Immersions associated with holomorphic germs

Abstract

A holomorphic germ $\Phi :({\mathbb{C}}^2,0)\to ({\mathbb{C}}^3,0)$ , singular only at the origin, induces at the links level an immersion of $S^3$ into $S^5$ . The regular homotopy type of immersions $S^3\looparrowright S^5$ are determined by their Smale invariant, defined up to a sign ambiguity. In this paper we fix a sign of the Smale invariant and we show that for immersions induced by holomorphic gems the sign-refined Smale invariant $\Omega$ is the negative of the number of cross caps appearing in a generic perturbation of $\Phi$ . Using the algebraic method we calculate $\Omega$ for some families of singularities, among others the A-D-E quotient singularities. As a corollary, we obtain that the regular homotopy classes which admit holomorphic representatives are exactly those, which have non-positive sign-refined Smale invariant. This answers a question of Mumford regarding exactly this correspondence. We also determine the sign ambiguity in the topological formulae of Hughes–Melvin and Ekholm–Szűcs connecting the Smale invariant with (singular) Seifert surfaces. In the case of holomorphic realizations of Seifert surfaces, we also determine their involved invariants in terms of holomorphic geometry.