Fixed Point Sets of Compact Abelian Lie Groups of Transformations

Abstract

P. A. Smith [11, 12] has formulated and studied the problem of determining the extent to which the fixed point set F of a periodic map of the n-sphere Sn resembles a sphere of lower dimension. For n = 3, Smith proved [10] that F is actually a lower dimensional sphere; Brouwer [1] had previously proved the corresponding theorem for n = 2. For general n, the Smith special homology theory yielded that, if the period is a power of a prime p, then F has the mod p homology properties of a sphere. In the case of composite period it is a wellknown question of Smith [2, Problem 45] as to whether F has the homology of groups of a sphere. In Section 2, we construct a simply connected compact triangulable manifold X having the (reals mod 1) homology groups of an nsphere; there is also constructeda simplicial periodic map of period 6 on X whose fixed point set does not have the homology groups of a sphere (over any non-trivial coefficient group). While this does not answer the question of Smith, it indicates strongly that the answer is in the negative.' The construction of the example leans heavily on an earlier example of the author [6]. In Section 3, we turn to the action of a circle group T operating on a homological n-sphere X. In this case, the fixed point set F is considerably simpler. For example, F has the reals mod 1 homology groups of an Sr. Smith has proved earlier [8] that F has the mod p homology of an Srp for all prime p.