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A complement to the theory of equivariant finiteness obstructions. (English) Zbl 0882.57018

The finiteness obstruction \(w(X)\in K_0 (\mathbb{Z} [\pi_1(X)])\) of C. T. C. Wall answers the question when a finitely dominated CW-complex \(X\) is homotopy finite. In the equivariant setting for a compact Lie group \(G\) acting on \(X\), the author showed in [P. Andrzejewski, Lect. Notes Math. 1217, 11-25 (1986; Zbl 0621.57009)] that a construction of equivariant finiteness obstructions leads to a family \(\{w^H_\alpha (X)\}\) of elements of the groups \(K_0(\mathbb{Z} [\pi_0 (WH(X)^*_\alpha)])\), where \(H\) runs over the closed subgroups of \(G \) and the index \(\alpha\) indicates the different components of \(X^H\). In the present paper the author proves that every family \(\{w_\alpha^H\}\) of such elements can be realized as the family of equivariant finiteness obstructions of an appropriate finitely dominated \(G\)-complex. This result is used to show the existence of a natural equivalence between the geometric finiteness obstruction introduced in [W. Lück, Transformation groups and algebraic \(K\)-theory, Lect. Notes Math. 1408 (1989; Zbl 0679.57022)] and the obstructions \(\{w_\alpha^H (X)\}\).

MSC:

57Q12 Wall finiteness obstruction for CW-complexes
57S10 Compact groups of homeomorphisms
55S91 Equivariant operations and obstructions in algebraic topology
19J05 Finiteness and other obstructions in \(K_0\)
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