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Licensed Unlicensed Requires Authentication Published by De Gruyter May 17, 2006

Infinite-dimensional homotopy space forms

Jang Hyun Jo EMAIL logo and Jong Bum Lee
From the journal

Abstract

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 Sm, 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 Hm+1(Γ,ℤ) → Ĥm+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≅Γ0H where Γ0is torsion-free and H is isomorphic to ℤ/2, resΓH(Ω) ≠ 0, and m is even, or else

(3) all the torsion elements of Γ lie in Γ0, and Ω is mapped to Ωˆ0for which some power of Ωˆ0is an invertible element of the generalized Farrell-Tate cohomology ring of Γ0, and m is odd.


(Communicated by Frederick R. Cohen)


Received: 2003-05-18
Published Online: 2006-05-17
Published in Print: 2006-03-21

© Walter de Gruyter

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