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On the dimension of infinite covers

Author(s): W. G. Dwyer; S. Stolz; L. R. Taylor
Journal: Proc. Amer. Math. Soc. 124 (1996), 2235-2239.
MSC (1991): Primary 55U15, 57P10
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Abstract: We prove the following theorem and some generalizations.

Theorem.. Let $X$ be a connected CW complex which satisfies Poincaré duality of dimension $n\ge 4$. For any subgroup $H$ of $\pi _1(X)$, let $X_H$ denote the cover of $X$ corresponding to $H$. If $H$ has infinite index in $\pi _1(X)$, then $X_H$ is homotopy equivalent to an $(n-1)$-dimensional CW complex.

References:

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W. Browder, Poincaré spaces, their normal fibrations and surgery, Invent. Math. 17 (1972), 191--202. MR 48:5086

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K. S. Brown, Cohomology of groups, Graduate Texts in Math., vol. 87, Springer-Verlag, New York, 1982. MR 83k:20002

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H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, NJ, 1956. MR 17:1040e

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J. A. Hillman, A homotopy fibration theorem, Topology Appl. 33 (1989), 151--161. MR 90k:57023

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R. Strebel, A remark on subgroups of infinite index in Poincaré duality groups, Comment Math. Helv. 52 (1977), 317--324. MR 56:15793

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C. T. C. Wall, Finiteness conditions for CW complexes, Ann. of Math. (2) 81 (1965), 56--69. MR 30:1515

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Additional Information:

W. G. Dwyer
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: dwyer.1@nd.edu

S. Stolz
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: stolz.1@nd.edu

L. R. Taylor
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: taylor.2@nd.edu

DOI: 10.1090/S0002-9939-96-03250-9
PII: S 0002-9939(96)03250-9
Keywords: Infinite covers, dimension
Received by editor(s): May 10, 1994
Received by editor(s) in revised form: November 18, 1994
Additional Notes: Partially supported by the National Science Foundation.
Communicated by: Thomas Goodwillie
Copyright of article: Copyright 1996, American Mathematical Society

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