Lukas Schneider

Title:	Topological K-theory and the image of the J-homomorphism
Speaker:	Lukas Schneider
Time:	Tuesday, 22.04.2025, 14:00
Place:	SR 2.059

Despite their rather simple definition, calculating the higher homotopy groups $\pi_r(X)$ for a space $X$ is difficult. Even for spheres these groups are still unknown in general. The (reduced) suspension functor $\Sigma$ induces a homomorphism $\Sigma\colon \pi_r(X)\to \pi_{r+1}(\Sigma X)$ . Iterating suspension gives the sequence $\pi_r(X)\to\pi_{r+1}(\Sigma X)\to \pi_{r+2}(\Sigma^2X)\to\dotsb$

It follows from the Freudenthal suspension theorem (1937) that this sequence stabilizes, i.e. the map $\pi_{r+n}(\Sigma^n X)\to \pi_{r+n+1}(\Sigma^{n+1}X)$ is an isomorphism, for large enough $n$ . The group at which this sequence stabilizes is denoted by $\pi_r^s(X)$ and is called $r$ -th stable homotopy of $X$ . The discovery of this stabilization phenomena was the starting point of an area which is nowadays known as stable homotopy theory.

A lot of work has been done to calculate the stable homotopy groups of interesting spaces. Nevertheless the stable homotopy groups of spheres $\pi_r^s:=\pi_r^s(S^0)$ are only known up to roughly $r\leq90$ . The first calculations were done by Adams in 1966. He calculated the image of the so-called $J$ -homomorphism $J:\pi_r(\mathrm{SO})\to\pi_r^s$ discovering cyclic direct summands of $\pi_r^s$ . The main tool of Adams’s proofs is topological $K$ -theory.

In this talk we will introduce the stable homotopy groups and the $J$ -homomorphism. After that we will discuss topological $K$ -theory to then sketch the proof of the above mentioned calculation of the image of the $J$ -homomorphism.