  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.
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