Fundamental group and analytic disks
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- by Dayal Dharmasena and Evgeny A. Poletsky PDF
- Trans. Amer. Math. Soc. 371 (2019), 709-728 Request permission
Abstract:
Let $W$ be a domain in a connected complex manifold $M$ and let $w_0\in W$. Let $\mathcal {A}_{w_0}(W,M)$ be the space of all continuous mappings of a closed unit disk $\overline {\mathbb D}$ into $M$ that are holomorphic on the interior of $\overline {\mathbb D}$, and let $f(\partial \mathbb {D})\subset W$ 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.
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 $b,c,d,g\in S$.
Let $\mathcal {R}_{w_0}(W,M)$ be the space of all continuous mappings of $\overline {\mathbb D}$ 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 $\mathcal {A}_{w_0}(W,M)$.
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Additional Information
- Dayal Dharmasena
- Affiliation: Department of Mathematics, Syracuse University, 215 Carnegie Hall, Syracuse, New York 13244
- Address at time of publication: Department of Mathematics, Faculty of Science, University of Colombo, Colombo 03, Sri Lanka
- Email: dayaldh@sci.cmb.ac.lk
- Evgeny A. Poletsky
- Affiliation: Department of Mathematics, Syracuse University, 215 Carnegie Hall, Syracuse, New York 13244
- MR Author ID: 197859
- Email: eapolets@syr.edu
- Received by editor(s): August 15, 2016
- Received by editor(s) in revised form: May 22, 2017
- Published electronically: June 20, 2018
- Additional Notes: The second author was partially supported by a grant from the Simons Foundation.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 709-728
- MSC (2010): Primary 32Q55; Secondary 32H02, 32E30
- DOI: https://doi.org/10.1090/tran/7323
- MathSciNet review: 3885158