# On the third homotopy group of Orr's space

## TL;DR

This paper proves that Orr's space has an infinitely generated third homotopy group, advancing understanding of its algebraic structure, though the realization of elements as links remains unresolved.

## Contribution

It establishes that the third homotopy group of Orr's space is infinitely generated, a significant step in understanding its topological properties.

## Key findings

- Third homotopy group of Orr's space is infinitely generated
- The non-triviality of this group is confirmed
- Open question remains on realizing elements as links

## Abstract

K. Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space $K_\omega.$ The problem of non-triviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.