Taming Free Circle Actions

It is shown that an arbitrary free action of the circle group on a closed manifold of dimension at least six is concordant to a "tame" action (so that the orbit space is a manifold). A consequence is that the concordance classification of arbitrary free actions of the circle on a simply connected manifold is the same as the equivariant homeomorphism classification of free tame actions. Consider a free action of a compact Lie group G of positive dimension on a topological manifold M. Such an action is called tame if the orbit space M/G is a manifold and wild otherwise. Starting with a tame free action on M one may obtain uncountably many inequivalent wild actions on M by collapsing out noncellular arcs in M/G, provided M/G has dimension at least three. For an exposition of this construction see L. Lininger [9]. The existence of such wild actions makes classification theorems difficult without a tameness hypothesis. In this paper a standard concordance equivalence relation on the set of free actions on a manifold is considered. At least for the circle group Sx, the concordance equivalence relation makes tame classification theorems applicable in the general case. Define a G-concordance to be a G-action on M X [0,1] preserving M X 0 and M X 1. Actions 6, then any free S -action on M is S -concordant to a tame action. There are both a relative version and a suitable uniqueness statement. One might conjecture a similar result for arbitrary compact Lie groups. C.T.C. Wall [12] has classified equivariant homeomorphism classes of tame free s'-actions on spheres S2n+X, n > 3, by classifying the possible orbit spaces using the theory of surgery. Wall conjectured [12, p. 192] that even in the wild case this calculation can be interpreted to give a correct result. The Main Theorem provides an affirmative answer to this conjecture. See Corollary 3.3. Received by the editors April 12, 1976. AMS (MOS) subject classifications (1970). Primary 57E10, 57A99.