Summary: A note on Contact Structures
S. Akbulut and R. Matveyev
We construct two contact structures on a homology sphere M, that are
homotopic through 2-plane fields, but are not isotopic as contact struc-
tures. A diffeomorphism of M permutes these two contact structures.
In [LM] Lisca and Mati´c gave examples of homotopic but not isotopic contact
structures. In this note we give a simple construction of another such example.
Unlike the examples of Lisca and Mati´c, in our example the group of diffeo-
morphism of the 3-manifold acts nontrivially on the set of path components
of oriented contact structures of the manifold, while the action of this group is
trivial on the set of path components of the oriented tangent plane distributions.
Our construction of contact structures uses theorem of Y. Eliashberg, charac-
terizing symplectic 4-manifolds with pseudo-convex boundary (PC -manifolds).
Theorem 1 ([E]) : Let X = B4
(1 - handles) (2 - handles) be four-
dimensional handlebody with one 0-handle and no 3- or 4-handles. Then:
· The standard PC structure on B4
can be extended over 1-handles so that
manifold X1 = B4