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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.

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