Abstract: In the reference [1], Kontsevich constructed a family of characteristic classes for framed smooth fiber bundles by developing the notion of graph complex and the technique of the configuration space integral.  The goal of this course is to introduce Kontsevich’s characteristic classes for framed smooth fiber bundles and Watanabe’s discovery that these classes can detect the surprising difference between some i-th rational homotopy groups of the diffeomorphism group of the n-sphere and the orthogonal group O(n+1) in nonstable range, which is roughly i>n/3. As the background, we will also review historically the study of the homotopy type of the diffeomorphism groups.

Prerequisites: Elementary algebraic topology and differential topology

Reference for the course:
[1] M. Kontsevich, Feynman diagrams and low-dimensional topology. First European Congress of Mathematics, Vol. II (Paris, 1992), 97–121, Progr. Math., 120, Birkhäuser, Basel, 1994.
[2] Watanabe, Tadayuki On Kontsevich's characteristic classes for higher dimensional sphere bundles. I. The simplest class. Math. Z. 262 (2009), no. 3, 683–712.
[3] Watanabe, Tadayuki On Kontsevich's characteristic classes for higher-dimensional sphere bundles. II. Higher classes. J. Topol. 2 (2009), no. 3, 624–660.