New sheaf theoretic methods in differential topology

Michael Weiss

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 5, page 549-567
  • ISSN: 0044-8753

Abstract

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The Mumford conjecture predicts the ring of rational characteristic classes for surface bundles with oriented connected fibers of large genus. The first proof in [11] relied on a number of well known but difficult theorems in differential topology. Most of these difficult ingredients have been eliminated in the years since then. This can be seen particularly in [7] which has a second proof of the Mumford conjecture, and in the work of Galatius [5] which is concerned mainly with a “graph” analogue of the Mumford conjecture. The newer proofs emphasize Tillmann’s theorem [23] as well as some sheaf-theoretic concepts and their relations with classifying spaces of categories. These notes are an overview of the shortest known proof, or more precisely, the shortest known reduction of the Mumford conjecture to the Harer-Ivanov stability theorems for the homology of mapping class groups. Some digressions on the theme of classifying spaces and sheaf theory are included for motivation.

How to cite

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Weiss, Michael. "New sheaf theoretic methods in differential topology." Archivum Mathematicum 044.5 (2008): 549-567. <http://eudml.org/doc/250306>.

@article{Weiss2008,
abstract = {The Mumford conjecture predicts the ring of rational characteristic classes for surface bundles with oriented connected fibers of large genus. The first proof in [11] relied on a number of well known but difficult theorems in differential topology. Most of these difficult ingredients have been eliminated in the years since then. This can be seen particularly in [7] which has a second proof of the Mumford conjecture, and in the work of Galatius [5] which is concerned mainly with a “graph” analogue of the Mumford conjecture. The newer proofs emphasize Tillmann’s theorem [23] as well as some sheaf-theoretic concepts and their relations with classifying spaces of categories. These notes are an overview of the shortest known proof, or more precisely, the shortest known reduction of the Mumford conjecture to the Harer-Ivanov stability theorems for the homology of mapping class groups. Some digressions on the theme of classifying spaces and sheaf theory are included for motivation.},
author = {Weiss, Michael},
journal = {Archivum Mathematicum},
keywords = {surface bundle; sheaf; classifying space; homological stability; surface bundle; sheaf; classifying space; homological stability},
language = {eng},
number = {5},
pages = {549-567},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {New sheaf theoretic methods in differential topology},
url = {http://eudml.org/doc/250306},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Weiss, Michael
TI - New sheaf theoretic methods in differential topology
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 5
SP - 549
EP - 567
AB - The Mumford conjecture predicts the ring of rational characteristic classes for surface bundles with oriented connected fibers of large genus. The first proof in [11] relied on a number of well known but difficult theorems in differential topology. Most of these difficult ingredients have been eliminated in the years since then. This can be seen particularly in [7] which has a second proof of the Mumford conjecture, and in the work of Galatius [5] which is concerned mainly with a “graph” analogue of the Mumford conjecture. The newer proofs emphasize Tillmann’s theorem [23] as well as some sheaf-theoretic concepts and their relations with classifying spaces of categories. These notes are an overview of the shortest known proof, or more precisely, the shortest known reduction of the Mumford conjecture to the Harer-Ivanov stability theorems for the homology of mapping class groups. Some digressions on the theme of classifying spaces and sheaf theory are included for motivation.
LA - eng
KW - surface bundle; sheaf; classifying space; homological stability; surface bundle; sheaf; classifying space; homological stability
UR - http://eudml.org/doc/250306
ER -

References

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