Fundamental group and analytic disks

Dayal Dharmasena, Evgeny A. Poletsky

Abstract: Let W be a domain in a connected complex manifold M and w_0\in W. Let {\mathcal A}_{w_0}(W,M) be the space of all continuous mappings of a closed unit disk \overline D into M that are holomorphic on the interior of \overline D, f(\partial\mathbb D)\subset W and f(1)=w_0. On the homotopic equivalence classes \eta_1(W,M,w_0) of {\mathcal A}_{w_0}(W,M) we introduce a binary operation \star so that \eta_1(W,M,w_0) becomes a semigroup and the natural mappings \iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0) and \delta_1:\,\eta_1(W,M,w_0)\to\pi_2(M,W,w_0) are homomorphisms. \par We show that if W is a complement of an analytic variety in M and if S=\delta_1(\eta_1(W,M,w_0)), then S\cap S^{-1}=\{e\} and any element a\in\pi_2(M,W,w_0) can be represented as a=bc^{-1}=d^{-1}g, where b,c,d,g\in S. \par Let {\mathcal R}_{w_0}(W,M) be the space of all continuous mappings of \overline D into M such that f(\partial{\mathbb D})\subset W and f(1)=w_0. We describe its open dense subset {\mathcal R}^{\pm}_{w_0}(W,M) such that any connected component of {\mathcal R}^{\pm}_{w_0}(W,M) contains at most one connected component of {\mathcal A}_{w_0}(W,M).

Journal: Trans. Amer. Math. Soc.

DOI: 10.1090/tran/7323