Secondary characteristic classes of surface bundles

Galatius, Søren

doi:10.2140/agt.2009.9.293

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Secondary characteristic classes of surface bundles

Søren Galatius

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 $E \to B$ a class $κ_{i} (E) \in H^{2 i} (B; ℤ)$ . It was proved by the author, Madsen and Tillman [J. Amer. Math. Soc. 19 (2006) 759-779] that the mod $p$ reduction $κ_{i} (E) \in H^{2 i} (B; ℤ ∕ p)$ vanishes when $i + 1$ is divisible by $(p - 1)$ . In this note we prove that the $p^{2}$ reduction $κ_{i} (E) \in H^{2 i} (B; ℤ ∕ p^{2})$ vanishes when $i + 1$ is divisible by $p (p - 1)$ . We also define for each integer $i \geq 1$ a characteristic class $λ_{i} (E) \in H^{2 i (p - 1) - 2} (B; ℤ ∕ p)$ which satisfies $p λ_{i} (E) = κ_{i (p - 1) - 1} (E) \in H^{*} (B; ℤ ∕ p^{2})$ .

Keywords

mapping class group, characteristic class, Toda bracket

Mathematical Subject Classification 2000

Primary: 55R40

References

Publication

Received: 27 February 2008

Revised: 26 September 2008

Accepted: 29 September 2008

Published: 19 February 2009

Authors

Søren Galatius
Department of Mathematics Stanford University Stanford, CA 94305 USA

Volume 9, issue 1 (2009)