classical invariants - reminiscent of Betti numbers and Reidemeister torsion - are outlined for compact areas and finite team activities. they are often generalized utilizing von Neumann algebras and their lines, and utilized additionally to non-compact areas and countless teams. those new L2-invariants include very fascinating and novel info and will be utilized to difficulties coming up in topology, K-Theory, differential geometry, non-commutative geometry and spectral thought.

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The preimage of a path component of G\(Xn − Xn−1 ). The closure of an equivariant open n-dimensional cell is called an equivariant closed n-dimensional cell . 25, then the equivariant closed n-dimensional cells are just the G-subspaces Qi (G/Hi × Dn ). If X is a G-CW -complex, then G\X is a CW -complex. If G is discrete or if G is a Lie group and H ⊂ G is compact, then the H-fixed point set X H inherits a WH-CW -complex structure. Here and in the sequel NH = {g ∈ G | gHg −1 = H} is the normalizer of H in G and WH denotes the Weyl group NH/H of H in G.

Let βp be the number of p-cells in X. Then Tf n has a CW -structure with βp + βp−1 cells of dimension p. Hence the von Neumann dimension of the cellular Hilbert N (Gn )-chain module Cp (Tf n ) is βp + βp−1 . 12 (2) that (2) bp (Tf n ) ≤ βp + βp−1 . We have shown 0 ≤ b(2) p (Tf ) ≤ βp + βp−1 . n Since βp + βp−1 is independent of n, the claim follows by taking the limit for n → ∞. e. 2 on page 53]). This is a weaker notion than the notion of a (locally trivial) fiber bundle with typical fiber F [269, chapter 4, section 5].

J∈J A directed set I is a non-empty set with a partial ordering ≤ such that for two elements i0 and i1 there exists an element i with i0 ≤ i and i1 ≤ i. A net 18 1. L2 -Betti Numbers (xi )i∈I in a topological space is a map from a directed set to the topological space. The net (xi )i∈I converges to x if for any neighborhood U of x there is an index i(U ) ∈ I such that xi ∈ U for each i ∈ I with i(U ) ≤ i. A net (fi )i∈I in B(H) converges strongly to f ∈ B(H) if for any v ∈ H the net (fi (v))i∈I converges to f (v) in H.