A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8.

Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants of powers of certain Euler classes in the RO(Pin(2))-graded equivariant stable homotopy groups of spheres. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. For the proof, we use the technique of cell-diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence.

This is joint work with Michael Hopkins, Jianfeng Lin and XiaoLin Danny Shi.

Professor Zhouli Xu is a C.L.E. Moore instructor at Department of Mathematics at Massachusetts Institute of Technology. He received his Ph. D. from the University of Chicago in 2017, under the supervision of J. Peter May, Daniel C. Isaksen, and Mark Mahowald. His research interests include Classical, motivic and equivariant stable homotopy groups of spheres, with connections and applications to geometric topology and chromatic homotopy theory. He received 2018 ICCM Best Paper Award, joint with Guozhen Wang on the paper "The triviality of the 61-stem in the stable homotopy groups of spheres".