Which Finite Groups Act Freely on Spheres?
نویسنده
چکیده
For those who know about group cohomology will know that if a group acts freely on sphere, then it has periodic cohomology. Now the group Zp×Zp does not have periodic cohomology, (just use the Künneth formula again) therefore it cannot act freely on any sphere. For those who do not know about group cohomology a finite group having periodic cohomology is equivalent to all the abelian subgroups being cyclic. Is periodic cohomology a sufficient condition for a finite group to act freely on a sphere? For example, S3 (the symmetric group on three elements) has periodic cohomology, so can it act freely on some sphere. We shall see that the answer to this question is no.
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