Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tropical singular intersection homology

Junta Kamiya

Abstract

We introduce tropical singular intersection homologies (non-GM and GM) with the tropical coefficients on rational polyhedral spaces using their filtrations. We investigate their fundamental properties. In the non-GM case, we give a Poincaré duality and two bilinear pairings analogous to the cup and cap products under some assumptions. We compute the homologies in some cases.

Tropical singular intersection homology

Abstract

We introduce tropical singular intersection homologies (non-GM and GM) with the tropical coefficients on rational polyhedral spaces using their filtrations. We investigate their fundamental properties. In the non-GM case, we give a Poincaré duality and two bilinear pairings analogous to the cup and cap products under some assumptions. We compute the homologies in some cases.

Paper Structure

This paper contains 29 sections, 39 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Tropical singular intersection homology
  3. Basic properties
  4. Poincaré duality in non-GM (co)homology
  5. Background and related works
  6. Structure of the paper
  7. Definitions
  8. Preliminaries
  9. Tropical geometry
  10. Filtered space
  11. Filtered space in tropical geometry
  12. Other preliminaries
  13. Tropical intersection (co)homology
  14. Properties of tropical intersection (co)homology
  15. Basic properties
  16. ...and 14 more sections

Key Result

Proposition A

The homology $_{(\mathop{\mathrm{GM}}\nolimits)}\mathop{\mathrm{I^{\bar{\mathsf{p}}}H}}\nolimits_{p,q}(X,\mathfrak{X}, Q)$ is independent of the choice of an open face structure on $X$.

Theorems & Definitions (120)

  • Proposition A: $=$Proposition \ref{['prop:indep']}
  • Proposition B: $=$Proposition \ref{['prop:tropstr']}
  • Proposition C: $=$Proposition \ref{['prop:sheaf']}
  • Theorem D: $=$Theorem \ref{['thm:genpoin']}, Poincaré duality
  • Definition 2.1: Rational polyhedral space MR3894860
  • Remark 2.2
  • Definition 2.3: (Open) face structure MR3894860
  • Remark 2.4
  • Definition 2.5: Filtered space Friedman_2020
  • Definition 2.6: Stratified map, stratified homeomorphism Friedman_2020
  • ...and 110 more