Seminar: THE KERVAIRE INVARIANT AND MANIFOLDS WITH CORNERS

Time:

Venue/Location: Phòng C101, VIASM

Time: Every Wednesday, starting from October 22, 2025 to December 10, 2025, from 10:00 to 11:45.

Venue: Room C101, Vietnam Institute for Advanced Study in Mathematics (VIASM), Hanoi

Speaker: Prof. Jean Lannes, Université Paris Diderot, France

Abstract: In 1956, John Milnor showed that in dimension 7, there exist manifolds that have the same topology as an ordinary seven-dimensional sphere, but have a different differential structure. Such manifolds are called exotic spheres. In the 1960s, Milnor, Michel Kervaire, and William Browder showed that in almost all dimensions, every manifold — or more precisely, any framed manifold — can be converted into an exotic sphere by performing a process called surgery. In surgery, a part of the manifold gets drilled out, and a new piece is glued in along the boundary of the excised piece. The work of Milnor, Kervaire and Browder left an important gap. They didn’t know what happens in dimensions of the form \(2^{n} – 2\), for whole-number values of n: that is, dimensions 2, 6, 14, 30, 62, 126, 254, 510, etc. In these dimensions, it was conceivable that some manifolds might exist that could not be converted into spheres via surgery. Building on algebra developed by the late Turkish mathematician Cahit Arf, Kervaire defined an invariant of a framed manifold — that is, a number determined by the manifold’s topology — that measures whether the manifold could be surgically converted into a sphere. This number, called the Arf-Kervaire invariant, evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. In any given dimension, there are only two possibilities: either all manifolds have Arf-Kervaire invariant equal to 0, or half have Arf-Kervaire invariant 0 and the other half have Arf-Kervaire invariant 1. In 1969, Browder proved that there was a framed manifold with nontrivial Arf-Kervaire invariant if and only if a certain element in the Adams spectral sequence survives. In 2008, Jean Lannes and Haynes Miller obtained a proof of Browder's theorem using manifolds with corners. The aim of the course would be to describe this proof.

Mode of participation: Hybrid

Link Zoom: Join Zoom Meeting

https://zoom.us/j/96139101608?pwd=bTxkJk9vT7sGmfera0UZa0eJRzZ1uN.1 

Meeting ID: 961 3910 1608

Passcode: 833851 

Contact

Ms. Nguyễn Hồng Anh, email: nguyenhonganh@viasm.edu.vn