Free actions of -groups on products of lens spaces

Author:
Ergün Yalçin

Journal:
Proc. Amer. Math. Soc. **129** (2001), 887-898

MSC (2000):
Primary 57S25; Secondary 20J06, 20D15

DOI:
https://doi.org/10.1090/S0002-9939-00-05756-7

Published electronically:
September 20, 2000

MathSciNet review:
1792188

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Let be an odd prime number. We prove that if acts freely on a product of equidimensional lens spaces, then . This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian -groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A -group can act freely on a product of lens spaces with a trivial action on homology if and only if and has the -extension property. The main technique is to study group extensions associated to free actions.

**1.**A. Adem,*actions on*, Trans. A.M.S.**300**(1987), 791-809. MR**88b:57037****2.**A. Adem and J. Smith,*On spaces with periodic cohomology*, preprint.**3.**A. Adem and D.J. Benson,*Abelian groups acting on products of spheres*, Math. Z.**228**(1998), 705-712. MR**99k:57033****4.**A. Adem and W. Browder,*The free rank of symmetry on*,

Invent. Math.**92**(1988), 431-440. MR**89e:57034****5.**A. Adem and E. Yalçin,*On some examples of group actions and group extensions*, Journal of Group Theory**2**(1999), 69-79. MR**2000b:57050****6.**C. Allday,*Elementary abelian**-group actions on lens spaces*, Topology Hawaii (Honolulu, HI, 1990), 1-11, World Sci. Publishing, River Edge, NJ, 1992. MR**93e:57068****7.**K. Alzubaidy,*Free actions of**-groups**on*, Glasgow Math. J.**23**(1982), 97-101. MR**83i:57032****8.**J.F. Carlson,*Depth and transfer maps in the cohomology of groups*, Math. Z.**218**(1995), 461-468. MR**95m:20058****9.**G. Carlsson,*On the non-existence of free actions of elementary abelian groups on products of spheres*, Amer. J. Math.**102**(1980), 1147-1157. MR**82a:57038****10.**G. Carlsson,*On the rank of abelian groups acting freely on*, Invent. Math.**69**(1982), 393-400. MR**84e:57033****11.**A. Heller,*A note on spaces with operators*, Ill. J. Math.**3**(1959), 98-100. MR**21:6588****12.**G. Lewis,*Free actions on*, Trans. A.M.S.**132**(1968), 531-540. MR**37:4809****13.**B. Oliver,*Free compact group actions on products of spheres*, Algebraic Topology, Aarhus 1978, Lecture Notes in Math.**763**, Springer-Verlag, 1979, 539-548. MR**81k:55005****14.**A. Yu. Ol'shanskii,*The number of generators and orders of abelian subgroups of finite**-groups*, Math. Notes**23**(1978), 183-185.**15.**U. Ray,*Free linear actions of finite groups on products of spheres*, J. Algebra**147**(1992), 456-490. MR**93d:20019****16.**T.S. Weigel,*Combinatorial properties of**-central groups*, preprint.**17.**T.S. Weigel,*-Central groups and Poincaré duality*, to appear in Trans. A.M.S. CMP**98:12****18.**N. Yagita,*On the dimension of spheres whose product admits a free action by a non-abelian group*, Quart. J. Math.**36**(1985), 117-127. MR**86h:57041****19.**E. Yalçin,*Group actions and group extensions*, Trans. A.M.S.**352**(2000), no. 6, 2689-2700. CMP**2000:10**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
57S25,
20J06,
20D15

Retrieve articles in all journals with MSC (2000): 57S25, 20J06, 20D15

Additional Information

**Ergün Yalçin**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Address at time of publication:
Department of Mathematics, Bilkent University, Ankara, Turkey 06533

Email:
yalcine@math.mcmaster.ca

DOI:
https://doi.org/10.1090/S0002-9939-00-05756-7

Keywords:
Group actions,
products of lens spaces,
group extensions

Received by editor(s):
May 12, 1999

Published electronically:
September 20, 2000

Communicated by:
Ralph Cohen

Article copyright:
© Copyright 2000
American Mathematical Society