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Bulletin of the American Mathematical Society

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Chern-Weil forms and abstract homotopy theory


Authors: Daniel S. Freed and Michael J. Hopkins
Journal: Bull. Amer. Math. Soc. 50 (2013), 431-468
MSC (2010): Primary 58Axx; Secondary 53C05, 53C08, 55U35
DOI: https://doi.org/10.1090/S0273-0979-2013-01415-0
Published electronically: April 17, 2013
MathSciNet review: 3049871
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Abstract: We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal $ G$-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebra as the de Rham complex of a specific simplicial sheaf, and similarly give a new interpretation of the Weil model in equivariant de Rham theory. There is an appendix proving a general theorem about set-theoretic transformations of polynomial functors.


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Additional Information

Daniel S. Freed
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
Email: dafr@math.utexas.edu

Michael J. Hopkins
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: mjh@math.harvard.edu

DOI: https://doi.org/10.1090/S0273-0979-2013-01415-0
Received by editor(s): January 24, 2013
Published electronically: April 17, 2013
Additional Notes: The work of the first author was supported by the National Science Foundation under grants DMS-0603964, DMS-1207817, and DMS-1160461
The work of the second author was supported by the National Science Foundation under grants DMS-0906194, DMS-0757293, DMS-1158983
Dedicated: In memory of Dan Quillen
Article copyright: © Copyright 2013 American Mathematical Society

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