The topology of spaces of rational functions

where a, and b t are complex numbers, defines a continuous map of degree n from the Riemann sphere S2=CtJ co to itself. I f the coefficients (a 1 .. . . . an; bl . . . . . bn) va ry cont inuously in C 2" the m a p ] varies continuously providing the polynomials p and q have no root in common; but the topological degree of the map / jumps when a root of p moves into coincidence with a root of q. Let F* denote the open set of (~" consisting of pairs of monic polynomials (p, q) of degree n with no common root. F* is the complement of an algebraic hypersurface, the "resul tant locus", in (~2~. On the other hand it can be identified with a subspace of the space M~* of maps $ 2 ~ S 2 which take c~ to 1 and have degree n. I n this paper I shall prove tha t when n is large the 2n-dimensional complex var ie ty F* is a good approximat ion to the homotopy type of the space M*, or, more precisely

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