Speaker: Gabriel Angelini-Knoll, Wayne State University

Periodicity is a highly studied phenomenon in homotopy theory. For example, R. Bott showed that the homotopy groups of the classifying space of the infinite orthogonal group are periodic with period eight. This periodicity is also reflected in the stable homotopy groups of spheres using the J homomorphism, a map from the homotopy groups of the infinite orthogonal group to the stable homotopy groups of spheres. Using this map J.F. Adams produced a “height one” periodic family in the homotopy groups of spheres.
The image of J can be realized as a topological space and at odd primes this space is equivalent to algebraic K-theory of certain finite fields after p-completion. Therefore, a periodic family of height one in the language of chromatic homotopy theory is detected by algebraic K-theory of finite fields. In my talk, I will describe a higher height version of this phenomenon that I prove in my thesis. In particular, I demonstrate that a periodic family of height two is detected in mod (p, v_1)-homotopy of iterated algebraic K-theory of a finite field of order q, where q is a generator of the units in the p-adic integers. This result gives some evidence for a red-shift type phenomenon, which loosely states that algebraic K-theory increases the wavelength of periodicity.