# Contractible 3-manifold and Positive scalar curvature (II)

## TL;DR

This paper investigates whether all complete contractible 3-manifolds with positive scalar curvature are topologically equivalent to Euclidean 3-space, focusing on the fundamental group at infinity and its implications.

## Contribution

It proves that such manifolds with trivial fundamental group at infinity are homeomorphic to , advancing understanding of the topology of positive scalar curvature manifolds.

## Key findings

- Complete contractible 3-manifolds with positive scalar curvature and trivial  are homeomorphic to .
- The fundamental group at infinity plays a key role in classifying these manifolds.
- The study links geometric curvature conditions with topological classification.

## Abstract

In this article, we are interested in the question whether any complete contractible $3$-manifold of positive scalar curvature is homeomorphic to $\mathbb{R}^{3}$. We study the fundamental group at infinity, $\pi_{1}^{\infty}$, and its relationship with the existence of complete metrics of positive scalar curvature. We prove that a complete contractible $3$-manifold with positive scalar curvature and trivial $\pi^{\infty}_{1}$ is homeomorphic to $\mathbb{R}^{3}$.