Algebraic & Geometric Topology 9 (2009) 293–303

DOI: 10.2140/agt.2009.9.293

Abstract

The Miller–Morita–Mumford classes associate to an oriented surface bundle E → B a class κ_i(E) in H²ⁱ(B; Z). It was proved by the author, Madsen and Tillman [J. Amer. Math. Soc. 19 (2006) 759-779] that the mod p reduction κ_i(E) in H²ⁱ(B; Z ∕ p) vanishes when i + 1 is divisible by (p− 1). In this note we prove that the p² reduction κ_i(E) in H²ⁱ(B; Z ∕ p²) vanishes when i + 1 is divisible by p(p− 1). We also define for each integer i ≥ 1 a characteristic class λ_i(E) in H^{2i(p−1)−2}(B; Z ∕ p) which satisfies pλ_i(E) = κ_i(p−1)−1(E) in H^*(B; Z ∕ p²).

Keywords

mapping class group, characteristic class, Toda bracket

Mathematical Subject Classification

Primary: 55R40

References

Publication

Received: 27 February 2008
Revised: 26 September 2008
Accepted: 29 September 2008
Published: 19 February 2009

Authors