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Jang Hyun Jo, Jong Bum Lee

Infinite-dimensional homotopy space forms

A free Γ-complex is a connected complex X together with an action of Γ which permutes freely the cells of X. Let Γ be a group in the class and X be an infinite-dimensional free Γ-complex which is homotopy equivalent to some sphere S^m, m > 1, and let Ω be the Euler class of X/Γ. Then we prove the following main results:

Theorem B. Suppose Γ induces a trivial action on H* (X). Then X/Γ is homotopy equivalent to a finite-dimensional complex if and only if Γ is torsion-free, or else the natural map H^m+1(Γ,ℤ) → Ĥ^m+1(Γ,ℤ) sends Ω to Ωˆ, which is an invertible element of the generalized Farrell-Tate cohomology ring of Γ, and m is odd.

Theorem C. Suppose Γ induces a nontrivial action on H* (X). Then X/Γ is homotopy equivalent to a finite-dimensional complex if and only if either

(1) Γ is torsion-free,

(2) Γ≅Γ₀ ⋊H where Γ₀ is torsion-free and H is isomorphic to ℤ/2, res^Γ_H(Ω) ≠ 0, and m is even, or else

(3) all the torsion elements of Γ lie in Γ₀, and Ω is mapped to Ωˆ₀ for which some power of Ωˆ₀ is an invertible element of the generalized Farrell-Tate cohomology ring of Γ₀, and m is odd.

Forum Mathematicum, Walter de Gruyter

Print ISSN: 0933-7741
Volume: 18, 03/2006
Pages: 305 - 322

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