Knots, links, and 4-manifolds

On the number of components of the symplectic representatives of the canonical class

“…In order to study this construction, we will present an elliptic surface E(n + 2) as link surgery manifold (see [FS2] for the definition) obtained from the Hopf link. This presentation will help us identify some symplectic submanifolds (spheres and tori) in the elliptic surface E(n + 2), for n ≥ 0, that we will use in our construction.…”

Double node neighborhoods and families of simply connected 4-manifolds with 𝑏⁺=1

“…We first recall the result of [S] and [FS3] concerning the manifolds Y n , n > 0. Recall that since b − Y n = 9, the manifold Y n has a well-defined Seiberg-Witten invariant.…”

“…So the classes in question die in π 1 (E(1) \ (F 1 ∪ F 2 )), and we see that π 1 (V n ) = 1. Using [FS3], one computes the Seiberg-Witten invariants of V n : up to sign the only classes in H 2 (V n ; Z) with nontrivial Seiberg-Witten invariants are T and 3T , and |SW V n (±3T )| = n, |SW V n (±T )| = 2n − 1.…”

“…The two pieces are glued together in such a way that the homology class [pt × ∂D 2 ] is identified with [pt × λ], where λ is the class of a longitude of K. This latter condition does not, in general, completely determine the diffeomorphism type of X K ; however if we take X K to be any manifold constructed in this fashion and if, for example, T has a cusp neighborhood, then the Seiberg-Witten invariant of X K is completely determined by the Seiberg-Witten invariant of X and the Alexander polynomial of K [FS3]. Furthermore, if X and X \ T are simply connected, then so is X K .…”

Simply connected symplectic 4-manifolds with b2+=1 and c12=2

“…Though the complete answer is far from reach, gauge theory makes us to answer partially these questions ([FS1], [FS2], [G], [Sz], [T]). But most known results are in the case of simply connected 4-manifolds with either b + 2 > 1 odd or b + 2 = 1 and c 2 1 ≤ 0.…”