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A note on pullbacks and blowups of Lie algebroids, singular foliations, and Dirac structures

Andreas Schüßler, Marco Zambon

TL;DR

The paper studies how pullbacks and blowups of Dirac structures, Lie algebroids, and singular foliations behave under maps with constant rank or transversality. It establishes that, under smoothness or transversality assumptions, the foliation of a pulled-back Lie algebroid equals the pullback of the original foliation, and that backward Dirac maps lift Dirac structures compatibly, with canonical isomorphisms when transversality holds. The blowup analysis extends these ideas to Blup(A,C) and blows up along submanifolds, giving explicit descriptions of the resulting foliations in terms of base data and two key cases (restricted and isotropy) that involve the b-tangent and edge structures. Together, these results unify pullback and blowup operations across Dirac structures, Lie algebroids, and singular foliations, and clarify when the expected equalities and isomorphisms hold or fail, supported by concrete examples and literature context.

Abstract

Lie algebroids, singular foliations, and Dirac structures are closely related objects. We examine the relation between their pullbacks under maps satisfying a constant rank or transversality assumption. A special case is given by blowdown maps. In that case, we also establish the relation between the blowup of a Lie algebroid and its singular foliation.

A note on pullbacks and blowups of Lie algebroids, singular foliations, and Dirac structures

TL;DR

The paper studies how pullbacks and blowups of Dirac structures, Lie algebroids, and singular foliations behave under maps with constant rank or transversality. It establishes that, under smoothness or transversality assumptions, the foliation of a pulled-back Lie algebroid equals the pullback of the original foliation, and that backward Dirac maps lift Dirac structures compatibly, with canonical isomorphisms when transversality holds. The blowup analysis extends these ideas to Blup(A,C) and blows up along submanifolds, giving explicit descriptions of the resulting foliations in terms of base data and two key cases (restricted and isotropy) that involve the b-tangent and edge structures. Together, these results unify pullback and blowup operations across Dirac structures, Lie algebroids, and singular foliations, and clarify when the expected equalities and isomorphisms hold or fail, supported by concrete examples and literature context.

Abstract

Lie algebroids, singular foliations, and Dirac structures are closely related objects. We examine the relation between their pullbacks under maps satisfying a constant rank or transversality assumption. A special case is given by blowdown maps. In that case, we also establish the relation between the blowup of a Lie algebroid and its singular foliation.

Paper Structure

This paper contains 13 sections, 10 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Definitions of pullbacks
  3. Pullbacks of Lie algebroids
  4. Pullbacks of singular foliations
  5. Pullbacks of Dirac structures
  6. Examples
  7. Lie algebroids and singular foliations
  8. Dirac structures and Lie algebroids
  9. Blowups
  10. Lifted Dirac structures on the blowup
  11. Blowups and Lie algebroids
  12. The singular foliation $\mathcal{F}_\mathrm{Blup}$ of $\mathrm{Blup}(A,C)$ in terms of a singular foliation on $M$
  13. The singular foliation $\mathcal{F}_\mathrm{Blup}$ of $\mathrm{Blup}(A,C)$ in terms of a singular foliation on $B$

Key Result

Lemma 2.1

Let $f\colon B\to M$ be a smooth map, and $A$ a Lie algebroid over $M$. Then $f^!A$ is smooth iff the following subspaces have constant rank for all $x\in B$: In turn, this is equivalent to $f^!A$ being a vector subbundle of $f^*A\oplus TB$.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Example 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Proposition 4.3
  • proof
  • ...and 14 more