Geometry and Topology Seminar

Monday, October 29, 2018 at 3:30pm to 5:30pm


Zhouli Xu (MIT) and Jianfeng Lin (MIT)

Talk 1 - 3:30pm

The geography problem on 4-manifolds: 10/8 + 4

Abstract: 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. 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. This is joint work with Mike Hopkins and XiaoLin Danny Shi.


Talk 2 - 4:30pm

The Pin(2)-Equivariant Mahowald Invariant

Abstract: The existence of Pin(2)-equivariant stable maps between representation spheres has deep applications in 4-dimensional topology. We will sketch the proof for our main result on the Pin(2)-equivariant Mahowald invariants. More specifically, we will discuss the Pin(2)-equivariant Mahowald invariants of powers of certain Euler classes in the RO(Pin(2))-graded equivariant stable homotopy groups of spheres. The proof analyzes maps between certain finite spectra arising from BPin(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. This is joint work with Mike Hopkins and XiaoLin Danny Shi. This talk provides the homotopy theory required by the first talk but will not depend upon it.


(Please note unusual time and location)


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