On the rank of a product of manifolds

TL;DR

This paper provides an example of closed smooth manifolds where the rank of their product exceeds the sum of their individual ranks, challenging previous assumptions about rank behavior under products.

Contribution

It presents a counterexample demonstrating that the rank of a product of manifolds can be greater than the sum of their ranks, which was previously unknown.

Findings

Counterexample showing rank(M×N) > rank(M) + rank(N)

Challenges existing assumptions about manifold rank behavior

Advances understanding of manifold product properties

Abstract

This note gives an example of closed smooth manifolds $M$ and $N$ for which the rank of $M\times N$ is strictly greater than $rank M + rank N$.