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

Hopkins, Michael; Lin, Jianfeng; Shi, XiaoLin Danny; Xu, Zhouli

doi:10.1090/cams/4

Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant
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by Michael J. Hopkins, Jianfeng Lin, XiaoLin Danny Shi and Zhouli Xu

Comm. Amer. Math. Soc. 2 (2022), 22-132

DOI: https://doi.org/10.1090/cams/4

Published electronically: February 23, 2022

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Abstract:

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of $\operatorname {Pin}(2)$-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 $\operatorname {Pin}(2)$-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 $B\operatorname {Pin}(2)$ 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 $j$-based Atiyah–Hirzebruch spectral sequence.

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