ℤ2 surgery theory and smooth involutions on homotopy complex projective spaces

Abstract

Let a group act smoothly on a manifold M. One of the fundamental problems in transformation groups is to study relations between the global invariants of M (e.g. Pontrjagin classes) and invariants of the fixed point set. The Atiyah-Singer index theorem gives profound answers to this problem, which are necessary conditions of the action. Conversely it is interesting to ask if those are sufficient conditions. In other words, to what extent are there actions realizing such relations ? In this paper we deal with the realization problem of this kind for smooth involutions on homotopy complex projective spaces. Let X be a 2(N-l)-dimensional closed smooth manifold homotopy equivalent to the complex projective space p(~N). We call such X a homotopy P(C N) briefly. Suppose that X supports a smooth involution, that is to say, an order two group (denoted by G throughout this paper) acts on X. Then Bredon-Su's Fixed Point Theorem (see p.382 of [B]) describes the oohomologieal nature of the fixed point set X G of X. It depends on the number of connected components of X G :