Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles (Q1392742)

In previous work, the author obtained an integer-valued homomorphism \(d\) on the mapping class group \({\mathcal M}_{g,1}\) of a closed orientable surface of genus \(g\) (relative to an embedded disk). When restricted to the subgroup \({\mathcal K}_{g,1}\) of \({\mathcal M}_{g,1}\) generated by Dehn twists around seperating simple closed curves, this invariant \(d\) is basically equivalent to the Casson invariant of the homology 3-sphere obtained by using the mapping class as gluing map in a Heegaard splitting (there is a second term which, in contrast to \(d\), is easy to compute). The main purpose of the present paper is to give a geometrical interpretation of the invariant \(d\): it is proved that \(d\) is equal to Hirzebruch's signature defect of the mapping torus associated to the mapping class, with respect to a certain canonical framing of its tangent bundle. Then this is generalized from the setting of 3-manifolds to the case of higher dimensions considering surface bundles whose holonomy is contained in \({\mathcal K}_{g,1}\). This leads to the definition of certain secondary characteristic classes of surface bundles \(d_i\in H^{4i-3} ({\mathcal K}_{g,1}, \mathbb{Q})\), where the first class \(d_1\) is the above invariant \(d\). ``We expect that other secondary characteristic classes of higher degrees will also reflect certain unknown structure of the mapping class group''.