A note on the characteristic rank and related numbers

TL;DR

This paper establishes a precise inequality linking the characteristic rank of vector bundles, Betti numbers, and partition counts, with applications to real Grassmann manifolds, enhancing understanding of topological invariants.

Contribution

It introduces a sharp inequality connecting characteristic rank, Betti numbers, and partition sums, providing new insights into topological invariants and their relationships.

Findings

Derived a sharp inequality relating key topological quantities

Connected partition counts to Betti numbers of Grassmannians

Enhanced understanding of the interplay between vector bundle invariants and topology

Abstract

This note quantifies, via a sharp inequality, an interplay between (a) the characteristic rank of a vector bundle over a topological space X, (b) the Z/2Z-Betti numbers of X, and (c) sums of the numbers of certain partitions of integers. In a particular context, (c) is transformed into a sum of the readily calculable Betti numbers of the real Grassmann manifolds.