# free action on mod p cohomology sphere

It is well known that the group $G=\mathbb Z_2\oplus\mathbb Z_2$ cannot act freely on mod 2 cohomology n-sphere.

Is it also true that this group $G$ cannot act freely on any mod p cohomology n-sphere where p is an odd prime. ?

• Is there any reason to think it might be true? – HJRW Oct 23 '13 at 12:23
• proof of this result is given in "compact transformation groups book by breadon" (chap III sec 8) with little modification in the proof I feel this should be true.!!! cross checking needed!! – user168639 Oct 29 '13 at 17:18