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Normal crossing immersions, cobordisms and flips

Karim Adiprasito, Gaku Liu

Abstract

We study various analogues of theorems from PL topology for cubical complexes. In particular, we characterize when two PL homeomorphic cubulations are equivalent by Pachner moves by showing the question to be equivalent to the existence of cobordisms between generic immersions of hypersurfaces. This solves a question and conjecture of Habegger and Funar.

Normal crossing immersions, cobordisms and flips

Abstract

We study various analogues of theorems from PL topology for cubical complexes. In particular, we characterize when two PL homeomorphic cubulations are equivalent by Pachner moves by showing the question to be equivalent to the existence of cobordisms between generic immersions of hypersurfaces. This solves a question and conjecture of Habegger and Funar.

Paper Structure

This paper contains 9 sections, 13 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic definitions
  4. Cubical Pachner moves
  5. Cubical stellar and cubical derived subdivisions
  6. From cubical-stellar to stellar and back: the confinement map
  7. A non-useful subdivision
  8. Whitehead's neighborhood theorem
  9. Cubical Pachner Theorem

Key Result

Theorem 1.1

For two PL cubulations $X_0$, $X_1$ of the same manifold $M$, the following three conditions are equivalent:

Figures (1)

  • Figure 2.1: A cubical stellar subdivision of a complex $C$, seen as a subcomplex of the product of $C$ with an interval.

Theorems & Definitions (25)

  • Theorem 1.1
  • Definition 2.1: Cubical Pachner moves
  • Lemma 2.2
  • proof
  • Proposition 2.3: BruggesserMani
  • Definition 2.4: Cubical stellar subdivision
  • Definition 2.5: Cubical derived subdivision
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • ...and 15 more