Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots.

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. (M_1 - i_1(0)) \sqcup (M_2 - i_2(0)) by identifying i_1(tu) with i_2((1 - t)u) for each unit vector u \in S^{n-1} and each 0 . Choose the orientation for M_1 \mathbin{\#} M_2 which is compatible with M_1 and M_2 . The fact that this construction is well-defined depends crucially on the disc theorem, which is not at all obvious. For further details, see Kosinski, Differential Manifolds. The operation of connected sum is denoted by \# . The operation of connected sum has the sphere S^m as an identity; that is, M \mathbin{\#} S^m is homeomorphic (or diffeomorphic) to M . The classification of closed surfaces, a foundational and historically significant result in topology, states that any closed surface can be expressed as the connected sum of a sphere with some number g of tori and some number k of real projective planes. \psi: N_{M_1} V \to N_{M_2} V that reverses the orientation on each fiber. Then \psi induces an orientation-preserving diffeomorphism N_1 \setminus V \cong N_{M_1} V \setminus V \to N_{M_2} V \setminus V \cong N_2 \setminus V, where each normal bundle N_{M_i} V is diffeomorphically identified with a neighborhood N_i of V in M_i , and the map N_{M_1} V \setminus V \to N_{M_2} V \setminus V is the orientation-reversing diffeomorphic involution v \mapsto v / |v|^2 on normal vectors. The connected sum of M_1 and M_2 along V is then the space (M_1 \setminus V) \bigcup_{N_1 \setminus V = N_2 \setminus V} (M_2 \setminus V) obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted (M_1, V) \mathbin{\#} (M_2, V). Its diffeomorphism type depends on the choice of the two embeddings of V and on the choice of \psi . Loosely speaking, each normal fiber of the submanifold V contains a single point of V , and the connected sum along V is simply the connected sum as described in the preceding section, performed along each fiber. For this reason, the connected sum along V is often called the fiber sum. The special case of V a point recovers the connected sum of the preceding section. e\left(N_{M_1} V\right) = -e\left(N_{M_2} V\right). Furthermore, in this case the structure group of the normal bundles is the circle group SO(2) ; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from V to the circle, which in turn equals the first integral cohomology group H^1(V) . So the diffeomorphism type of the sum depends on the choice of \psi and a choice of element from H^1(V) . A connected sum along a codimension-two V can also be carried out in the category of symplectic manifolds; this elaboration is called the symplectic sum. If one does not take into account the orientations of the knots, the connected sum operation is not well-defined on isotopy classes of (nonoriented) knots. To see this, consider two noninvertible knots K, L which are not equivalent (as unoriented knots); for example take the two pretzel knots K = P(3, 5, 7) and L = P(3, 5, 9). Let K + and K − be K with its two inequivalent orientations, and let L + and L − be L with its two inequivalent orientations. There are four oriented connected sums we may form: A = K + # L + B = K − # L − C = K + # L − D = K − # L + The oriented ambient isotopy classes of these four oriented knots are all distinct, and, when one considers ambient isotopy of the knots without regard to orientation, there are two distinct equivalence classes: {A ~ B} and {C ~ D}. To see that A and B are unoriented equivalent, simply note that they both may be constructed from the same pair of disjoint knot projections as above, the only difference being the orientations of the knots. Similarly, one sees that C and D may be constructed from the same pair of disjoint knot projections. ==References==

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