(American Mathematical Society (AMS), 2022-02-23) Lin, Jianfeng; Shi, XiaoLin Danny; Xu, Zhouli; Michael, Hopkins; Hopkins, Michael
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.
