#### Volume 9, issue 1 (2009)

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Secondary characteristic classes of surface bundles

### Søren Galatius

Algebraic & Geometric Topology 9 (2009) 293–303
##### Abstract

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

##### Keywords
mapping class group, characteristic class, Toda bracket
Primary: 55R40