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4:00 pm Monday, March 4, 2013 Topology Seminar: Relations among characteristic classes of manifold bundlesby Ilya Grigoriev (Stanford) in HB 427- The characteristic classes of surface bundles with fiber of genus g coincide with the elements of the cohomology of the classifying space of surface bundles, denoted BDiff \Sigma_g. The n-th cohomology of this space is known in the "stable range" (n <= (2g-2)/3) by theorems of Madsen-Weiss, Harer, and others. In this range, the map from a free algebra generated by the so-called "kappa-classes" to H^*(BDiff \Sigma_g) is an isomorphism. Recently, Soren Galatius and Oscar Randal-Williams have obtained similar results for the case where the surface \Sigma_g is replaced with a certain high-dimensional manifold, namely the connect sum of g copies of the product of spheres S^k x S^k. Outside the stable range, the kernel of the above-mentioned map for surfaces has been studied by Morita, Faber, Looijenga, Pandharipande and many others. In this talk, I will describe a vast family of elements in the kernel that also works in the the high-dimensional case (for odd k >= 1). The kernel is large enough to imply that the image of this map ("the tautological subring") is finitely-generated for all odd k, rationally, even though there are infinitely many kappa classes. It also implies upper bounds on the stable range of cohomology for fixed g and k.
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