Absolute neighbourhood retracts and spaces of holomorphic maps from Stein manifolds to Oka manifolds
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Abstract:
The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is $\mathcal {O}(S,X)$ a deformation retract of $\mathcal {C}(S,X)$? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points, in particular, if $S$ is affine algebraic. The only property of $X$ used in the proof is the parametric Oka property with approximation with respect to finite polyhedra, so our theorem holds under the weaker assumption that $X$ is an Oka manifold. Our main tool, apart from Oka theory itself, is the theory of absolute neighbourhood retracts. We also make use of the mixed model structure on the category of topological spaces.References
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Additional Information
- Finnur Lárusson
- Affiliation: School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia
- MR Author ID: 347171
- Email: finnur.larusson@adelaide.edu.au
- Received by editor(s): June 23, 2013
- Published electronically: October 16, 2014
- Additional Notes: The author was supported by Australian Research Council grant DP120104110.
The author is grateful to Jaka Smrekar for helpful discussions. - Communicated by: Franc Forstneric
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 1159-1167
- MSC (2010): Primary 32E10; Secondary 32H02, 32Q28, 54C35, 54C55, 55M15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12335-5
- MathSciNet review: 3293731