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## Details

Material Type: | Document, Thesis/dissertation, Internet resource |
---|---|

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Ilya Grigoriev; Søren Galatius; Ralph L Cohen; Ravi Vakil; Stanford University. Department of Mathematics. |

OCLC Number: | 855466364 |

Notes: | Submitted to the Department of Mathematics. |

Description: | 1 online resource. |

Responsibility: | Ilya Grigoriev. |

### Abstract:

We study a generalization of the tautological subring of the cohomology of the moduli space of Riemann surfaces to manifold bundles. The infinitely many "generalized Miller-Morita-Mumford classes" determine a map R from a free polynomial algebra to the cohomology of the classifying space of manifold bundles. In the case when M is the connected sum of g copies of the product of spheres (S^d times S^d), with d odd, we find numerous polynomials in the kernel of the map R and show that the image of R is a finitely generated ring. Some of the elements in the kernel do not depend on d. Our results contrast with the fact that the map R is an isomorphism in a range of cohomological degrees that grows linearly with g. This is known from theorems of Madsen-Weiss and Harer for the case of surfaces (d=1) and from the recent work of Soren Galatius and Oscar Randal-Williams in higher dimensions. For surfaces, the image of the map R coincides with the classical tautological ring, as introduced by Mumford.

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