Homotopy theoretical considerations of the Bauer-Furuta stable homotopy Seiberg-Witten invariants

TL;DR

This paper explores the limitations of Bauer-Furuta stable homotopy Seiberg-Witten invariants using homotopy theory, demonstrating their role in adjunction inequalities and explaining why they cannot resolve the 11/8-conjecture.

Contribution

It provides a homotopy-theoretic analysis of the invariants, highlighting their non-existence results and explaining their insufficiency for certain conjectures.

Findings

Non-existence results are crucial for applications of the invariants.

A unified proof of the adjunction inequalities is presented.

Nilpotency explains the limitations of the invariants in proving the 11/8-conjecture.

Abstract

We show the "non-existence" results are essential for all the previous known applications of the Bauer-Furuta stable homotopy Seiberg-Witten invariants. As an example, we present a unified proof of the adjunction inequalities. We also show that the nilpotency phenomenon explains why the Bauer-Furuta stable homotopy Seiberg-Witten invariants are not enough to prove 11/8-conjecture.