| 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 for a space
is difficult. Even for spheres these groups are still unknown in general. The (reduced) suspension functor
induces a homomorphism
. Iterating suspension gives the sequence
It follows from the Freudenthal suspension theorem (1937) that this sequence stabilizes, i.e. the map is an isomorphism, for large enough
. The group at which this sequence stabilizes is denoted by
and is called
-th stable homotopy of
. 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 are only known up to roughly
. The first calculations were done by Adams in 1966. He calculated the image of the so-called
-homomorphism
discovering cyclic direct summands of
. The main tool of Adams’s proofs is topological
-theory.
In this talk we will introduce the stable homotopy groups and the -homomorphism. After that we will discuss topological
-theory to then sketch the proof of the above mentioned calculation of the image of the
-homomorphism.