By Wolfgang Lück

In algebraic topology a few classical invariants - reminiscent of Betti numbers and Reidemeister torsion - are outlined for compact areas and finite staff activities. they are often generalized utilizing von Neumann algebras and their lines, and utilized additionally to non-compact areas and limitless teams. those new L2-invariants comprise very attention-grabbing and novel details and will be utilized to difficulties bobbing up in topology, K-Theory, differential geometry, non-commutative geometry and spectral thought. it really is fairly those interactions with diversified fields, that make L2-invariants very robust and intriguing. The e-book provides a accomplished creation to this zone of analysis, in addition to its most modern effects and advancements. it truly is written in a fashion which allows the reader to select his favorite subject and to discover the outcome he or she is attracted to speedy and with out being pressured to head via different fabric.

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**Example text**

We use induction over k. The beginning k = −1 is trivial, the induction step from k − 1 to k ≥ 0 is done as follows. Suppose that the k-skeleton Zk is obtained from the (k − 1)-skeleton Zk−1 by attaching equivariant cells S 1 /Hi × S k−1 −−−−→ Zk−1 i∈I i∈I S 1 /Hi × Dk −−−−→ Zk where Hi ⊂ S 1 is a finite subgroup for each i ∈ I. Then ES 1 ×S 1 Zk is the pushout i∈I ES 1 ×S 1 (S 1 /Hi × S k−1 ) −−−−→ ES 1 ×S 1 Zk−1 i∈I ES 1 ×S 1 (S 1 /Hi × Dk ) −−−−→ ES 1 ×S 1 Zk and S 1 \Zk is the pushout: S k−1 −−−−→ S 1 \Zk−1 i∈I i∈I Dk −−−−→ S 1 \Zk The projections ES 1 ×S 1 Y → S 1 \Y for Y = Zk−1 , i∈I S 1 /Hi × S k−1 and i∈I S 1 /Hi × Dk are rational cohomology equivalences by the induction hypothesis and because BHi → {∗} is one for each i ∈ I.

In particular a free G-CW -complex is always proper. However, not every free G-space is proper. A G-space is called cocompact if G\X is compact. A G-CW -complex X is finite if X has only finitely many equivariant cells. A G-CW -complex is finite if and only if it is cocompact. A G-CW -complex X is of finite type if each n-skeleton is finite. It is called of dimension ≤ n if X = Xn and finite dimensional if it is of dimension ≤ n for some integer n. A free G-CW -complex X is the same as a regular covering X → Y of a CW -complex Y with G as group of deck transformations.

For an interval I ⊂ [0, 1] let χI ∈ N (Z) = L∞ (S 1 ) be the characteristic function of the subset {exp(2πit) | t ∈ I}. Define two Hilbert N (Z)-modules by the orthogonal Hilbert sums ∞ U= im(χ[0,2−n ] ); n=1 ∞ V = im(χ[1/(n+1),1/n] ), n=1 where im(χI ) is the direct summand in l2 (Z) = L2 (S 1 ) given by the projection χI ∈ N (Z). 11 imply dimN (Z) (U ) = dimN (Z) (V ) = 1. We want to show that U is not finitely generated. This is not obvious, for instance, V is isomorphic to the Hilbert N (Z)-module l2 (Z) and in particular finitely generated, although it is defined as an infinite Hilbert sum of nontrivial Hilbert N (Z)-modules.