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**Sample 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.