# A note on the Petersen-Wilhelm conjecture

## TL;DR

This paper proves the Petersen-Wilhelm Conjecture for all known compact manifolds with positive curvature by analyzing submersions from certain homotopy equivalent spaces.

## Contribution

It demonstrates that nontrivial submersions from specific positively curved manifolds have base dimensions exceeding fiber dimensions, confirming the conjecture for all known cases.

## Key findings

- Nontrivial submersions have larger base than fiber dimensions.
- Confirms Petersen-Wilhelm Conjecture for all known positively curved compact manifolds.
- Supports the conjecture's validity in the context of Eschenburg and Bazaikin spaces.

## Abstract

In this note we consider submersions from compact manifolds, homotopy equivalent to the Eschenburg or Bazaikin spaces of positive curvature. We show that if the submersion is nontrivial, the dimension of the base is greater than the dimension of the fiber. Together with previous results, this proves the Petersen-Wilhelm Conjecture for all the known compact manifolds with positive curvature.