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Finite group actions on products of spheres. (English) Zbl 0872.57041
Summary: We make some contributions to the theory of actions of finite groups on products of spheres. Suppose that the group $$Z_q^r$$ acts freely on the product of $$k$$ copies of spheres. Question: Is $$r\leq k$$? We solve this question for several values of $$r$$ and $$k$$.

##### MSC:
 57S17 Finite transformation groups
##### Keywords:
actions; finite groups; products of spheres
Full Text:
##### References:
 [1] A. Adem and W. Browder,The Free Rank of Symmetry of (S n)k, Invent. Math.92 (1988), 431–440. · Zbl 0644.57022 · doi:10.1007/BF01404462 [2] W. Browder,Cohomology and Group Actions, Invent. Math.71 (1983), 599–607. · Zbl 0529.57021 · doi:10.1007/BF02095996 [3] G. Carlsson,On the Rank of Abelian Groups Acting Freely on (S n)k, Invent Math.69 (1982), 393–400. · Zbl 0517.57020 · doi:10.1007/BF01389361 [4] G. Carlsson,On the Non-Existence of Free Actions of Elementary Abelian Groups on Products of Spheres, Am. J. Math.102 (1980), 1147–1157. · Zbl 0454.57028 · doi:10.2307/2374182 [5] H. Cartan and S. Eilenberg,Homological Algebra, Princeton University Press (1956). [6] P. Conner,On the Action of a Finite Group on S n$$\times$$Sn, Ann. Math.66 (1957), 586–588. · Zbl 0079.38904 · doi:10.2307/1969910 [7] A. Heller,A Note on Spaces With Operators, Ill. J. Math.3, (1959), 98–100. · Zbl 0084.38803 [8] H. Nagao and T. Nakayama,On the Structure of (M 0)- and (M$$\mu$$)-modules, Math. Zeit.,59 (1953), 164–170. · Zbl 0051.26305 · doi:10.1007/BF01180248
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