GROUP-ACTIONS AND THE SINGULAR SET

For any fibration there is a number we call the trace which measures the best natural transfer that exists. For a group G acting on a space X we get the fibration from the Borel construction and we say the trace for that fibration is the trace of the action, written tr(G, X). The singular set of an action is the subspace of X where G is not acting freely. The main theorem states that the trace of an action is equal to the trace of the action on its singular set. We use the techniques arising in the proof of the main theorem to study the size of the smallest orbit of the action. Finally we compare the trace of an action with the closely related exponent of the action as studied by W. Browder and A. Adem.