Boundary framings for locally conformally symplectic four-manifolds

TL;DR

This paper develops a rational homotopy model for classifying locally conformally symplectic structures on four-manifolds, introduces a cobordism category for three-manifolds with principal bundle structures, and explores advanced cohomology theories for their study.

Contribution

It introduces a new homotopy-theoretic model for classifying locally conformally symplectic structures and defines a cobordism category for anchored three-manifolds, extending contact structure theory.

Findings

Constructed a rational homotopy model for classifying spaces.

Defined a cobordism category for three-manifolds with principal bundles.

Applied $sl_2$-valued Hodge-Lefschetz cohomology to study these structures.

Abstract

We construct a rational homotopy-theoretic model for a classifying space of locally conformally symplectic structures on four-manifolds, and use it to definition a cobordism category of three-manifolds `anchored' by principal $\Omega^2 S^2$ - bundles ($\S2$, generalizing contact structures). Powerful $sl_2$ - representation-valued Hodge-Lefschetz cohomology (going back to Chern and Weil), taking values in the $\mathbb{Z}$-graded category of bidifferential modules of Angella, Otiman, and Tardini is available for its study. This is an extended revision with a detailed introduction replacing the final section. The original concern of the paper was a characteristic two issue which remains unchanged.