Subject: Answer to question on Cohomology of Symmetric groups Date: Thu, 20 Dec 101 11:12:31 +0000 (GMT) From: Chris T Stretch To: dmd1@lehigh.edu Re: Cohomology of infinite symmetric group The restriction of the second Chern class (or the first Pontryagin class) has order either 12 or 24 in H^4(S_\infinity). The class can be detected by restricting to the cyclic groups C_n contained in S_n as the powers of an n-cycle. H^*(C_n,Z)=Z[x]/(nx) with x in dimension 2. The standard representation of U_n restricts to the regular representation of C_n, with Chern polynomial \prod_{j=0}^{n-1} (1+jxt) The required restriction is the coefficient of t^2, this is 2x^2 mod 3, for n=3 and 3x^2 mod 4 for n=4. Thus the Chern class has order at least 12. The Chern classes of any rational representation of a finite group have order dividing the denominators of Bernoulli numbers by Galois invariance. For the second Chern class the order divides 24. Chris Stretch University of Ulster ct.stretch@ulst.ac.uk