# A higher chromatic analogue of the image of J.

*Algebra/Geometry/Topology Seminar*

*by Craig Westerland*

*Institution:*University of Melbourne

*Date: Fri 4th May 2012*

*Time: 3:15 PM*

*Location: 213 Richard Berry*

*Abstract*: Different cohomology theories "see" different parts of the stable homotopy groups of spheres. Singular cohomology, for instance, detects all maps from a sphere to itself using the notion of degree. K-theory detects a large swath of homotopy known as the "image of J," which can be described very geometrically using the relation between a vector bundle and its unit sphere-bundle. In this talk, which covers work that is very much in progress, I will discuss an analogue of the image of J for higher chromatic homotopy theory -- the part seen by the Morava K-theories, K(n). The result lacks the charming geometry of the J-homomorphism, and stretches the notion of "homotopy group," but will hopefully give us insight into the K(n)-local homotopy category.