All Finite Groups Act on Fake Complex Projective Spaces

We prove the result asserted in the title. The object of this note is to prove the assertion of its title in a slightly more precise form that shows that one can somewhat control the normal representation at a fixed point and that there is an infinite amount of choice for the rational Pontrjagin classes. This result is in sharp contrast to a conjecture of Ted Pétrie [5] that only homotopy CP"'s with the same Pontrjagin classes as standard CPn can have smooth S actions. Our method, relying on the surgery theory of Browder and Quinn, implies nothing about his case. This should not be viewed as support for the conjecture since the method also fails in the PL locally linear case, for which counterexamples have been produced in profusion [2]. Many cases of our theorem have been proven, in some cases in a more precise form; in particular M. Hughes has proven the assertion of our title for all odd order groups by a different method. See [3, 4] and the references given there. Definition. The graph of a group action has as labeled vertices the components of fixed sets of the subgroups that label the subgroup, and as edges maps induced by inclusions. If all components are simply connected and have codimension at least three in one another, the Browder-Quinn surgery group of the situation [1] just depends (modulo dimensions mod 4) on the graph of the action. We call this hypothesis H. Proposition. For any graph there is an N, such that if G acts smoothly on CPn , n> N, with given graph and satisfying hypothesis H, then G acts smoothly on infinitely many homotopy CP" 's in an equivariantly homotopy equivalent way, and with the same normal representations. Received by the editors September 7, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 57S17, 57S30.