Abstract: We consider low dimensional homology groups of the smooth (or real analytic), orientation preserving diffeomorphism group of S^1. The Euler class is a second cohomology class and a fundamental characteristic class.
It is an open problem whether some power of the rational Euler class vanishes for real analytic flat S^1 bundles. On the other hand, it is well known that any power of the rational Euler class never vanishes in the smooth case.
In this talk we discuss that this problem is related with the homological images of lens spaces in the homology groups of the diffeomorphism group. Along this line we can give a new proof for the non-triviality of any power of the rational Euler class in the smooth case. This talk will be based on a joint work with Shigeyuki Morita and Yoshihiko Mitsumatsu.
