# Path and quasihomotopy for Sobolev maps between manifolds

## TL;DR

This paper investigates the relationship between quasihomotopy and path homotopy for Sobolev maps between manifolds, revealing nuanced differences and conditions under which they coincide or differ.

## Contribution

It establishes that in the critical exponent case, path homotopy implies quasihomotopy, and shows that quasihomotopic maps need not be path homotopic, especially in certain target manifolds.

## Key findings

- Path homotopy implies quasihomotopy in the critical case.
- Quasihomotopic maps may not be path homotopic.
- Differences depend on the target manifold's properties.

## Abstract

We study the relationship between quasihomotopy and path homotopy for Sobolev maps between manifolds. We employ singular integrals on manifolds to show that, in the critical exponent case, path homotopy implies quasihomotopy - and observe the rather surprising fact that $n$-quasihomotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with nonpositive sectional curvature, and the contrasting case of the target being a sphere.