# Borsuk-Ulam theorems for products of spheres and Stiefel manifolds revisited

## TL;DR

This paper presents a new, more accessible proof of a generalized Borsuk-Ulam theorem for products of spheres and extends these results to Stiefel manifolds, providing alternative proofs and new insights.

## Contribution

It offers a novel proof technique for Borsuk-Ulam theorems on spheres and Stiefel manifolds, simplifying existing proofs and extending their applicability.

## Key findings

- New proof of Borsuk-Ulam theorem for products of spheres
- Extension of Borsuk-Ulam results to Stiefel manifolds
- Alternative proofs of Fadell and Husseini's results

## Abstract

We give a different and possibly more accessible proof of a general Borsuk--Ulam theorem for a product of spheres, originally due to Ramos. That is, we show the non-existence of certain $(\mathbb{Z}/2)^k$-equivariant maps from a product of $k$ spheres to the unit sphere in a real $(\mathbb{Z}/2)^k$-representation of the same dimension. Our proof method allows us to derive Borsuk--Ulam theorems for certain equivariant maps from Stiefel manifolds, from the corresponding results about products of spheres, leading to alternative proofs and extensions of some results of Fadell and Husseini.