Enhanced Bruhat decomposition and Morse theory

TL;DR

This paper explores the properties of Bruhat numbers associated with Morse functions, linking them to Reidemeister torsion, and extends classical Bruhat decomposition to a new linear algebraic setting.

Contribution

It introduces a novel interpretation of Bruhat numbers as Reidemeister torsion and extends Bruhat decomposition to classify matrices up to unitriangular transformations.

Findings

Product of Bruhat numbers is invariant and relates to Reidemeister torsion.

Proves a mod 2 equation for generic paths in one-parameter Morse theory.

Provides a new linear algebraic framework generalizing Bruhat decomposition.

Abstract

Morse function is called strong if all its critical values are pairwise distinct. Given such a function $f$ and a field $\mathbb{F}$ Barannikov constructed a pairing of some of the critical points of $f$, which is now also known as barcode. With every Barannikov pair we naturally associate (up to sign) an element of $\mathbb{F} \setminus \{0\}$; we call it Bruhat number. The paper is devoted to the study of these Bruhat numbers. We investigate several situations where the product of all these numbers (some being raised to the power $-1$) is independent of $f$ and interpret it as a Reidemeister torsion. We apply our results in the setting of one-parameter Morse theory by proving that generic path of functions must satisfy a certain equation mod 2 (this was initially proven in \cite{Akhm} under additional assumptions). On the linear-algebraic level our constructions are served by the following variation of a classical Bruhat decomposition for $GL(\mathbb{F})$. A unitriangular matrix is an upper triangular one with 1's on the diagonal. Consider all rectangular matrices over $\mathbb{F}$ up to left and right multiplication by unitriangular ones. Enhanced Bruhat decomposition describes canonical representative in each equivalence class.