Geometry & Topology Monographs 11 (2007) 179–200

DOI: 10.2140/gtm.2007.11.179

arXiv: 0903.4973

Abstract

Let P be the extraspecial p–group of order p²ⁿ⁺¹, of p–rank n + 1, and of exponent p if p > 2. Let Z be the center of P and let κ_n,r be the characteristic classes of degree 2ⁿ − 2^r (resp. 2(pⁿ − p^r)) for p = 2 (resp. p > 2), 0 ≤ r ≤ n − 1, of a degree pⁿ faithful irreducible representation of P. It is known that, modulo nilradical, the ιth powers of the κ_n,r’s belong to T = Im(H^∗(P∕Z,F_p)∕H^∗(P,F_p)∕), with ι = 1 if p = 2, ι = p if p > 2. We obtain formulae in H^∗(P,F_p)∕ relating the κ_n,r^ι terms to the ones of fewer variables. For p > 2 and for a given sequence r₀,…,r_n−1 of non-negative integers, we also prove that, modulo-nilradical, the element ∏ _riκ_n,i^r_i belongs to T if and only if either r₀ ≥ 2, or all the r_i are multiple of p. This gives the determination of the subring of invariants of the symplectic group Sp_2n(F_p) in T .

Keywords

Extraspecial p–groups, Chern classes, Stiefiel–Whitney classes, Evens norm, transfer

Mathematical Subject Classification

Primary: 20J06

Secondary: 55S10

References

Publication

Received: 30 November 2004
Accepted: 23 January 2005

Authors