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| In [[mathematics]], specifically in [[symplectic geometry]], the '''symplectic sum''' is a geometric modification on [[symplectic manifold]]s, which glues two given manifolds into a single new one. It is a symplectic version of [[connected sum]]mation along a submanifold, often called a fiber sum.
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| The symplectic sum is the inverse of the [[symplectic cut]], which decomposes a given manifold into two pieces. Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to [[blowing up|deformation to the normal cone]] in [[algebraic geometry]].
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| The symplectic sum has been used to construct previously unknown families of symplectic manifolds, and to derive relationships among the [[Gromov-Witten invariant]]s of symplectic manifolds.
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| == Definition ==
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| Let <math>M_1</math> and <math>M_2</math> be two symplectic <math>2n</math>-manifolds and <math>V</math> a symplectic <math>(2n - 2)</math>-manifold, embedded as a submanifold into both <math>M_1</math> and <math>M_2</math> via
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| :<math>j_i : V \hookrightarrow M_i,</math>
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| such that the [[Euler class]]es of the [[normal bundle]]s are opposite:
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| :<math>e(N_{M_1} V) = -e(N_{M_2} V).</math>
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| In the 1995 paper that defined the symplectic sum, [[Robert Gompf]] proved that for any [[orientation (mathematics)|orientation]]-reversing isomorphism
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| :<math>\psi : N_{M_1} V \to N_{M_2} V</math>
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| there is a canonical [[Ambient isotopy|isotopy]] class of symplectic structures on the connected sum
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| :<math>(M_1, V) \# (M_2, V)</math> | |
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| meeting several conditions of compatibility with the summands <math>M_i</math>. In other words, the theorem defines a '''symplectic sum''' operation whose result is a symplectic manifold, unique up to isotopy.
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| To produce a well-defined symplectic structure, the connected sum must be performed with special attention paid to the choices of various identifications. Loosely speaking, the isomorphism <math>\psi</math> is composed with an orientation-reversing symplectic involution of the normal bundles of <math>V</math> (or rather their corresponding punctured unit disk bundles); then this composition is used to [[quotient space|glue]] <math>M_1</math> to <math>M_2</math> along the two copies of <math>V</math>.
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| == Generalizations ==
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| In greater generality, the symplectic sum can be performed on a single symplectic manifold <math>M</math> containing two disjoint copies of <math>V</math>, gluing the manifold to itself along the two copies. The preceding description of the sum of two manifolds then corresponds to the special case where <math>X</math> consists of two connected components, each containing a copy of <math>V</math>.
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| Additionally, the sum can be performed simultaneously on submanifolds <math>X_i \subseteq M_i</math> of equal dimension and meeting <math>V</math> [[Transversality (mathematics)|transversally]].
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| Other generalizations also exist. However, it is not possible to remove the requirement that <math>V</math> be of codimension two in the <math>M_i</math>, as the following argument shows.
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| A symplectic sum along a submanifold of codimension <math>2k</math> requires a symplectic involution of a <math>2k</math>-dimensional annulus. If this involution exists, it can be used to patch two <math>2k</math>-dimensional balls together to form a symplectic <math>2k</math>-dimensional [[sphere]]. Because the sphere is a [[compact space|compact]] manifold, a symplectic form <math>\omega</math> on it induces a nonzero [[cohomology]] class
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| :<math>[\omega] \in H^2(\mathbb{S}^{2k}, \mathbb{R}).</math>
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| But this second cohomology group is zero unless <math>2k = 2</math>. So the symplectic sum is possible only along a submanifold of codimension two.
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| == Identity element ==
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| Given <math>M</math> with codimension-two symplectic submanifold <math>V</math>, one may projectively complete the normal bundle of <math>V</math> in <math>M</math> to the <math>\mathbb{CP}^1</math>-bundle
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| :<math>P := \mathbb{P}(N_M V \oplus \mathbb{C}).</math>
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| This <math>P</math> contains two canonical copies of <math>V</math>: the zero-section <math>V_0</math>, which has normal bundle equal to that of <math>V</math> in <math>M</math>, and the infinity-section <math>V_\infty</math>, which has opposite normal bundle. Therefore one may symplectically sum <math>(M, V)</math> with <math>(P, V_\infty)</math>; the result is again <math>M</math>, with <math>V_0</math> now playing the role of <math>V</math>:
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| :<math>(M, V) = ((M, V) \# (P, V_\infty), V_0).</math>
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| So for any particular pair <math>(M, V)</math> there exists an [[identity (mathematics)|identity element]] <math>P</math> for the symplectic sum. Such identity elements have been used both in establishing theory and in computations; see below.
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| == Symplectic sum and cut as deformation ==
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| It is sometimes profitable to view the symplectic sum as a family of manifolds. In this framework, the given data <math>M_1</math>, <math>M_2</math>, <math>V</math>, <math>j_1</math>, <math>j_2</math>, <math>\psi</math> determine a unique smooth <math>(2n + 2)</math>-dimensional symplectic manifold <math>Z</math> and a [[fibration]]
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| :<math>Z \to D \subseteq \mathbb{C}</math>
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| in which the central fiber is the singular space
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| :<math>Z_0 = M_1 \cup_V M_2</math>
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| obtained by joining the summands <math>M_i</math> along <math>V</math>, and the generic fiber <math>Z_\epsilon</math> is a symplectic sum of the <math>M_i</math>. (That is, the generic fibers are all members of the unique isotopy class of the symplectic sum.)
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| Loosely speaking, one constructs this family as follows. Choose a nonvanishing holomorphic section <math>\eta</math> of the trivial complex line bundle
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| :<math>N_{M_1} V \otimes_\mathbb{C} N_{M_2} V.</math>
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| Then, in the direct sum
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| :<math>N_{M_1} V \oplus N_{M_2} V,</math>
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| with <math>v_i</math> representing a normal vector to <math>V</math> in <math>M_i</math>, consider the locus of the quadratic equation
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| :<math>v_1 \otimes v_2 = \epsilon \eta</math>
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| for a chosen small <math>\epsilon \in \mathbb{C}</math>. One can glue both <math>M_i \setminus V</math> (the summands with <math>V</math> deleted) onto this locus; the result is the symplectic sum <math>Z_\epsilon</math>.
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| As <math>\epsilon</math> varies, the sums <math>Z_\epsilon</math> naturally form the family <math>Z \to D</math> described above. The central fiber <math>Z_0</math> is the symplectic cut of the generic fiber. So the symplectic sum and cut can be viewed together as a quadratic deformation of symplectic manifolds.
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| An important example occurs when one of the summands is an identity element <math>P</math>. For then the generic fiber is a symplectic manifold <math>M</math> and the central fiber is <math>M</math> with the normal bundle of <math>V</math> "pinched off at infinity" to form the <math>\mathbb{CP}^1</math>-bundle <math>P</math>. This is analogous to deformation to the normal cone along a smooth [[divisor (algebraic geometry)|divisor]] <math>V</math> in algebraic geometry. In fact, symplectic treatments of Gromov-Witten theory often use the symplectic sum/cut for "rescaling the target" arguments, while algebro-geometric treatments use deformation to the normal cone for these same arguments.
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| However, the symplectic sum is not a complex operation in general. The sum of two [[Kaehler manifold|Kähler]] manifolds need not be Kähler.
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| == History and applications ==
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| The symplectic sum was first clearly defined in 1995 by Robert Gompf. He used it to demonstrate that any [[finitely presented group]] appears as the [[fundamental group]] of a symplectic four-manifold. Thus the [[category (mathematics)|category]] of symplectic manifolds was shown to be much larger than the category of Kähler manifolds.
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| Around the same time, Eugene Lerman proposed the symplectic cut as a generalization of symplectic blow up and used it to study the [[symplectic quotient]] and other operations on symplectic manifolds.
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| A number of researchers have subsequently investigated the behavior of [[pseudoholomorphic curve]]s under symplectic sums, proving various versions of a symplectic sum formula for Gromov-Witten invariants. Such a formula aids computation by allowing one to decompose a given manifold into simpler pieces, whose Gromov-Witten invariants should be easier to compute. Another approach is to use an identity element <math>P</math> to write the manifold <math>M</math> as a symplectic sum
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| :<math>(M, V) = (M, V) \# (P, V_\infty).</math>
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| A formula for the Gromov-Witten invariants of a symplectic sum then yields a recursive formula for the Gromov-Witten invariants of <math>M</math>.
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| == References ==
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| * Robert Gompf: A new construction of symplectic manifolds, ''Annals of Mathematics'' 142 (1995), 527-595
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| * Dusa McDuff and Dietmar Salamon: ''Introduction to Symplectic Topology'' (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9
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| * Dusa McDuff and Dietmar Salamon: ''J-Holomorphic Curves and Symplectic Topology'' (2004) American Mathematical Society Colloquium Publications, ISBN 0-8218-3485-1
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| [[Category:Symplectic topology]]
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