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We define a generalized notion of cohomological periodicity for a connected CW-complex $X$, and show that it is equivalent to the existence of an oriented spherical fibration over $X$ with total space homotopy equivalent to a finite dimensional complex. As applications we characterize discrete groups which can act freely and properly on some $\mathbb R^n\times \mathbb S^m$, show that every rank two $p$-group acts freely on a homotopy product of two spheres and construct exotic free actions of many simple groups on such spaces.

Copyright 2000 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 06 (2000), pp. 1-6
• Publisher Identifier: S 1079-6762(00)00074-3
• 2000 Mathematics Subject Classification. Primary 57S30; Secondary 20J06
• Key words and phrases. Group cohomology, periodic complex
• Received by the editors October 27, 1999
• Posted on January 31, 2000
• Communicated by Dave J. Benson
• Comments (When Available)

Alejandro Adem

Mathematics Department, University of Wisconsin, Madison, Wisconsin 53706

E-mail address: adem@math.wisc.edu

Jeff H. Smith

Mathematics Department, Purdue University, West Lafayette, Indiana 47907

E-mail address: jhs@math.purdue.edu

Both authors were partially supported by grants from the NSF