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Deformation spaces and normal forms around transversals
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 4 / April 2020
- Published online by Cambridge University Press:
- 17 February 2020, pp. 697-732
- Print publication:
- April 2020
-
- Article
- Export citation
-
Given a manifold
$M$ with a submanifold 
$N$, the deformation space 
${\mathcal{D}}(M,N)$ is a manifold with a submersion to 
$\mathbb{R}$ whose zero fiber is the normal bundle 
$\unicode[STIX]{x1D708}(M,N)$, and all other fibers are equal to 
$M$. This article uses deformation spaces to study the local behavior of various geometric structures associated with singular foliations, with 
$N$ a submanifold transverse to the foliation. New examples include 
$L_{\infty }$-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theorem around 
$N$, in terms of a model structure over 
$\unicode[STIX]{x1D708}(M,N)$.
