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An effective Mayer-Vietoris Theorem for discrete Morse homology

Sajal Mukherjee, Pritam Chandra Pramanik, Arundhati Rakshit

Abstract

The Mayer-Vietoris theorem is known for its wide applications, especially in determining homology. In fact, this theorem provides us with a long exact sequence, where the underlying homology groups fit in. However, this theorem does not provide an explicit way to compute homology. In this paper we prove an ``effective" version of the Mayer-Vietoris theorem using discrete Morse theory. Suppose, we have a Mayer-Vietoris type setup, i.e., let $X$ be a simplicial complex and $A$ and $B$ be two subcomplexes of $X$, such that $A \cup B=X$. Moreover, let $\mathcal{W}_A$, $\mathcal{W}_{B}$ and $\mathcal{W}_{A \cap B}$ be gradient vector fields on $A$, $B$ and $A \cap B$ respectively (which need not be ``coherent", i.e., they do not need to coincide on their intersection). Then, the main theorem of our paper provides an explicit way to compute the homology groups of $X$, using the combinatorial information regarding the trajectories of the aforementioned gradient vector fields, we do not even need to know the individual homology groups $H_{*}(A)$, $H_{*}(B)$ and $H_{*}(A \cap B)$. In principle, the homology of $X$ can always be computed explicitly using our theorem irrespective of the choice of the gradient vector fields. Further, if we choose the subcomplexes $A$ and $B$ wisely so that each of $A$, $B$ and $A \cap B$ admits an efficient gradient vector field, then the computation of the homology groups is considerably reduced.

An effective Mayer-Vietoris Theorem for discrete Morse homology

Abstract

The Mayer-Vietoris theorem is known for its wide applications, especially in determining homology. In fact, this theorem provides us with a long exact sequence, where the underlying homology groups fit in. However, this theorem does not provide an explicit way to compute homology. In this paper we prove an ``effective" version of the Mayer-Vietoris theorem using discrete Morse theory. Suppose, we have a Mayer-Vietoris type setup, i.e., let be a simplicial complex and and be two subcomplexes of , such that . Moreover, let , and be gradient vector fields on , and respectively (which need not be ``coherent", i.e., they do not need to coincide on their intersection). Then, the main theorem of our paper provides an explicit way to compute the homology groups of , using the combinatorial information regarding the trajectories of the aforementioned gradient vector fields, we do not even need to know the individual homology groups , and . In principle, the homology of can always be computed explicitly using our theorem irrespective of the choice of the gradient vector fields. Further, if we choose the subcomplexes and wisely so that each of , and admits an efficient gradient vector field, then the computation of the homology groups is considerably reduced.
Paper Structure (8 sections, 12 theorems, 86 equations, 2 figures)

This paper contains 8 sections, 12 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem
  4. Construction of a new simplicial complex $\widetilde{X}$
  5. Constructing a gradient vector field on $\widetilde{X}$
  6. Isomorphism between simplicial chain complex of $X$ and Thom-Smale complex of $\widetilde{X}$ with respect to $\mathcal{V}$
  7. Construction of the gradient vector field $\mathcal{W}$ on $\widetilde{X}$ using $\mathcal{W}_{\bar{A}}$, $\mathcal{W}_{\bar{B}}$ and $\mathcal{W}_{\overline{A \cap B}}$
  8. Isomorphism between the simplicial homology of $\widetilde{X}$ and the homology groups of $\mathcal{D}_{\#}(X)$

Key Result

Theorem 1.1

The Thom-Smale chain complex of $X$ (with respect to a gradient vector field $\mathcal{V}$ on $X$), $(C^\mathcal{V}_{\#}(X),\partial^\mathcal{V}_{\#})$ is homotopy equivalent to the simplicial chain complex of $X$, $(C_{\#}(X), \partial_{\#})$. Hence $H^{\mathcal{V}}_{\#}(X) \cong H_{\#}(X)$.

Figures (2)

  • Figure 1: $S_0 * S_0 * S_0$, and the disjoint copies of its subcomplexes $A$, $B$ and $A \cap B$. The arrows represent the respective gradient vector fields while the critical simplices have been marked in bold.
  • Figure 2: An illustration of $\mathbb{P}(X)$.

Theorems & Definitions (31)

  • Definition 1.0.1
  • Definition 1.0.2
  • Theorem 1.1: Forman1Forman2
  • Definition 1.1.1
  • Theorem 1.2
  • Definition 2.0.1
  • Definition 2.0.2
  • Definition 2.0.3
  • Remark 3.0.1
  • Theorem 3.1
  • ...and 21 more