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The Weak b-principle: Mumford Conjecture

  Published:2015-09-07
 Printed: Apr 2016
  • Rustam Sadykov,
    Department of Mathematics , CINVESTAV, Av. Instituto Politecnico Nacional 2508, Col. San Pedro Zacatenco, Mexico, D.F. CP 07360
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Abstract

In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension $2$ approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension $2$ to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem.
Keywords: generalized cohomology theories, fold singularities, h-principle, infinite loop spaces generalized cohomology theories, fold singularities, h-principle, infinite loop spaces
MSC Classifications: 55N20, 53C23 show english descriptions Generalized (extraordinary) homology and cohomology theories
Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces
55N20 - Generalized (extraordinary) homology and cohomology theories
53C23 - Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces
 

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