Editing Connected sum (section)

== Connected sum along a submanifold ==

The connected sum can be defined along a submanifold.<ref>{{cite journal |last1=Gompf |first1=Robert E. |title=A New Construction of Symplectic Manifolds |journal=The Annals of Mathematics |date=November 1995 |volume=142 |issue=3 |pages=527 |doi=10.2307/2118554}}</ref>{{rp|§1}}

Let <math>M_1</math> and <math>M_2</math> be two smooth, oriented manifolds of equal dimension and <math>V</math> a smooth, closed, oriented manifold, embedded as a submanifold into both <math>M_1</math> and <math>M_2.</math> Suppose furthermore that there exists an [[isomorphism]] of [[normal bundle]]s

:<math>\psi: N_{M_1} V \to N_{M_2} V</math>

that reverses the orientation on each fiber. Then <math>\psi</math> induces an orientation-preserving diffeomorphism

:<math>N_1 \setminus V \cong N_{M_1} V \setminus V \to N_{M_2} V \setminus V \cong N_2 \setminus V,</math>

where each normal bundle <math>N_{M_i} V</math> is diffeomorphically identified with a [[neighborhood (mathematics)|neighborhood]] <math>N_i</math> of <math>V</math> in <math>M_i</math>, and the map

:<math>N_{M_1} V \setminus V \to N_{M_2} V \setminus V</math>

is the orientation-reversing diffeomorphic [[involution (mathematics)|involution]]

:<math>v \mapsto v / |v|^2</math>

on [[normal vector]]s. The '''connected sum''' of <math>M_1</math> and <math>M_2</math> along <math>V</math> is then the space

:<math>(M_1 \setminus V) \bigcup_{N_1 \setminus V = N_2 \setminus V} (M_2 \setminus V)</math>

obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted

:<math>(M_1, V) \mathbin{\#} (M_2, V).</math>

Its diffeomorphism type depends on the choice of the two embeddings of <math>V</math> and on the choice of <math>\psi</math>.

Loosely speaking, each normal fiber of the submanifold <math>V</math> contains a single point of <math>V</math>, and the connected sum along <math>V</math> is simply the connected sum as described in the preceding section, performed along each fiber. For this reason, the connected sum along <math>V</math> is often called the '''fiber sum'''.

The special case of <math>V</math> a point recovers the connected sum of the preceding section.