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Commentarii Mathematici Helvetici

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Volume 96, Issue 1, 2021, pp. 1–45
DOI: 10.4171/CMH/505

Published online: 2021-03-12

Geometric cycles and characteristic classes of manifold bundles

Bena Tshishiku[1]

(1) Harvard University, Cambridge, USA

We produce new cohomology for non-uniform arithmetic lattices $\Gamma < \mathrm {SO}(p,q)$ using a technique of Millson–Raghunathan. From this, we obtain new characteristic classes of manifold bundles with fiber a closed $4k$-dimensional manifold $M$ with indefinite intersection form of signature $(p,q)$. These classes are defined on finite covers of $B$ Diff $(M)$ and are shown to be nontrivial for $M=\#_g(S^{2k}\times S^{2k})$. In this case, the classes produced live in degree $g$ and are independent from the algebra generated by the stable (i.e. MMM) classes. We also give an application to bundles with fiber a K3 surface.

Keywords: Arithmetic groups, characteristic classes, manifold bundles

Tshishiku Bena: Geometric cycles and characteristic classes of manifold bundles. Comment. Math. Helv. 96 (2021), 1-45. doi: 10.4171/CMH/505