  On subsets E of the Mandelbrot set M, homeomorphisms are constructed by
quasi-conformal surgery. When the dynamics of quadratic polynomials is changed
piecewise by a combinatorial construction, a general theorem yields the
corresponding homeomorphism h: E to E in the parameter plane. Each h has two
fixed points in E, and a countable family of mutually homeomorphic fundamental
domains. Possible generalizations to other families of polynomials or rational
mappings are discussed. The homeomorphisms on subsets E of M constructed by
surgery are extended to homeomorphisms of M, and employed to study groups of
non-trivial homeomorphisms h: M to M . It is shown that these groups have the
cardinality of the continuum, and they are not compact.
