A closed manifold is a fat CW complex

TL;DR

This paper introduces fat CW complexes, a smooth variant that encompasses all closed manifolds, verifies de Rham theorem applicability, and establishes homotopy equivalence with topological CW complexes.

Contribution

It presents a new smooth CW complex version called fat CW complexes, including all closed manifolds, and proves several key properties and equivalences.

Findings

Fat CW complexes include all closed manifolds.

De Rham theorem holds for fat CW complexes.

Any topological CW complex is homotopy equivalent to a fat CW complex.

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.