Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a homology of foliations defined by non-singular Morse-Smale flows

Masato Akizawa, Ryosuke Furuta, Shigeaki Miyoshi

TL;DR

The paper defines a homology theory for one-dimensional foliations arising from non-singular Morse-Smale flows by combining round handle decompositions with Conley index techniques to produce a foliation-sensitive boundary operator. It introduces $C^{ ext{NMS}}_{k}(\mathcal{F})$ generated by index-$k$ leaves and proves a chain complex whose homology $H^{ ext{NMS}}_{ullet}(\mathcal{F})$ matches the homology of the associated $RHD$, providing a dynamics-aware invariant. A concrete calculation for $NMS$ foliations tied to Seifert fibrations shows $H^{ ext{NMS}}_{*}(\mathcal{F})$ recovers the base surface topology $H_{*}(oldsymbol{oldsymbol{\Sigma}})$ under coprime invariants, highlighting the approach’s topological relevance. Overall, the work connects foliation dynamics, Morse-Smale theory, and 3-manifold topology to yield a robust invariant of $NMS$ foliations.

Abstract

We propose a definition of a homology of a one-dimensional foliation defined by a non-singular Morse-Smale flow. We also show the calculation of the homology of such a foliation which is naturally associated with Seifert fibration.

On a homology of foliations defined by non-singular Morse-Smale flows

TL;DR

The paper defines a homology theory for one-dimensional foliations arising from non-singular Morse-Smale flows by combining round handle decompositions with Conley index techniques to produce a foliation-sensitive boundary operator. It introduces generated by index- leaves and proves a chain complex whose homology matches the homology of the associated , providing a dynamics-aware invariant. A concrete calculation for foliations tied to Seifert fibrations shows recovers the base surface topology under coprime invariants, highlighting the approach’s topological relevance. Overall, the work connects foliation dynamics, Morse-Smale theory, and 3-manifold topology to yield a robust invariant of foliations.

Abstract

We propose a definition of a homology of a one-dimensional foliation defined by a non-singular Morse-Smale flow. We also show the calculation of the homology of such a foliation which is naturally associated with Seifert fibration.
Paper Structure (9 sections, 14 theorems, 16 equations, 3 figures)

This paper contains 9 sections, 14 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-singular Morse-Smale flows and round handle decompositions
  3. A homology of round handle decompositions
  4. Conley index and boundary operators
  5. Index pair and Conley index
  6. The explicit construction of index pairs for the boundary operator
  7. The explicit construction of index pairs in the general case
  8. A boundary operator along flow-lines
  9. Calculations for NMS-foliations associated with Seifert fibrations

Key Result

Proposition 2.1

Let $(M, \partial_{-}M)$ is a manifold pair with $\chi (M, \partial_{-}M) = 0$. Then, $(M, \partial_{-}M)$ has an RHD if $\mathrm{dim}M \neq 3$. Moreover, if $M$ is not a Möbius band $($and $\mathrm{dim}\neq 3)$, then $(M, \partial_{-}M)$ has a simple RHD.

Figures (3)

  • Figure 1: A round subhandle and connecting annuli
  • Figure 2: A local view of $\partial_{-}Q^{k}$
  • Figure 3: A handle decomposition of $\Sigma\setminus\mathrm{Int}(h^{2})$ for a Seifert fibration $p:M\rightarrow\Sigma$

Theorems & Definitions (35)

  • Proposition 2.1: Asimov Asimov_RHD
  • Proposition 2.2: Morgan Morgan
  • Proposition 2.3: Asimov Asimov_RHD
  • Proposition 2.4: Morgan Morgan
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Theorem I
  • Definition 3.4
  • ...and 25 more