Vertical flows and a general currential homotopy formula

TL;DR

This paper extends transgression formulas for flowing differential forms using vertical vector fields, including Morse-Bott-Smale fields, with broad applications such as answering Quillen's question and refining classical theorems.

Contribution

It introduces a highly general transgression formula applicable to non-compact cases and diverse vector fields, advancing the theoretical framework of differential geometry.

Findings

Complete answer to Quillen's question

Construction of the Maslov spark

Short proof of a refined Chern-Gauss-Bonnet theorem

Abstract

We generalize some of the results of Harvey, Lawson and Latschev about transgression formulas. The focus here is on flowing forms via vertical vector fields, especially Morse-Bott-Smale vector fields. We prove a very general transgression formula including also a version covering non-compact situations. Among applications, we completely answer a question of Quillen, construct the Maslov spark, give a very short proof of a refined Chern- Gauss-Bonnet theorem, and reprove some theorems of Nicolaescu and Getzler. A discussion about odd Chern-Weil theory is also included.