For periodic instantons, interesting results have been obtained by Garland and Murray [3], [4]. On the other hand gauge anomalies have been investigated by Alvarez-Gaume and Ginsparg [1] and others. By their work we know that gauge anomalies are deeply concerned with homotopy theory.
　In this article we discuss periodic instantons which satisfy suitable conditions((3), (4) in Section 2). In Section 2, using a conformal compactification [7] the action is identified with the gauge anomaly in [1] up to the constant 4π2. In Section 3 we consider periodic instantons with even topological charges. There werefer to [6] for lifting group actions on bundles. For the SU(2) instantons we can directly verify the existence of a lifting action, but for a use in future we discuss from a general view point.
　Troughout the article S1 denotes the circle with length 2π, and w.n. means a winding number.