Semi-free ? p -Actions on highly-connected manifolds

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O. Introduction In this paper we study semi-free cyclic group actions on highly-connected manifolds whose dimensions are a multiple of 4. The actions we consider are required to induce the identity map on homology, and to have fixed-point sets either a union of isolated points or a highly-connected submanifold. We consider both smooth and piecewise-linear actions. There are two basic problems in the study of such actions- when they exist and what they look like. We attack these problems in three stages: First, we construct specific examples of manifolds with actions on them. Second we show that a salient characteristic of these actions holds generally, so that any action resembles one we construct. Third, we show how to modify specific actions in order to obtain actions in great generality. More specifically, we proceed as follows: In Chapter I we construct actions on highly-connected manifolds via an equivariant plumbing technique. That is, we first construct bundles with actions on them, and then plumb them together respecting the actions. In order to use this technique to construct actions with isolated fixed points, we must first find a basis for the middle-dimensional homology with respect to which the cupproduct form has a matrix of a certain type. An algebraic argument shows one always exists. Its existence also helps in constructing actions with positivedimensional fixed-point sets. By construction, these actions have invariant spheres forming a basis for the middle-dimensional homology. In Chapter II we show that such invariant spheres always exist for cyclic group actions which have isolated fixed points and act trivially on homology. (Although the specific calculations are for oddorder groups, the same proof works for arbitrary cyclic groups.) Thus the actions we construct differ from those in the general case only in that we begin with linear actions on normal bundles to these spheres, rather than on arbitrary PL actions on regular neighborhoods of these spheres. Chapter III is devoted to showing how to modify actions (such as, but by no means only, those constructed in Chapter I) to obtain new actions on new manifolds. Here the group actions we consider are those of cyclic groups of