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A special type of [[Surgery|surgery]] on a (strict) contact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202001.png" /> (i.e. a smooth manifold admitting a (strict) contact structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202003.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202004.png" />-form satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202005.png" />), which results in a new contact manifold.
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In topological terms, surgery on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202006.png" /> denotes the replacement of an embedded copy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202007.png" />, a tubular neighbourhood of an embedded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202008.png" />-sphere with trivial normal bundle, by a copy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202009.png" />, with the obvious identification along the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020010.png" />. Alternatively, one can attach a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020011.png" />-handle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020012.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020013.png" /> to a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020014.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020015.png" />, and the new boundary will be the result of performing surgery on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020016.png" />.
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As shown by Y. Eliashberg [[#References|[a2]]] and A. Weinstein [[#References|[a11]]], contact surgery is possible along spheres which are isotropic submanifolds (cf. also [[Ising model|Isotropic submanifold]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020017.png" /> and have trivial normal bundle. The choice of framing, i.e. trivialization of the normal bundle, for which contact surgery is possible is restricted.
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A special type of [[Surgery|surgery]] on a (strict) contact manifold $( M ^ { 2 n - 1 } , \xi )$ (i.e. a smooth manifold admitting a (strict) contact structure $\xi = \operatorname{ker} \alpha$, where $\alpha$ is a $1$-form satisfying $\alpha \wedge ( d \alpha ) ^ { n - 1 } \neq 0$), which results in a new contact manifold.
  
A contact manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020018.png" /> may be regarded as the strictly pseudo-convex boundary of an almost-complex (in fact, symplectic) manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020020.png" /> is given by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020021.png" />-invariant subspace of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020022.png" />. Contact surgery on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020023.png" /> can then be interpreted as the attaching of an almost-complex or symplectic handle to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020024.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020025.png" />, and the framing condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020026.png" /> is given by requiring the almost-complex structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020027.png" /> to extend over the handle. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020028.png" /> the situation is more subtle, see [[#References|[a2]]], [[#References|[a5]]]. Weinstein formulates his construction in terms of symplectic handle-bodies, Eliashberg (whose results are somewhat stronger) in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020030.png" />-convex Morse functions on almost-complex manifolds (cf. also [[Almost-complex structure|Almost-complex structure]]; [[Morse function|Morse function]]).
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In topological terms, surgery on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c1202006.png"/> denotes the replacement of an embedded copy of $S ^ { k } \times D ^ { m - k }$, a tubular neighbourhood of an embedded $k$-sphere with trivial normal bundle, by a copy of $D ^ { k + 1 } \times S ^ { m - k - 1 }$, with the obvious identification along the boundary $S ^ { k } \times S ^ { m - k - 1 }$. Alternatively, one can attach a $( k + 1 )$-handle $D ^ { k + 1 } \times D ^ { m - k }$ along $S ^ { k } \times D ^ { m - k }$ to a manifold $W ^ { m + 1 }$ with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020015.png"/>, and the new boundary will be the result of performing surgery on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020016.png"/>.
  
A [[Stein manifold|Stein manifold]] of real dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020031.png" /> has the homotopy type of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020032.png" />-dimensional [[CW-complex|CW-complex]], cf. [[#References|[a8]]], p. 39. Eliashberg uses his construction to show that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020033.png" /> this is indeed the only topological restriction on a Stein manifold, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020034.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020035.png" />-dimensional smooth manifold with an almost-complex structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020036.png" /> and a proper Morse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020037.png" /> with critical points of Morse index at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020038.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020039.png" /> is homotopic to a genuine complex structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020040.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020041.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020042.png" />-convex and, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020043.png" /> is Stein.
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As shown by Y. Eliashberg [[#References|[a2]]] and A. Weinstein [[#References|[a11]]], contact surgery is possible along spheres which are isotropic submanifolds (cf. also [[Ising model|Isotropic submanifold]]) of $( M , \xi )$ and have trivial normal bundle. The choice of framing, i.e. trivialization of the normal bundle, for which contact surgery is possible is restricted.
  
The usefulness of contact surgery in this and other applications rests on the fact that there is an [[H-principle|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020044.png" />-principle]] for isotropic spheres. This allows one to replace a given embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020045.png" /> by an isotropic embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020046.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020047.png" />) that is isotopic to the initial one, provided only an obvious necessary bundle condition is satisfied: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020048.png" /> is an isotropic embedding, then its differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020049.png" /> extends to a complex bundle monomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020051.png" /> inherits a complex structure from the (conformal) symplectic structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020052.png" />. The relevant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020053.png" />-principle says that, conversely, the existence of such a bundle mapping covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020054.png" /> is sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020055.png" /> to be isotopic to an isotropic embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020056.png" />.
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A contact manifold $( M , \xi = \operatorname { ker } \alpha )$ may be regarded as the strictly pseudo-convex boundary of an almost-complex (in fact, symplectic) manifold $W = ( M \times ( 0,1 ] , J )$ such that $\xi $ is given by the $J$-invariant subspace of the tangent bundle $T M$. Contact surgery on $M$ can then be interpreted as the attaching of an almost-complex or symplectic handle to $W$ along $M$, and the framing condition for $n &gt; 2$ is given by requiring the almost-complex structure on $W$ to extend over the handle. For $n = 2$ the situation is more subtle, see [[#References|[a2]]], [[#References|[a5]]]. Weinstein formulates his construction in terms of symplectic handle-bodies, Eliashberg (whose results are somewhat stronger) in terms of $J$-convex Morse functions on almost-complex manifolds (cf. also [[Almost-complex structure|Almost-complex structure]]; [[Morse function|Morse function]]).
  
This allows one to use topological structure theorems, such as Barden's classification of simply-connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020058.png" />-manifolds [[#References|[a1]]], to construct contact structures on a wide class of higher-dimensional manifolds, see [[#References|[a3]]].
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A [[Stein manifold|Stein manifold]] of real dimension $2 n$ has the homotopy type of an $n$-dimensional [[CW-complex|CW-complex]], cf. [[#References|[a8]]], p. 39. Eliashberg uses his construction to show that for $n &gt; 2$ this is indeed the only topological restriction on a Stein manifold, that is, if $W$ is a $2 n$-dimensional smooth manifold with an almost-complex structure $J$ and a proper Morse function $\varphi$ with critical points of Morse index at most $n$, then $J$ is homotopic to a genuine complex structure $J ^ { \prime }$ such that $\varphi$ is $J ^ { \prime }$-convex and, in particular, $( W , J ^ { \prime } )$ is Stein.
  
In dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020059.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020060.png" />) there is a different notion of contact surgery, due to R. Lutz and J. Martinet [[#References|[a7]]]; it allows surgery along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020061.png" />-spheres embedded transversely to a contact structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020062.png" />. This was used by Lutz and Martinet to show the existence of a contact structure on any closed, orientable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020063.png" />-manifold and in any homotopy class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020064.png" />-plane fields. For applications of other topological structure theorems (such as branched coverings or open book decompositions, cf. also [[Open book decomposition|Open book decomposition]]) to the construction of contact manifolds, see [[#References|[a4]]] and references therein.
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The usefulness of contact surgery in this and other applications rests on the fact that there is an [[H-principle|$h$-principle]] for isotropic spheres. This allows one to replace a given embedding $\iota : S ^ { k } \rightarrow ( M ^ { 2 n - 1 } , \xi )$ by an isotropic embedding $\iota_0$ (for $k \leq n - 1$) that is isotopic to the initial one, provided only an obvious necessary bundle condition is satisfied: If $\iota_0$ is an isotropic embedding, then its differential $T _ { \iota 0 }$ extends to a complex bundle monomorphism $T S ^ { k } \otimes \mathbf{C} \rightarrow \xi$, where $\xi $ inherits a complex structure from the (conformal) symplectic structure $d \alpha |_\xi$. The relevant $h$-principle says that, conversely, the existence of such a bundle mapping covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020054.png"/> is sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020055.png"/> to be isotopic to an isotropic embedding $\iota_0$.
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This allows one to use topological structure theorems, such as Barden's classification of simply-connected $5$-manifolds [[#References|[a1]]], to construct contact structures on a wide class of higher-dimensional manifolds, see [[#References|[a3]]].
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In dimension $3$ ($n = 2$) there is a different notion of contact surgery, due to R. Lutz and J. Martinet [[#References|[a7]]]; it allows surgery along $1$-spheres embedded transversely to a contact structure $\xi $. This was used by Lutz and Martinet to show the existence of a contact structure on any closed, orientable $3$-manifold and in any homotopy class of $2$-plane fields. For applications of other topological structure theorems (such as branched coverings or open book decompositions, cf. also [[Open book decomposition|Open book decomposition]]) to the construction of contact manifolds, see [[#References|[a4]]] and references therein.
  
 
Other types of surgery compatible with some geometric structure include surgery on manifolds of positive [[Scalar curvature|scalar curvature]] ([[#References|[a6]]], [[#References|[a9]]]) and surgery on manifolds of positive [[Ricci curvature|Ricci curvature]] ([[#References|[a10]]], [[#References|[a12]]]).
 
Other types of surgery compatible with some geometric structure include surgery on manifolds of positive [[Scalar curvature|scalar curvature]] ([[#References|[a6]]], [[#References|[a9]]]) and surgery on manifolds of positive [[Ricci curvature|Ricci curvature]] ([[#References|[a10]]], [[#References|[a12]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Barden, "Simply connected five-manifolds" ''Ann. of Math.'' , '''82''' (1965) pp. 365–385 {{MR|0184241}} {{ZBL|0136.20602}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Y. Eliashberg, "Topological characterization of Stein manifolds of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020065.png" />" ''Internat. J. Math.'' , '''1''' (1990) pp. 29–46 {{MR|1044658}} {{ZBL|0699.58002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> H. Geiges, "Applications of contact surgery" ''Topology'' , '''36''' (1997) pp. 1193–1220 {{MR|1452848}} {{ZBL|0912.57019}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Geiges, "Constructions of contact manifolds" ''Math. Proc. Cambridge Philos. Soc.'' , '''121''' (1997) pp. 455–464 {{MR|1434654}} {{ZBL|0882.57007}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R.E. Gompf, "Handlebody construction of Stein surfaces" ''Ann. of Math.'' , '''148''' (1998) pp. 619–693 {{MR|1668563}} {{ZBL|0919.57012}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Gromov, H.B. Lawson Jr., "The classification of simply connected manifolds of positive scalar curvature" ''Ann. of Math.'' , '''111''' (1980) pp. 423–434 {{MR|0577131}} {{ZBL|0463.53025}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Martinet, "Formes de contact sur les variétés de dimension<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020066.png" />" , ''Proc. Liverpool Singularities Sympos. II'' , ''Lecture Notes Math.'' , '''209''' , Springer (1971) pp. 142–163 {{MR|0350771}} {{ZBL|0215.23003}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> J. Milnor, "Morse theory" , Princeton Univ. Press (1963) {{MR|0163331}} {{ZBL|0108.10401}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Schoen, S.T. Yau, "On the structure of manifolds with positive scalar curvature" ''Manuscripta Math.'' , '''28''' (1979) pp. 159–183 {{MR|0535700}} {{ZBL|0423.53032}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> J.-P. Sha, D.-G. Yang, "Positive Ricci curvature on the connected sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120200/c12020067.png" />" ''J. Diff. Geom.'' , '''33''' (1991) pp. 127–137 {{MR|1085137}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Weinstein, "Contact surgery and symplectic handlebodies" ''Hokkaido Math. J.'' , '''20''' (1991) pp. 241–251 {{MR|1114405}} {{ZBL|0737.57012}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> D. Wraith, "Surgery on Ricci positive manifolds" ''J. Reine Angew. Math.'' , '''501''' (1998) pp. 99–113 {{MR|1637825}} {{ZBL|0915.53018}} </TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top"> D. Barden, "Simply connected five-manifolds" ''Ann. of Math.'' , '''82''' (1965) pp. 365–385 {{MR|0184241}} {{ZBL|0136.20602}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> Y. Eliashberg, "Topological characterization of Stein manifolds of dimension $&gt; 2$" ''Internat. J. Math.'' , '''1''' (1990) pp. 29–46 {{MR|1044658}} {{ZBL|0699.58002}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> H. Geiges, "Applications of contact surgery" ''Topology'' , '''36''' (1997) pp. 1193–1220 {{MR|1452848}} {{ZBL|0912.57019}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> H. Geiges, "Constructions of contact manifolds" ''Math. Proc. Cambridge Philos. Soc.'' , '''121''' (1997) pp. 455–464 {{MR|1434654}} {{ZBL|0882.57007}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> R.E. Gompf, "Handlebody construction of Stein surfaces" ''Ann. of Math.'' , '''148''' (1998) pp. 619–693 {{MR|1668563}} {{ZBL|0919.57012}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> M. Gromov, H.B. Lawson Jr., "The classification of simply connected manifolds of positive scalar curvature" ''Ann. of Math.'' , '''111''' (1980) pp. 423–434 {{MR|0577131}} {{ZBL|0463.53025}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> J. Martinet, "Formes de contact sur les variétés de dimension$3$" , ''Proc. Liverpool Singularities Sympos. II'' , ''Lecture Notes Math.'' , '''209''' , Springer (1971) pp. 142–163 {{MR|0350771}} {{ZBL|0215.23003}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> J. Milnor, "Morse theory" , Princeton Univ. Press (1963) {{MR|0163331}} {{ZBL|0108.10401}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> R. Schoen, S.T. Yau, "On the structure of manifolds with positive scalar curvature" ''Manuscripta Math.'' , '''28''' (1979) pp. 159–183 {{MR|0535700}} {{ZBL|0423.53032}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> J.-P. Sha, D.-G. Yang, "Positive Ricci curvature on the connected sum of $S ^ { n } \times S ^ { m }$" ''J. Diff. Geom.'' , '''33''' (1991) pp. 127–137 {{MR|1085137}} {{ZBL|}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> A. Weinstein, "Contact surgery and symplectic handlebodies" ''Hokkaido Math. J.'' , '''20''' (1991) pp. 241–251 {{MR|1114405}} {{ZBL|0737.57012}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> D. Wraith, "Surgery on Ricci positive manifolds" ''J. Reine Angew. Math.'' , '''501''' (1998) pp. 99–113 {{MR|1637825}} {{ZBL|0915.53018}} </td></tr></table>

Revision as of 16:56, 1 July 2020

A special type of surgery on a (strict) contact manifold $( M ^ { 2 n - 1 } , \xi )$ (i.e. a smooth manifold admitting a (strict) contact structure $\xi = \operatorname{ker} \alpha$, where $\alpha$ is a $1$-form satisfying $\alpha \wedge ( d \alpha ) ^ { n - 1 } \neq 0$), which results in a new contact manifold.

In topological terms, surgery on denotes the replacement of an embedded copy of $S ^ { k } \times D ^ { m - k }$, a tubular neighbourhood of an embedded $k$-sphere with trivial normal bundle, by a copy of $D ^ { k + 1 } \times S ^ { m - k - 1 }$, with the obvious identification along the boundary $S ^ { k } \times S ^ { m - k - 1 }$. Alternatively, one can attach a $( k + 1 )$-handle $D ^ { k + 1 } \times D ^ { m - k }$ along $S ^ { k } \times D ^ { m - k }$ to a manifold $W ^ { m + 1 }$ 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 $( M , \xi )$ 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 $( M , \xi = \operatorname { ker } \alpha )$ may be regarded as the strictly pseudo-convex boundary of an almost-complex (in fact, symplectic) manifold $W = ( M \times ( 0,1 ] , J )$ such that $\xi $ is given by the $J$-invariant subspace of the tangent bundle $T M$. Contact surgery on $M$ can then be interpreted as the attaching of an almost-complex or symplectic handle to $W$ along $M$, and the framing condition for $n > 2$ is given by requiring the almost-complex structure on $W$ to extend over the handle. For $n = 2$ 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 $J$-convex Morse functions on almost-complex manifolds (cf. also Almost-complex structure; Morse function).

A Stein manifold of real dimension $2 n$ has the homotopy type of an $n$-dimensional CW-complex, cf. [a8], p. 39. Eliashberg uses his construction to show that for $n > 2$ this is indeed the only topological restriction on a Stein manifold, that is, if $W$ is a $2 n$-dimensional smooth manifold with an almost-complex structure $J$ and a proper Morse function $\varphi$ with critical points of Morse index at most $n$, then $J$ is homotopic to a genuine complex structure $J ^ { \prime }$ such that $\varphi$ is $J ^ { \prime }$-convex and, in particular, $( W , J ^ { \prime } )$ is Stein.

The usefulness of contact surgery in this and other applications rests on the fact that there is an $h$-principle for isotropic spheres. This allows one to replace a given embedding $\iota : S ^ { k } \rightarrow ( M ^ { 2 n - 1 } , \xi )$ by an isotropic embedding $\iota_0$ (for $k \leq n - 1$) that is isotopic to the initial one, provided only an obvious necessary bundle condition is satisfied: If $\iota_0$ is an isotropic embedding, then its differential $T _ { \iota 0 }$ extends to a complex bundle monomorphism $T S ^ { k } \otimes \mathbf{C} \rightarrow \xi$, where $\xi $ inherits a complex structure from the (conformal) symplectic structure $d \alpha |_\xi$. The relevant $h$-principle says that, conversely, the existence of such a bundle mapping covering is sufficient for to be isotopic to an isotropic embedding $\iota_0$.

This allows one to use topological structure theorems, such as Barden's classification of simply-connected $5$-manifolds [a1], to construct contact structures on a wide class of higher-dimensional manifolds, see [a3].

In dimension $3$ ($n = 2$) there is a different notion of contact surgery, due to R. Lutz and J. Martinet [a7]; it allows surgery along $1$-spheres embedded transversely to a contact structure $\xi $. This was used by Lutz and Martinet to show the existence of a contact structure on any closed, orientable $3$-manifold and in any homotopy class of $2$-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 $> 2$" Internat. J. Math. , 1 (1990) pp. 29–46 MR1044658 Zbl 0699.58002
[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$3$" , Proc. Liverpool Singularities Sympos. II , Lecture Notes Math. , 209 , Springer (1971) pp. 142–163 MR0350771 Zbl 0215.23003
[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 $S ^ { n } \times S ^ { m }$" J. Diff. Geom. , 33 (1991) pp. 127–137 MR1085137
[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
How to Cite This Entry:
Contact surgery. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Contact_surgery&oldid=24403
This article was adapted from an original article by H. Geiges (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article