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Novikov type inequalities for orbifolds

Fabricio Valencia

TL;DR

This work extends classical Novikov theory from manifolds to proper Lie groupoids and differentiable stacks, introducing Novikov numbers $b_j(\xi)$ and $q_j(\xi)$ for basic cohomology classes of closed basic 1-forms on groupoids of finitely generated type. It develops the algebraic-topology toolkit on groupoids—Hurewicz $G$-homomorphisms, groupoid coverings, and groupoid homology with local coefficients—along with a Morita-invariant framework and Eilenberg-type isomorphisms, to define Novikov homology $H_j^{\text{tot}}(G,\mathcal{L}_\xi)$ and its invariants. The main result yields Novikov inequalities for compact orbifolds presented by proper étale groupoids: for a Morse-type closed basic 1-form $\omega$ on $X$, the zeros satisfy $c_j(\omega)\ge b_j(\xi)+q_j(\xi)+q_{j-1}(\xi)$, with several structural reductions and special cases (e.g., action groupoids, equivariant settings). The paper further connects these results to zeros of symplectic vector fields on orbifolds, highlighting potential applications in Hamiltonian dynamics on stacks and orbifolds, and outlining Morita-invariant Morse theory for stacky 1-forms. Overall, it provides a robust topological-meets-analytic framework to bound zeros of closed basic 1-forms and symplectic fields in the orbifold/differentiable-stack setting.

Abstract

We propose a natural extension of the Novikov numbers for the basic cohomology class of a closed basic $1$-form on a proper Lie groupoid of finitely generated type. As an application, we prove corresponding Novikov inequalities for compact orbifolds.

Novikov type inequalities for orbifolds

TL;DR

This work extends classical Novikov theory from manifolds to proper Lie groupoids and differentiable stacks, introducing Novikov numbers $b_j(\xi)$ and $q_j(\xi)$ for basic cohomology classes of closed basic 1-forms on groupoids of finitely generated type. It develops the algebraic-topology toolkit on groupoids—Hurewicz $G$-homomorphisms, groupoid coverings, and groupoid homology with local coefficients—along with a Morita-invariant framework and Eilenberg-type isomorphisms, to define Novikov homology $H_j^{\text{tot}}(G,\mathcal{L}_\xi)$ and its invariants. The main result yields Novikov inequalities for compact orbifolds presented by proper étale groupoids: for a Morse-type closed basic 1-form $\omega$ on $X$, the zeros satisfy $c_j(\omega)\ge b_j(\xi)+q_j(\xi)+q_{j-1}(\xi)$, with several structural reductions and special cases (e.g., action groupoids, equivariant settings). The paper further connects these results to zeros of symplectic vector fields on orbifolds, highlighting potential applications in Hamiltonian dynamics on stacks and orbifolds, and outlining Morita-invariant Morse theory for stacky 1-forms. Overall, it provides a robust topological-meets-analytic framework to bound zeros of closed basic 1-forms and symplectic fields in the orbifold/differentiable-stack setting.

Abstract

We propose a natural extension of the Novikov numbers for the basic cohomology class of a closed basic -form on a proper Lie groupoid of finitely generated type. As an application, we prove corresponding Novikov inequalities for compact orbifolds.
Paper Structure (10 sections, 17 theorems, 38 equations)

This paper contains 10 sections, 17 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Some algebraic topology ingredients
  3. Hurewicz $G$-homomorphism and coverings
  4. Groupoid homology with local coefficients
  5. Closed basic 1-forms
  6. G-homomorphisms of periods
  7. Closed basic 1-forms of Morse type
  8. The Novikov numbers
  9. Novikov inequalities
  10. Zeros of symplectic vector fields

Key Result

Theorem 1

Let $X$ be a compact orbifold and $\omega$ be a closed 1-form of Morse type on $X$. If $c_j(\omega)$ denotes the number of zeros of $\omega$ having Morse index $j$ then where $\xi\in H^1(X,\mathbb{R})$ stands for the cohomology class of $\omega$.

Theorems & Definitions (50)

  • Theorem
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.2: Hurewicz $G$-homomorphism
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • ...and 40 more