On spaces with periodic cohomology
Authors:
Alejandro Adem and Jeff H. Smith
Journal:
Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 1-6
MSC (2000):
Primary 57S30; Secondary 20J06
DOI:
https://doi.org/10.1090/S1079-6762-00-00074-3
Published electronically:
January 31, 2000
MathSciNet review:
1745517
Full-text PDF Free Access
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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.
- Francis X. Connolly and Stratos Prassidis, Groups which act freely on ${\bf R}^m\times S^{n-1}$, Topology 28 (1989), no. 2, 133β148. MR 1003578, DOI https://doi.org/10.1016/0040-9383%2889%2990016-5
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- Richard G. Swan, Periodic resolutions for finite groups, Ann. of Math. (2) 72 (1960), 267β291. MR 124895, DOI https://doi.org/10.2307/1970135
- C. T. C. Wall, Finiteness conditions for ${\rm CW}$ complexes. II, Proc. Roy. Soc. London Ser. A 295 (1966), 129β139. MR 211402, DOI https://doi.org/10.1098/rspa.1966.0230
- C. T. C. Wall, Periodic projective resolutions, Proc. London Math. Soc. (3) 39 (1979), no. 3, 509β553. MR 550082, DOI https://doi.org/10.1112/plms/s3-39.3.509
- Connolly, F. and Prassidis, S., Groups Which Act Freely on $R^m\times S^{n-1}$, Topology 28, pp. 133β148 (1989).
- Gorenstein, D., The Classification of Finite Simple Groups, Plenum Press (1983).
- Mislin, G. and Talelli, O., On Groups which Act Freely and Properly on Finite Dimensional Homotopy Spheres, preprint (1999).
- Oliver, R., Free Compact Group Actions on Products of Spheres, Springer-Verlag LNM 763, pp. 539β548 (Arhus 1978).
- Swan, R.G., Periodic Resolutions for Finite Groups, Annals of Mathematics 72, pp. 267β291 (1960).
- Wall, C.T.C., Finiteness Conditions for CW-complexes II, Proceedings Royal Society, Series A 295, pp. 129β139 (1966).
- Wall, C.T.C., Periodic Projective Resolutions, Proc. London Math. Soc. 39, pp. 509β533 (1979).
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Additional Information
Alejandro Adem
Affiliation:
Mathematics Department, University of Wisconsin, Madison, Wisconsin 53706
MR Author ID:
23100
Email:
adem@math.wisc.edu
Jeff H. Smith
Affiliation:
Mathematics Department, Purdue University, West Lafayette, Indiana 47907
Email:
jhs@math.purdue.edu
Keywords:
Group cohomology,
periodic complex
Received by editor(s):
October 27, 1999
Published electronically:
January 31, 2000
Additional Notes:
Both authors were partially supported by grants from the NSF
Communicated by:
Dave J. Benson
Article copyright:
© Copyright 2000
American Mathematical Society
