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Lectures on the triangulation conjecture

Ciprian Manolescu

Abstract

We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology.

Lectures on the triangulation conjecture

Abstract

We outline the proof that non-triangulable manifolds exist in any dimension greater than four. The arguments involve homology cobordism invariants coming from the Pin(2) symmetry of the Seiberg-Witten equations. We also explore a related construction, of an involutive version of Heegaard Floer homology.

Paper Structure

This paper contains 32 sections, 9 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Triangulations
  3. Basic definitions
  4. Triangulations of manifolds
  5. The Kirby-Siebenmann obstruction
  6. Triangulability of manifolds
  7. Seiberg-Witten theory
  8. Strategy of proof
  9. The Seiberg-Witten equations
  10. The Seiberg-Witten equations in Coulomb gauge
  11. Finite dimensional approximation
  12. The Conley index
  13. Seiberg-Witten Floer Homology
  14. Invariance
  15. The Seiberg-Witten Floer stable homotopy type
  16. ...and 17 more sections

Key Result

Theorem 1.1

There exist non-triangulable $n$-dimensional topological manifolds for every $n \geq 5$.

Theorems & Definitions (30)

  • Theorem 1.1: man13
  • Example 2.1
  • Example 2.2
  • Theorem 2.3: man13
  • Example 2.5
  • Theorem 3.1
  • Theorem 3.2: cf. man03
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 20 more