Seminar: What is ... the Kervaire invariant?

Benjamin Böhme will talk about what is the Kervaire invariant?

In 1963, Kervaire and Milnor classified "exotic" smooth structures on spheres, i.e. differentiable structures which do not come from restricting differentiation in Euclidean space to the unit sphere. Their solution depends on the "Kervaire invariant" which assigns either "0" or "1" to a (4k+2)-dimensional manifold. For about 50 years, finding manifolds of Kervaire invariant one has been one of the greatest open problems in geometric topology. It was solved by Hill, Hopkins and Ravenel in 2009, except in dimension 126, which remains a mystery until today.

In my talk, I will give a geometric description of the Kervaire invariant in low dimensions, compute some examples and provide a brief historical overview. If time permits, I will also explain Kervaire's construction of a 10-dimensional topological manifold which does not admit any smooth structure.

This lecture is part of the "What is..?" seminar.