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Morse-Bott cohomology from homological perturbation theory

Zhengyi Zhou

Abstract

In this paper, we construct cochain complexes generated by the cohomology of critical manifolds in the abstract setup of flow categories for Morse-Bott theories under minimum transversality assumptions. We discuss the relations between different constructions of Morse-Bott theories. In particular, we explain how homological perturbation theory is used in Morse-Bott theories, and both our construction and the cascades construction can be interpreted as applications of homological perturbations. In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximations of classifying spaces and the existence of the action spectral sequence are proven. We carry out our construction for Morse-Bott functions on closed manifolds and prove it recovers the regular cohomology. We outline the project of combining our construction with the polyfold theory.

Morse-Bott cohomology from homological perturbation theory

Abstract

In this paper, we construct cochain complexes generated by the cohomology of critical manifolds in the abstract setup of flow categories for Morse-Bott theories under minimum transversality assumptions. We discuss the relations between different constructions of Morse-Bott theories. In particular, we explain how homological perturbation theory is used in Morse-Bott theories, and both our construction and the cascades construction can be interpreted as applications of homological perturbations. In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximations of classifying spaces and the existence of the action spectral sequence are proven. We carry out our construction for Morse-Bott functions on closed manifolds and prove it recovers the regular cohomology. We outline the project of combining our construction with the polyfold theory.

Paper Structure

This paper contains 73 sections, 77 theorems, 257 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Cohomology of flow categories
  3. Equivariant theories
  4. Constructions of flow categories
  5. Organization of the paper
  6. Motivation From Homological Perturbation Theory
  7. Differential topology notation
  8. Manifolds and submanifolds with boundaries and corners
  9. Orientations
  10. Flow categories.
  11. Conventions for cochain complexes
  12. Review of existing constructions
  13. Austin-Braam's Morse-Bott cochain complex $(BC^{{\rm AB}}, d^{{\rm AB}})$
  14. Fukaya's Morse-Bott chain complex
  15. The cascades model
  16. ...and 58 more sections

Key Result

Theorem 1

To every oriented flow category, we can assign a minimal Morse-Bott cochain complex $({\rm BC},d_{{\rm BC}})$ over $\mathbb{R}$ generated by the cohomology of the object space (with a suitable completion) in a functorial way.

Figures (8)

  • Figure 1: A $2$-cascade
  • Figure 2: Graph of $\rho_n$
  • Figure 3: Graph of $\Delta^n$ near the boundary
  • Figure 4: Graph of $\rho_n$
  • Figure 5: Pullback of Thom classes
  • ...and 3 more figures

Theorems & Definitions (220)

  • Claim
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • ...and 210 more