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A closed manifold is a fat CW complex

Norio Iwase, Yuki Kojima

TL;DR

The paper introduces fat CW complexes as a smooth, diffeological analogue of CW complexes designed to include all closed manifolds, and develops a robust handle-theoretic framework using smooth blocks ${\mathbb{D}}^{n,m}$ with boundaries ${\mathbb{S}}^{n-1,m}$. It proves that fat CW complexes satisfy the de Rham theorem, possess partitions of unity, and have a well-behaved interior/boundary structure under regularity; crucially, it establishes that every regular fat CW complex of finite dimension is reflexive, and that regular CW complexes with finite dimension correspond to closed manifolds. The work also provides explicit smoothing/diffeomorphism results for attaching handles, clarifying when smooth models faithfully represent topological CW constructions, and it contrasts fat CW complexes with non-reflexive thin (or exotic) CW notions.

Abstract

The main purpose of this paper is to introduce a new smooth version of a CW complex named a fat CW complex, and to show that it includes all closed manifolds, because existing smooth versions of CW complexes (e.g. [Iwa22]) do not have such property. We also verify that de Rham theorem holds for a fat CW complex and that a regular CW complex is reflexive in the sense of Y. Karshon, J. Watts and P. I-Zemmour. Further, any topological CW complex is topologically homotopy equivalent to a fat CW complex. So, a fat CW complex enjoys many nice properties.

A closed manifold is a fat CW complex

TL;DR

The paper introduces fat CW complexes as a smooth, diffeological analogue of CW complexes designed to include all closed manifolds, and develops a robust handle-theoretic framework using smooth blocks with boundaries . It proves that fat CW complexes satisfy the de Rham theorem, possess partitions of unity, and have a well-behaved interior/boundary structure under regularity; crucially, it establishes that every regular fat CW complex of finite dimension is reflexive, and that regular CW complexes with finite dimension correspond to closed manifolds. The work also provides explicit smoothing/diffeomorphism results for attaching handles, clarifying when smooth models faithfully represent topological CW constructions, and it contrasts fat CW complexes with non-reflexive thin (or exotic) CW notions.

Abstract

The main purpose of this paper is to introduce a new smooth version of a CW complex named a fat CW complex, and to show that it includes all closed manifolds, because existing smooth versions of CW complexes (e.g. [Iwa22]) do not have such property. We also verify that de Rham theorem holds for a fat CW complex and that a regular CW complex is reflexive in the sense of Y. Karshon, J. Watts and P. I-Zemmour. Further, any topological CW complex is topologically homotopy equivalent to a fat CW complex. So, a fat CW complex enjoys many nice properties.
Paper Structure (10 sections, 19 theorems, 11 equations)

This paper contains 10 sections, 19 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Foundations
  3. Smooth Handles
  4. Fat CW Complex
  5. Basic Properties
  6. Smooth Functions
  7. Regular Complex
  8. Main Results
  9. Proof of Theorem \ref{['thm:diffeotypeofhandle']}
  10. Proof of Theorem \ref{['thm:smoothtypeofhandle']}

Key Result

Proposition 3.1

$\lambda'(t) \le \lambda'(\hbox{\footnotesize$1${ \small$/$ }\footnotesize$2$})=\frac{\ell(\hbox{\tiny$3${ \tiny$/$ }\tiny$2$})^{2}}{\alpha} < \hbox{\footnotesize$20${ \small$/$ }\footnotesize$11$}$ for all $t \in \mathbb{R}$.

Theorems & Definitions (38)

  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • Theorem 3.7
  • ...and 28 more