Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

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 )

\operatorname {Pin}(2)

-equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the

Pin ⁡ ( 2 )

\operatorname {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

B Pin ⁡ ( 2 )

B\operatorname {Pin}(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 j

-based Atiyah–Hirzebruch spectral sequence.