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The Weak b-principle: Mumford Conjecture
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- Journal:
- Canadian Journal of Mathematics / Volume 68 / Issue 2 / 01 April 2016
- Published online by Cambridge University Press:
- 20 November 2018, pp. 463-480
- Print publication:
- 01 April 2016
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- Article
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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.
