The rational homology ring of the based loop space of the gauge groups and the spaces of connections on a four-manifold

TL;DR

This paper uses rational homotopy theory to analyze the homotopy groups and Pontrjagin homology rings of based loop spaces related to gauge groups and connection spaces on simply connected four-manifolds, revealing dependence on the second Betti number.

Contribution

It provides explicit formulas for the rational Pontrjagin homology rings of these loop spaces, extending understanding of their algebraic topology.

Findings

Ranks of homotopy groups depend only on second Betti number

Explicit formulas for rational Pontrjagin homology rings obtained

Homotopy groups are characterized via rational homotopy theory

Abstract

We provide the rational-homotopic proof that the ranks of the homotopy groups of a simply connected four-manifold depend only on its second Betti number. We also consider the based loop spaces of the gauge groups and the spaces of connections of a simply connected four-manifold and, appealing to~\cite{TI} and using the models from the rational homotopy theory, we obtain the explicit formulas for their rational Pontrjagin homology rings.