Kervaire invariant one

Tags: #homotopy #open/conjectures

References

https://courses.math.rochester.edu/current/549/notes/

See also surgery.

Motivation

The Kervaire invariant is an invariant of a certain framed manifold.

In 1956, Milnor found a curious example of a manifold. He was studying sphere bundles over spheres, and found that there was a bundle of the form \(S^3\to X\to S^4\), and that \(X\) is homeomorphic to \(S^7\), but it is not diffeomorphic to \(S^7\). In other words, there exist exotic smooth structures on manifolds

When does there exist a manifold of 126.

The Kervaire invariant has to do with which stable homotopy groups can be represented by exotic spheres.

Setup

  • Define \(bP_{n+1} \leq \Theta_n\) the subgroup of spheres that bound parallelizable manifolds.

  • The Kervaire invariant is an invariant of a framed manifold that measures whether the manifold could be surgically converted into a sphere.

    • 0 if true, 1 otherwise.

  • Hill-Hopkins-Ravenel :

    • It equals 0 for \(n \geq 254\).
    • Kervaire invariant = 1 only in 2, 6, 14, 30, 62.
    • Open case: 126.

  • Punchline: there is a map \(\Theta_n/bP_{n+1} \to \pi_n^S/ J\), (to be defined) and the Kervaire invariant influences the size of \(bP_{n+1}\).

    • This reduces the differential topology problem of classifying smooth structures to (essentially) computing homotopy groups of spheres.

  • Open question: is there a manifold of dimension 126 with Kervaire invariant 1?

Links to this page
#homotopy #open/conjectures