A Hochschild homology Euler characteristic for circle actions
  We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for the Euler class (integer coefficients) of the normal bundle to a smooth circle action without fixed points on a manifold. In the special case of a 3-dimensional Seifert fibered space, this formula is particularly effective. \~ 
Ross Geoghegan, Andrew Nicas