Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant

TL;DR

This paper solves a key problem related to the 11/8-Conjecture in 4-manifold topology by analyzing Pin(2)-equivariant stable maps, leading to a new geometric inequality in the field.

Contribution

It provides a complete solution to the existence of certain Pin(2)-equivariant stable maps, introducing new techniques and results in equivariant stable homotopy theory.

Findings

Proves the '10/8+4'-Theorem in 4-dimensional topology.

Analyzes maps between finite spectra using cell diagrams and spectral sequences.

Advances understanding of the Mahowald invariant in the equivariant setting.

Abstract

In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the $j$-based Atiyah-Hirzebruch spectral sequence.