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Directed homological and cohomological operations

Eric Goubault

Abstract

In this short note, we present a persistence module approach to directed cohomology, dual to the directed homology introduced by the author in a previous article. We lay out the first properties of directed cohomology and in particular of cohomological operations, partially linked to some homological operations. We treat here both the case of a specific class of precubical sets and of general directed spaces.

Directed homological and cohomological operations

Abstract

In this short note, we present a persistence module approach to directed cohomology, dual to the directed homology introduced by the author in a previous article. We lay out the first properties of directed cohomology and in particular of cohomological operations, partially linked to some homological operations. We treat here both the case of a specific class of precubical sets and of general directed spaces.
Paper Structure (12 sections, 5 theorems, 37 equations)

This paper contains 12 sections, 5 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Contents
  3. Background on directed spaces
  4. Background on precubical sets and cube chains
  5. Coboundary maps
  6. Directed Cohomology bimodules of certain precubical sets
  7. Directed Homology operations
  8. Directed Cohomology operations
  9. Local cup-product
  10. Cohomological inner-tensor
  11. Homological and cohomological operations in the case of general $d$-spaces
  12. Examples of cohomological calculations

Key Result

Lemma 5.3

Let $X$ be a finite precubical set that is such that is has proper non-looping length covering, $a, b \in X_0$. Then the $R$-module $HM^n[X]^b_a$, $n \geq 1$, is the standard $(n-1)$th cohomology of the trace space $Tr(| X |)^b_a$ from $a$ to $b$.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Definition 3.6
  • ...and 24 more