A Hochschild Homology Euler Characteristic for Circle Actions

Abstract. We define an “S -Euler characteristic”, χ̃S1 (X) , of a circle action on a compact manifold or finite complex X . It lies in the first Hochschild homology group HH1(ZG) where G is the fundamental group of X . This χ̃S1 (X) 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.