By studying the example of smooth structures on CP^2#3(-CP^2) we illustrate how surgery on a single embedded nullhomologous torus can be utilized to change the symplectic structure, the Seiberg-Witten invariant, and hence the smooth structure on a 4-manifold.

In Floer's 1990 ICM talk ([F3]), he stated that «with luck, one may be able to analyse the change of the Donaldson polynomial under handle addition. The exact triangle for surgery on M may be considered as one step in this direction.» It is the purpose of this note to elaborate on this comment and to illustrate this circle of ideas by computing the O-degree Donaldson invariant for the K3 surface. This invariant was first calculated by Donaldson [D3] using stable bundles. It was recently calculated by Kronheimer [K] who reduced the calculation to a count of representations of a related orbifold fundamental group. Our purpose here is to illustrate how FIoer's exact triangle can be implemented to compute Donaldson invariants without resorting to any underlying complex structure. Such cut and paste techniques were also utilized in [FS4] where we constructed irreducible 4manifolds not homotopy equivalent to any complex surface. The methods in this note complement those of [FS3] where it is shown that, for any 4-manifold X homotopy equivalent to the K3 surface and containing the Brieskom homology 3sphere ~(2, 3, 7), certain values of the degree 10 Donaldson polynomial invariant are odd.

We present a method for finding embedded nullhomologous tori in standard 4-manifolds which can be utilized to change their smooth structure. As an application, we show how to obtain infinite families of simply connected smooth 4-manifolds with b^+ = 1 and b^- = 2,...,7, via surgery on nullhomologous tori embedded in the standard manifolds CP^2 # k (-CP^2), k=2,...,7.

Locally smooth S'-actions on simply connected 4-manifolds are studied in terms of their weighted orbit spaces. An equivariant classification theorem is proved, and the weighted orbit space is used to compute the quadratic form of a given simply connected 4-manifold with S '-action. This is used to show that a simply connected 4-manifold which admits a locally smooth 5'-action must be homotopy equivalent to a connected sum of copies of S4, CP2, CP2, and S2 x S2.

We produce infinite families of exotic actions of finite cyclic groups on simply connected smooth 4-manifolds with non-trivial Seiberg-Witten invariants.

For 5 <= k <= 8 we show that the infinite family of exotic smooth structures on CP^2# k(-CP^2) can be achieved by 1/n - surgeries on a single embedded nullhomologous torus in a manifold R_k which is homeomorphic to CP^2# k(-CP^2).

I compute the Lagrangian Floer cohomology groups of certain tori in closed simply connected symplectic 4-manifolds arising from Fintushel - Stern knot/link surgery. These manifolds are usually not symplectically aspherical. As a result of the computation we observe examples where HF(L0) ≅ HF( L1) and L0 and L 1 are smoothly and Lagrangian isotopic but L0 , L1 are not symplectically isotopic. In [Gi182], C. Giller proposed an invariant of ribbon 2-knots in S4 based on a "Conway calculus" of crossing changes in a projection to R 3. In certain cases, this invariant computes the Alexander polynomial. Giller's invariant is, however, a symmetric polynomial - which the Alexander polynomial of a 2-knot need not be. After modifying a 2-knot into a Montesinos twin in a natural way, we show that Giller's invariant is actually the Seiberg-Witten invariant of the exterior of the twin, glued to the complement of a fiber in E(2).

We introduce a general procedure called 'reverse engineering' that can be used to construct infinite families of smooth 4-manifolds in a given homeomorphism type. As one of the applications of this technique, we produce an infinite family of pairwise nondiffeomorphic 4-manifolds homeomorphic to Cp2#3CP(2).

We introduce a procedure called ‘reverse engineering’ which can be used to construct infinite families of smooth 4-manifolds in a given homeomorphism type. As one of the applications of this technique, we produce an infinite family of pairwise nondiffeomorphic 4-manifolds homeomorphic to S × S.

We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admit smoothly embedded spheres with self-intersection -1, i.e. they are minimal. In addition, none these smooth structures admit an underlying symplectic structure. Shortly after the appearance of a preliminary version of this article, Park, Stipsicz, and Szabo used the techniques described herein to show that the complex projective plane blown up at 5 points has infinitely many distinct smooth structures. In the final section of this paper we give a somewhat different construction of such a family of examples.

In this note we fill a gap in the proof of the main theorem (Theorem 1.2) of our paper 'Surfaces in 4-manifolds', Math. Res. Letters 4 (1997), 907-914.

We study the question of how many embedded symplectic or Lagrangian tori can represent the same homology class in a simply connected symplectic 4-manifold.

We define an simple invariant lambda(T) of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda( T) is actually a C-infinity invariant. In addition, this invariant is used to show that many symplectic 4-manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3-surface obtained from knot surgery using the trefoil knot [19].

This article presents several new constructions of infinite families of smooth 4-manifolds with the property that any two manifolds in the same family are homeomorphic. While the construction gives strong evidence that any two of these manifolds of are not diffeomorphic, they cannot be distinguished by Seiberg–Witten invariants. Whether these manifolds are, or are not, diffeomorphic seems to be a very difficult question to answer. For one of these constructions, each member of the family is symplectic with the further property that each contains nullhomologous tori with the property that infinitely many log transformations on these tori yield nonsymplectic 4-manifolds. This is detected by calculations of Seiberg–Witten invariants. The surgery in question can be performed on any 4-manifold which contains as a codimension 0 submanifold a punctured surface bundle over a punctured surface and a nontrivial loop in the base which has trivial monodromy. A starting point for another class of examples in this paper is a family of examples which show that the Parshin–Arakelov theorem for holomorphic Lefschetz fibrations is false in the symplectic category. Such families are constructed by means of knot surgery on ellipitic surfaces. It is shown that for a fixed homeomorphism type X (of a simply connected elliptic surface) and a fixed integer g⩾3, there are infinitely many genus g Lefschetz fibrations on nondiffeomorphic 4-manifolds, all homeomorphic to X.

We present constructions of simply connected symplectic 4manifolds which have (up to sign) one basic class and which fill up the geographical region between the half-Noether and Noether lines.

In this article we present examples of simply connected symplectic 4-manifolds X whose canonical classes are represented by complicated disjoint unions of symplectic submanifolds of X: Theorem. Given finite collections {gi}, {mi}, i=1,...,n, of positive integers, there is a minimal symplectic simply connected 4-manifold X whose canonical class is represented by a disjoint union of embedded symplectic surfaces K ~ Sg1,1 « ... « Sg1,m1 « ... « Sgn,1 «... « S{gn,mn} where Sgi,j is a surface of genus gi. Furthermore, c12(X) = ch(X) - (2+ b) where b= S{i=1}n mi is the total number of connected components of the symplectic representative of the canonical class.

It is the purpose of this paper to construct families of examples of nonsymplectic 4-manifolds which (up to sign) have just one Seiberg-Witten basic class.