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
Abstract |
References |
Similar Articles |
Additional Information
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
- Daniel Gorenstein, The classification of finite simple groups. Vol. 1, The University Series in Mathematics, Plenum Press, New York, 1983. Groups of noncharacteristic $2$ type. MR 746470
- Mislin, G. and Talelli, O., On Groups which Act Freely and Properly on Finite Dimensional Homotopy Spheres, preprint (1999).
- Robert Oliver, Free compact group actions on products of spheres, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 539β548. MR 561237
- 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).
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (2000):
57S30,
20J06
Retrieve articles in all journals
with MSC (2000):
57S30,
20J06
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