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Sheaf Theoretic Approach to Lefschetz Calculus

Alejandro O. Majadas-Moure, David Mosquera-Lois

Abstract

We lift the Lefschetz number from an algebraic invariant of maps between spaces to an invariant of morphisms of data over the spaces.

Sheaf Theoretic Approach to Lefschetz Calculus

Abstract

We lift the Lefschetz number from an algebraic invariant of maps between spaces to an invariant of morphisms of data over the spaces.

Paper Structure

This paper contains 13 sections, 16 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions of the Lefschetz number
  3. Definable cellular structures
  4. The combinatorial Lefschetz number
  5. $c$-constructible sheafs
  6. The associated $f$-$c$-constructible sheaf $\mathcal{F}'$
  7. The sheaf-theoretic Lefschetz number
  8. Well-definedness and topological invariance of the combinatorial Lefschetz number
  9. The sheaf-theoretic Lefschetz number as an integral
  10. $\mathbb{N}$-constructible functions and constructible sheaves
  11. Sheaf-theoretic Lefschetz number and Lefschetz calculus
  12. $\mathbb{Z}$-valuated constructible functions and Barrow's rule.
  13. Axiomatization of the sheaf-theoretic Lefschetz number

Key Result

Theorem 2.1

Let $X\subset \mathbb{R}^n$ be a definable set and let $\{X_i\}_{i=1}^{m}$ be a finite family of definable subsets of $X$. Then there exists a definable triangulation of $X$ compatible with the collection of subsets.

Figures (3)

  • Figure 1: Generalized simplicial complex.
  • Figure 2: Complex X.
  • Figure 3: Elements of the partition.

Theorems & Definitions (31)

  • Theorem 2.1: Definable triangulation theorem Dries
  • Lemma 2.2: Additivity of the combinatorial Lefschetz number
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • ...and 21 more