Balanced Splittings of Semi-free Actions on Homotopy Spheres

n+k Let 23 be a hornotopy (n+k)-sphere and p : G)< ~E~-2] a smooth semi-free action of a finite group G on 23 with fixed-point set a manifold F n of dimension no A decomposition of 23 into two G-invariant disks will be called a splitting of the action and the induced splitting of 2] G denoted F = FI<9 F 2. We ask whether every such action has a splitting with Hi(F1) ~Hi(F2) for i> 0 (these are called balanced spl[ttin~s)o One class of actions for which balanced splittings exist is obtained by the "twisted double" construction. Namely, let p : G XD n+k~ D n+k be a semi-free action of G on an (n+k)-disko Let Z = D<_) D where q~ : 8D-* 8D is an equivariant d[ffeomorphism. Our interest in the problem considered here arose from trying to understand the conditions under which a given seml-free action is a twisted double. An action that admits a balanced splitting resembles a twisted double at least homo-logically and thus exhibits some symmetry. On the other hand, an action with no balanced splitting is rather strongly asymmetrical. In this paper we introduce a semi-characteristic [nvar[ant of the action to detect the existence of balanced splittings and construct some examples of actions whose semi-characteristic invar[ant is nonzero. Such actions have no balanced splitting. For most of our results, the arguments are outlined here so that the reader who is familiar with work in this area [2]) can follow them. Full details will appear elsewhere. Before beginning a precise description of the results, we remark that the fixed-point set F will be assumed nonempty and connected throughout to avoid