Secondary characteristic classes of surface bundles

Galatius, Søren¹

¹Department of Mathematics, Stanford University, Stanford, CA 94305, USA

Algebraic and Geometric Topology, Tome 9 (2009) no. 1, pp. 293-303

Résumé

The Miller–Morita–Mumford classes associate to an oriented surface bundle E → B a class κi(E) ∈ 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 κi(E) ∈ H2i(B; ℤ∕p) vanishes when i + 1 is divisible by (p − 1). In this note we prove that the p2 reduction κi(E) ∈ 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 λi(E) ∈ H2i(p−1)−2(B; ℤ∕p) which satisfies pλi(E) = κi(p−1)−1(E) ∈ H∗(B; ℤ∕p2).

Détail

DOI : 10.2140/agt.2009.9.293

Keywords: mapping class group, characteristic class, Toda bracket

Bibliographie

[1] S Galatius, I Madsen, U Tillmann, Divisibility of the stable Miller–Morita–Mumford classes, J. Amer. Math. Soc. 19 (2006) 759

[2] I Madsen, U Tillmann, The stable mapping class group and $Q(\mathbb{C} P^\infty_+)$, Invent. Math. 145 (2001) 509

[3] H Toda, Composition methods in homotopy groups of spheres, Annals of Math. Studies 49, Princeton Univ. Press (1962)