Blowup and Fixed Points

Blowing up a point p in a manifold M builds a new manifold M̂ in which p is replaced by the projectivization of the tangent space TpM . This well–known operation also applies to fixed points of diffeomorphisms, yielding continuous homomorphisms between automorphism groups of M and M̂ . 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; C data are not sufficient to determine much about these issues. Topological generalizations of the blowup construction prove to be much less natural than the classical versions, and no lifting homomorphism for homeomorphism groups can be constructed.