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Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory

Markus Upmeier

TL;DR

The paper develops a comprehensive framework for pushforward operations in twisted K-theory, focusing on principal $BU(1)$-bundles with orientations and introducing the projective Euler $\pi_!^\theta$ and projective rank $s_\theta^*$ operations as fundamental generators of stable pushforwards. It establishes a precise exact-classification scheme via inverse-limit cohomology, proves the generation of all stable pushforwards by PE and rk in the rational setting, and extends the theory to a homological context with a homology-level Euler operation. These tools are then employed to construct a graded Lie bracket on the homology of moduli spaces with BU(1)-action, providing a robust algebraic framework for wall-crossing phenomena in enumerative geometry. Together, the results connect twisted K-theory orientations, transfer operations, and moduli-space Lie theory, enabling new computations and conceptual understandings of wall-crossing in geometric contexts.

Abstract

We develop a general theory of pushforward operations for principal $G$-bundles equipped with a certain type of orientation. In the case $G=BU(1)$ and orientations in twisted K-theory we construct two pushforward operations, the projective Euler operation, whose existence was conjectured by Joyce, and the projective rank operation. We classify all stable pushforward operations in this context and show that they are all generated by the projective Euler and rank operation. As an application, we construct a graded Lie algebra structure on the homology of a commutative H-space with a compatible $BU(1)$-action and orientation. These play an important role in the context of wall-crossing formulas in enumerative geometry.

Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory

TL;DR

The paper develops a comprehensive framework for pushforward operations in twisted K-theory, focusing on principal -bundles with orientations and introducing the projective Euler and projective rank operations as fundamental generators of stable pushforwards. It establishes a precise exact-classification scheme via inverse-limit cohomology, proves the generation of all stable pushforwards by PE and rk in the rational setting, and extends the theory to a homological context with a homology-level Euler operation. These tools are then employed to construct a graded Lie bracket on the homology of moduli spaces with BU(1)-action, providing a robust algebraic framework for wall-crossing phenomena in enumerative geometry. Together, the results connect twisted K-theory orientations, transfer operations, and moduli-space Lie theory, enabling new computations and conceptual understandings of wall-crossing in geometric contexts.

Abstract

We develop a general theory of pushforward operations for principal -bundles equipped with a certain type of orientation. In the case and orientations in twisted K-theory we construct two pushforward operations, the projective Euler operation, whose existence was conjectured by Joyce, and the projective rank operation. We classify all stable pushforward operations in this context and show that they are all generated by the projective Euler and rank operation. As an application, we construct a graded Lie algebra structure on the homology of a commutative H-space with a compatible -action and orientation. These play an important role in the context of wall-crossing formulas in enumerative geometry.

Paper Structure

This paper contains 15 sections, 26 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Pushforward operations and stability
  3. Unstable pushforward operations
  4. Stable pushforward operations
  5. Multiplicative properties
  6. A technical result
  7. Projective Euler operations
  8. Background on Chern classes and the Chern character
  9. Exact classification sequence
  10. Proof Theorem 1.1(a)-(f) and Theorem 1.2
  11. Module structure on group of pushforward operations
  12. Proof of Theorem 1.4
  13. Proof of Theorem 1.1(g)--(i)
  14. Projective Euler operation in homology
  15. Homological Lie brackets on moduli spaces

Key Result

Theorem 1.1

For each principal ${B\mathrm{U}(1)}$-bundle ${\pi\colon P\to B}$ and orientation $\theta\in K_P(B)$ in twisted K-theory there is a projective Euler operation where $r$ is the rank of $\theta,$ uniquely determined by the following properties.

Theorems & Definitions (65)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • ...and 55 more