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

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.**Alejandro Adem,*𝑍/𝑝𝑍 actions on (𝑆ⁿ)^{𝑘}*, Trans. Amer. Math. Soc.**300**(1987), no. 2, 791–809. MR**876479**, 10.1090/S0002-9947-1987-0876479-6**2.**A. Adem and J. Smith,*On spaces with periodic cohomology*, preprint.**3.**A. Adem and D. J. Benson,*Elementary abelian groups acting on products of spheres*, Math. Z.**228**(1998), no. 4, 705–712. MR**1644440**, 10.1007/PL00004637**4.**Alejandro Adem and William Browder,*The free rank of symmetry of (𝑆ⁿ)^{𝑘}*, Invent. Math.**92**(1988), no. 2, 431–440. MR**936091**, 10.1007/BF01404462**5.**Alejandro Adem and Ergün Yalçın,*On some examples of group actions and group extensions*, J. Group Theory**2**(1999), no. 1, 69–79. MR**1670325**, 10.1515/jgth.1999.007**6.**Christopher Allday,*Elementary abelian 𝑝-group actions on lens spaces*, Topology Hawaii (Honolulu, HI, 1990) World Sci. Publ., River Edge, NJ, 1992, pp. 1–11. MR**1181477****7.**Kahtan Alzubaidy,*Free actions of 𝑝-groups (𝑝>3) on 𝑆ⁿ×𝑆ⁿ*, Glasgow Math. J.**23**(1982), no. 2, 97–101. MR**663134**, 10.1017/S0017089500004857**8.**Jon F. Carlson,*Depth and transfer maps in the cohomology of groups*, Math. Z.**218**(1995), no. 3, 461–468. MR**1324540**, 10.1007/BF02571916**9.**Gunnar Carlsson,*On the nonexistence of free actions of elementary abelian groups on products of spheres*, Amer. J. Math.**102**(1980), no. 6, 1147–1157. MR**595008**, 10.2307/2374182**10.**Gunnar Carlsson,*On the rank of abelian groups acting freely on (𝑆ⁿ)^{𝑘}*, Invent. Math.**69**(1982), no. 3, 393–400. MR**679764**, 10.1007/BF01389361**11.**Alex Heller,*A note on spaces with operators*, Illinois J. Math.**3**(1959), 98–100. MR**0107866****12.**Gene Lewis,*Free actions on 𝑆ⁿ×𝑆ⁿ*, Trans. Amer. Math. Soc.**132**(1968), 531–540. MR**0229235**, 10.1090/S0002-9947-1968-0229235-4**13.**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. 539–548. MR**561237****14.**A. Yu. Ol'shanskii,*The number of generators and orders of abelian subgroups of finite**-groups*, Math. Notes**23**(1978), 183-185.**15.**Urmie Ray,*Free linear actions of finite groups on products of spheres*, J. Algebra**147**(1992), no. 2, 456–490. MR**1161304**, 10.1016/0021-8693(92)90216-9**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.**Nobuaki Yagita,*On the dimension of spheres whose product admits a free action by a nonabelian group*, Quart. J. Math. Oxford Ser. (2)**36**(1985), no. 141, 117–127. MR**780356**, 10.1093/qmath/36.1.117**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