Infinite-dimensional homotopy space forms

초록

A free Gamma-complex is a connected complex X together with an action of Gamma which permutes freely the cells of X. Let Gamma be a group in the class H1F(6) and X be an infinite-dimensional free Gamma-complex which is homotopy equivalent to some sphere S-m, m > 1, and let Omega be the Euler class of X/Gamma. Then we prove the following main results: Theorem B. Suppose Gamma induces a trivial action on H*(X). Then X/Gamma is homotopy equivalent to a finite-dimensional complex if and only if Gamma is torsion free, or else the natural map Hm+1(Gamma, Z) -> Hm+1(Gamma, Z) sends Omega to Omega, which is an invertible element of the generalized Farrell-Tate cohomology ring of Gamma, and m is odd. Theorem C. Suppose Gamma induces a nontrivial action on H*(X). Then X/Gamma is homotopy equivalent to a finite-dimensional complex if and only if either (1) Gamma is torsion free, (2) Gamma congruent to Gamma(0) x H where Gamma(0) is torsion free and His isomorphic to Z/2, res(H)(Gamma)(Omega) not equal 0, and m is even, or else (3) all the torsion elements of Gamma lie in Gamma(0), and Omega is mapped to Omega(0) for which some power of Omega(0) is an invertible element of the generalized Farrell-Tate cohomology ring of Gamma(0), and m is odd. 2000 Mathematics Subject Classification.

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