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We study the parameter space structure of degree $d \ge 3$ one complexvariable polynomials as dynamical systems acting on $\C$. We introduce andstudy {\it straightening maps}. These maps are a natural higher degreegeneralization of the ones introduced by Douady and Hubbard to prove theexistence of small copies of the Mandelbrot set inside itself. We establishthat straightening maps are always injective and that their image contains allthe corresponding hyperbolic systems. Also, we characterize straightening mapswith compact domain. Moreover, we give two classes of bijective straighteningmaps. The first produces an infinite collection of embedded copies of the$(d-1)$-fold product of the Mandelbrot set in the connectedness locus of degree$d \ge 3$. The second produces an infinite collection of full families ofquadratic connected filled Julia sets in the cubic connectedness locus, suchthat each filled Julia set is quasiconformally embedded.

Combinatorics and topology of straightening maps I: compactness and bijectivity