Combinatorial cell complexes and Poincare duality

TL;DR

This paper introduces combinatorial cell complexes as finite topological models for polyhedral spaces, establishing algebraic topology tools and proving Poincare duality within this combinatorial framework.

Contribution

It defines combinatorial cell complexes and develops their algebraic topology, including orientability, homology, cohomology, and proves Poincare duality for these structures.

Findings

Defined combinatorial cell complexes (c.c.c) as finite models of polyhedral spaces.

Developed homology and cohomology theories for c.c.c.

Proved Poincare duality for c.c.c satisfying regularity conditions.

Abstract

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c's and develop enough algebraic topology in this setting to prove the Poincare duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.