Contact surgery
A special type of surgery on a (strict) contact manifold (i.e. a smooth manifold admitting a (strict) contact structure
, where
is a
-form satisfying
), which results in a new contact manifold.
In topological terms, surgery on denotes the replacement of an embedded copy of
, a tubular neighbourhood of an embedded
-sphere with trivial normal bundle, by a copy of
, with the obvious identification along the boundary
. Alternatively, one can attach a
-handle
along
to a manifold
with boundary
, and the new boundary will be the result of performing surgery on
.
As shown by Y. Eliashberg [a2] and A. Weinstein [a11], contact surgery is possible along spheres which are isotropic submanifolds (cf. also Isotropic submanifold) of and have trivial normal bundle. The choice of framing, i.e. trivialization of the normal bundle, for which contact surgery is possible is restricted.
A contact manifold may be regarded as the strictly pseudo-convex boundary of an almost-complex (in fact, symplectic) manifold
such that
is given by the
-invariant subspace of the tangent bundle
. Contact surgery on
can then be interpreted as the attaching of an almost-complex or symplectic handle to
along
, and the framing condition for
is given by requiring the almost-complex structure on
to extend over the handle. For
the situation is more subtle, see [a2], [a5]. Weinstein formulates his construction in terms of symplectic handle-bodies, Eliashberg (whose results are somewhat stronger) in terms of
-convex Morse functions on almost-complex manifolds (cf. also Almost-complex structure; Morse function).
A Stein manifold of real dimension has the homotopy type of an
-dimensional CW-complex, cf. [a8], p. 39. Eliashberg uses his construction to show that for
this is indeed the only topological restriction on a Stein manifold, that is, if
is a
-dimensional smooth manifold with an almost-complex structure
and a proper Morse function
with critical points of Morse index at most
, then
is homotopic to a genuine complex structure
such that
is
-convex and, in particular,
is Stein.
The usefulness of contact surgery in this and other applications rests on the fact that there is an -principle for isotropic spheres. This allows one to replace a given embedding
by an isotropic embedding
(for
) that is isotopic to the initial one, provided only an obvious necessary bundle condition is satisfied: If
is an isotropic embedding, then its differential
extends to a complex bundle monomorphism
, where
inherits a complex structure from the (conformal) symplectic structure
. The relevant
-principle says that, conversely, the existence of such a bundle mapping covering
is sufficient for
to be isotopic to an isotropic embedding
.
This allows one to use topological structure theorems, such as Barden's classification of simply-connected -manifolds [a1], to construct contact structures on a wide class of higher-dimensional manifolds, see [a3].
In dimension (
) there is a different notion of contact surgery, due to R. Lutz and J. Martinet [a7]; it allows surgery along
-spheres embedded transversely to a contact structure
. This was used by Lutz and Martinet to show the existence of a contact structure on any closed, orientable
-manifold and in any homotopy class of
-plane fields. For applications of other topological structure theorems (such as branched coverings or open book decompositions, cf. also Open book decomposition) to the construction of contact manifolds, see [a4] and references therein.
Other types of surgery compatible with some geometric structure include surgery on manifolds of positive scalar curvature ([a6], [a9]) and surgery on manifolds of positive Ricci curvature ([a10], [a12]).
References
[a1] | D. Barden, "Simply connected five-manifolds" Ann. of Math. , 82 (1965) pp. 365–385 MR0184241 Zbl 0136.20602 |
[a2] | Y. Eliashberg, "Topological characterization of Stein manifolds of dimension ![]() |
[a3] | H. Geiges, "Applications of contact surgery" Topology , 36 (1997) pp. 1193–1220 MR1452848 Zbl 0912.57019 |
[a4] | H. Geiges, "Constructions of contact manifolds" Math. Proc. Cambridge Philos. Soc. , 121 (1997) pp. 455–464 MR1434654 Zbl 0882.57007 |
[a5] | R.E. Gompf, "Handlebody construction of Stein surfaces" Ann. of Math. , 148 (1998) pp. 619–693 MR1668563 Zbl 0919.57012 |
[a6] | M. Gromov, H.B. Lawson Jr., "The classification of simply connected manifolds of positive scalar curvature" Ann. of Math. , 111 (1980) pp. 423–434 MR0577131 Zbl 0463.53025 |
[a7] | J. Martinet, "Formes de contact sur les variétés de dimension![]() |
[a8] | J. Milnor, "Morse theory" , Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401 |
[a9] | R. Schoen, S.T. Yau, "On the structure of manifolds with positive scalar curvature" Manuscripta Math. , 28 (1979) pp. 159–183 MR0535700 Zbl 0423.53032 |
[a10] | J.-P. Sha, D.-G. Yang, "Positive Ricci curvature on the connected sum of ![]() |
[a11] | A. Weinstein, "Contact surgery and symplectic handlebodies" Hokkaido Math. J. , 20 (1991) pp. 241–251 MR1114405 Zbl 0737.57012 |
[a12] | D. Wraith, "Surgery on Ricci positive manifolds" J. Reine Angew. Math. , 501 (1998) pp. 99–113 MR1637825 Zbl 0915.53018 |
Contact surgery. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contact_surgery&oldid=16386