**Title:** Relations in the cohomology of classifying spaces of manifold bundles.\

**Speaker:** Ilya Grigoriev

**Speaker Info:** Stanford

**Brief Description:**

**Special Note**:

**Abstract:**

The n-th cohomology of the classifying space of surface bundles with fiber of genus g, denoted H^n(BDiff \Sigma_g), 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, Oscar Randal-Williams, Ib Madsen, and Alexander Berglund have obtained similar results for a high-dimensional generalization of this space, with the surface \Sigma_g replaced by the connect sum of g copies of the cross product 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 method for producing a vast family of elements in the kernel that also works in the the high-dimensional case (for odd k >= 1). This method is based on a refinement of results in the surface case due to Oscar Randal-Williams. 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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