By Luther Pfahler Eisenhart

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The space E(G; FIN ) for a discrete group G is also called the classifying space for proper G-spaces since for any numerably proper G-space X there is up to Ghomotopy precisely one G-map from X to E(G, FIN ). 6], [336] and [342]. 29. Let X be a free G-CW -complex of finite type. Define its cellular L2 -chain complex and its cellular L2 -cochain complex by (2) C∗ (X) := l2 (G) ⊗ZG C∗ (X); ∗ C(2) (X) := homZG (C∗ (X), l2 (G)), where C∗ (X) is the cellular ZG-chain complex. If we fix a cellular basis for Cn (X) we obtain explicit isomorphisms 34 1.

Then pr ◦f is injective with dense image. Hence f and pr ◦f are weak isomorphisms of Hilbert spaces but pr is not. View the following diagram as a short exact sequence of Hilbert chain complexes which are concentrated in dimensions 0 and 1 0 −−−−→ 0 id −−−−→ H −−−−→ f H pr ◦f −−−−→ 0 pr 0 −−−−→ spanC (u) −−−−→ H −−−−→ spanC (u)⊥ −−−−→ 0 All the L2 -homology groups are trivial except the zero-th L2 -homology group of the left Hilbert chain complex 0 → spanC (u). Hence there cannot be a long weakly exact homology sequence.

Let D∗ be the ZG-chain complex which is concentrated in dimension 0 and 1 and has as first differential s∈S rs−1 ZG −−−−−−−→ ZG, s∈S where rs−1 is right multiplication with s − 1. There is a ZG-chain map f∗ : D∗ → C∗ (X) which is 0-connected. Hence it suffices because of the ar(2) gument in the proof of (1) to show b0 (l2 (G) ⊗ZG D∗ ) = 0. We have already seen that this is the same as the zero-th L2 -Betti number of the associated Hilbert N (G)-cochain complex homZG (D∗ , l2 (G)) which is the dimension of l2 (G)G .

### An Introduction To Differential Geometry With Use Of Tensor Calculus by Luther Pfahler Eisenhart

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