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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Symplectic forms invariant under free circle actions on 4-manifolds


Authors: Boguslaw Hajduk and Rafal Walczak
Journal: Trans. Amer. Math. Soc. 358 (2006), 1953-1970
MSC (2000): Primary 53D05; Secondary 57S25
Published electronically: December 20, 2005
MathSciNet review: 2197437
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Abstract: Let $ M$ be a smooth closed 4-manifold with a free circle action generated by a vector field $ X.$ Then for any invariant symplectic form $ \omega$ on $ M$ the contracted form $ \iota_X\omega$ is non-vanishing. Using the map $ \omega \mapsto \iota_X\omega$ and the related map to $ H^1(M \slash S^1,\mathbb{R})$ we study the topology of the space $ S_{inv}(M)$ of invariant symplectic forms on $ M.$ For example, the first map is proved to be a homotopy equivalence. This reduces examination of homotopy properties of $ S_{inv}$ to that of the space $ \mathcal{N}_L$ of non-vanishing closed 1-forms satisfying certain cohomology conditions. In particular we give a description of $ \pi_0S_{inv}(M)$ in terms of the unit ball of Thurston's norm and calculate higher homotopy groups in some cases. Our calculations show that the homotopy type of the space of non-vanishing 1-forms representing a fixed cohomology class can be non-trivial for some torus bundles over the circle. This provides a counterexample to an open problem related to the Blank-Laudenbach theorem (which says that such spaces are connected for any closed 3-manifold). Finally, we prove some theorems on lifting almost complex structures to symplectic forms in the invariant case.


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Additional Information

Boguslaw Hajduk
Affiliation: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland – and – Department of Mathematics and Information Technology, University of Warmia and Mazury, Żołnierska 14A, 10-561 Olsztyn, Poland
Email: hajduk@math.uni.wroc.pl

Rafal Walczak
Affiliation: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland – and – Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-956 Warsaw, Poland
Email: rwalc@math.uni.wroc.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-05-04140-1
PII: S 0002-9947(05)04140-1
Keywords: Circle action, symplectic form, Thurston norm
Received by editor(s): December 23, 2003
Published electronically: December 20, 2005
Additional Notes: Both authors were partially supported by Grant 2 P03A 036 24 of the Polish Committee of Sci. Research.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.