## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.math.psu.edu/era/.
**

On spaces with periodic cohomology
**This journal is archived by the American Mathematical
Society. The master copy is available at
http://www.ams.org/era/
**

## On spaces with periodic cohomology

### Alejandro Adem and Jeff H. Smith

**Abstract.**
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*

**Retrieve entire article **

#### 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

*Electronic Research Announcements of the AMS *Home page