Cartan's method and its applications in sheaf cohomology

TL;DR

This paper applies Cartan's method to prove vanishing theorems for sheaf cohomology on closed cubes, offering alternative proofs and insights into the topological dimension of manifolds.

Contribution

It introduces a new proof approach using Cartan's method for sheaf cohomology vanishing theorems and relates topological dimension to real dimension via Godement's argument.

Findings

Sheaf cohomology vanishes on closed cubes when degree exceeds dimension

Sheaf cohomology vanishes for locally constant sheaves in positive degree

Topological dimension of paracompact manifolds is bounded by their real dimension

Abstract

This paper aims to use Cartan's original method in proving Theorem A and B on closed cubes to provide a different proof of the vanishing of sheaf cohomology over a closed cube if either (i) the degree exceeds its real dimension or (ii) the sheaf is (locally) constant and the degree is positive. In the first case, we can further use Godement's argument to show the topological dimension of a paracompact topological manifold is less than or equal to its real dimension.