Blowup and Fixed Points

Abstract

Blowing up a point p in a manifold M builds a new manifold M' in which p is replaced by the projectivization of the tangent space of M at p. This well-known operation also applies to fixed points of diffeomorphisms, yielding continuous homomorphisms between automorphism groups of M and M'. The construction for maps involves a loss of regularity and is not unique at the lowest order of differentiability. Fixed point sets and other aspects of blownup dynamics at the singular locus are described in terms of derivative data; continuous data are not sufficient to determine much about these issues.