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

DOI:
https://doi.org/10.1090/S0002-9947-05-04140-1

Published electronically:
December 20, 2005

MathSciNet review:
2197437

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Abstract: Let be a smooth closed 4-manifold with a free circle action generated by a vector field Then for any invariant symplectic form on the contracted form is non-vanishing. Using the map and the related map to we study the topology of the space of invariant symplectic forms on For example, the first map is proved to be a homotopy equivalence. This reduces examination of homotopy properties of to that of the space of non-vanishing closed 1-forms satisfying certain cohomology conditions. In particular we give a description of 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:
https://doi.org/10.1090/S0002-9947-05-04140-1

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.