Enhanced Bruhat decomposition and Morse theory

Persistence of Conley-Morse Graphs in Combinatorial Dynamical Systems

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invaria…

Ascending and descending regions of a discrete Morse function

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of th…

Decomposing Perfect Discrete Morse Functions on Connected Sums of 3-manifolds

In this paper, we show that if a closed, connected, oriented 3-manifold M = M1#M2 admits a perfect discrete Morse function, then one can decompose this function as perfect discrete Morse functions on M_1 and M_2. We also give an explicit construction of a separating sphere on M corresponding to such a decomposition.

通过更新矩阵分解高效计算Conley-Morse持续同调条形码

Computing Conley-Morse Persistence Barcode Efficiently by Updating Matrix Decompositions

本文针对Conley-Morse持续同调条形码计算成本高的问题，提出用更简单的块取代指数对，通过更新矩阵分解的方法实现高效计算，优化了现有算法。

Cellular Homology of Real Flag Manifolds

Let $\mathbb{F}_{Θ}=G/P_{Θ}$ be a generalized flag manifold, where $G$ is a real noncompact semi-simple Lie group and $P_{Θ}$ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow $\mathbb{F}_Θ$ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in $\mathbb{R}^{n}$ and use them to compute the boundary operator $\partial $ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients o…

A nerve lemma for gluing together incoherent discrete Morse functions

Two of the most useful tools in topological combinatorics are the nerve lemma and discrete Morse theory. In this note we introduce a theorem that interpolates between them and allows decompositions of complexes into non-contractible pieces as long as discrete Morse theory ensures that they behave well enough. The proof is based on diagrams of spaces, but that theory is not needed for the formulation or applications of the theorem.

Morse functions and cohomology of homogeneous spaces

This article arose from a series of three lectures given at the Banach Center, Warsaw, during period of 24 March to 13 April, 2003. Morse functions are useful tool in revealing the geometric formation of its domain manifolds $M$. They define the handle decompositions of $M$ from which the additive homologies $H_{\ast}(M)$ may be constructed. In these lectures two further questions were emphasized. (1) How to find a Morse function on a given manifold? (2) From Morse functions can one derive the multiplicative cohomology rather than the additive homology?