Manolescu Invariants of Connected Sums

TL;DR

This paper establishes inequalities for Manolescu invariants under connected sums, computes these invariants for specific spaces, and uses them to prove Furuta's Theorem, revealing new insights into the homology cobordism group.

Contribution

It introduces inequalities for Manolescu invariants under connected sums and provides the first monopole-based proof of Furuta's Theorem.

Findings

Inequalities for Manolescu invariants under connected sum

Computed invariants for sums of Seifert fiber spaces

Proved Furuta's Theorem using monopoles

Abstract

We give inequalities for the Manolescu invariants $\alpha,\beta,\gamma$ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a $\mathbb{Z}^\infty$ subgroup of the homology cobordism group. To our knowledge, this is the first proof of Furuta's Theorem using monopoles. We also provide information about Manolescu invariants of the connected sum of $n$ copies of a three-manifold $Y$, for large $n$.