A Hochschild Homology Euler Characteristic for Circle Actions

We define an S¹-Euler characteristic, _S¹(X), of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group HH₁(G) where G is the fundamental group of X. This _S¹(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 three-dimensional Seifert fibered space, this formula is particularly effective.