On the geometric nature of characteristic classes of surface bundles (Q2914381)

A surface bundle is a fiber bundle \(E\to B\) with fiber \(\Sigma_g\), the closed oriented surface of genus \(g\), for \(g \geq 2\). When the base space \(B\) is a closed oriented \(i\)-dimensional manifold, a characteristic cohomology class \(c\) of degree \(i\) can be paired with the fundamental homology class of \(B\) to yield a numerical invariant, the associated characteristic number.NEWLINENEWLINEThe authors consider the characteristic numbers associated to the Miller-Morita-Mumford (MMM) classes for surface bundles, characteristic classes defined using the Euler classes of vector bundles constructed from the surface bundle. Based on the results of \textit{I. Madsen} and \textit{M. Weiss} [Ann. Math. (2) 165, No. 3, 843--941 (2007; Zbl 1156.14021)], all characteristic classes for surface bundles that are stable under fiberwise connected sums with trivial \(\Sigma_1\)-bundles are polynomials in the MMM classes.NEWLINENEWLINEThe authors indicate a general construction of closed, oriented manifolds that fiber as a surface bundle in more than one way and consider the question of whether the characteristic numbers depend on the choice of fibering. A characteristic class is called \textit{geometric} if the associated characteristic number depends only on the diffeomorphism class of the total space.NEWLINENEWLINEThe first main result of the paper is that odd MMM classes are geometric. In fact, it is demonstrated that odd MMM classes are \textit{geometric with respect to smooth cobordism}, meaning that the associated characteristic numbers depend only on the smooth cobordism class of the total space. As a corollary, the odd MMM classes are demonstrated to be obstructions to the existence of a fibering with fiber a surface of small genus for any element of a cobordism class.NEWLINENEWLINEIn contrast to the odd case, it is observed that no polynomial in the even MMM classes is cobordism invariant. However, it is demonstrated that if one restricts to holomorphic fibrations that are topologically surface bundles, every MMM class is geometric with respect to complex cobordism.NEWLINENEWLINEIn addition, the authors discuss several related open questions and possible generalizations to generalized MMM classes and vector bundles.