Constructions of Smooth 4-Manifolds

Ronald Fintushel^{1}and Ronald J. Stern^{2}

Abstract. We describe a collection of constructions which illustrate a panoply of “exotic” smooth 4-manifolds.

1991 Mathematics Subject Classification: 57R55

1. Introduction

At the time of the previous (1994) International Congress of Mathematicians, steady, but slow, progress was being made on the classification of simply connected closed smooth 4-manifolds. In particular, the Donaldson invariants had begun to take a particularly nice form [13] (also [4]), their computations were becoming more routine [3], and their behavior under blowing up (i.e. taking connected sum with \overline{CP}^{2}) was well understood [2]. Due to the complexity of the Donaldson invariants, great hope was held out that an even better understanding of these invariants would close the books on the classification of simply connected 4-manifolds.

A few short months after the 1994 ICM, the 4-manifold community was blindsided by the introduction of the now famous Seiberg-Witten equations [28]. Most of the results obtained by using Donaldson theory were found to have quicker, and sometimes more general, counterparts using the Seiberg-Witten technology. The potential applications of the difficult Donaldson technology became much more transparent using these new equations. As of July 1998, there is good news as well as bad news. The good news is that many of the earlier focus problems have been solved. In particular, the Thom conjecture [14] and its natural generalizations have been verified [20, 21]; also the study of symplectic 4-manifolds has taken a more central role [23, 24, 25, 26]. The bad news is that recent constructions and computations indicate that the Seiberg-Witten and Donaldson theories are too weak to distinguish simply connected smooth 4-manifolds [6]. It is these latter constructions and computations that we will discuss at this 1998 International Congress of Mathematicians. It is becoming more apparent that we are seeing only a small constellation of 4-dimensional manifolds. More seriously, we are lacking a reasonable conjectural classification of simply connected closed smooth 4-manifolds.

Current technology has given us many more 4-manifolds than had been expected in 1994. The authors hope that during the 2002 ICM the construction of large classes of new 4-manifolds will be discussed; in particular, they hope that a sufficiently large collection of 4-manifolds will have been discovered so as to allow for some general patterns to emerge and, at least, a conjectural classification to again be on the books.

1Partially supported by NSF Grant DMS9704927

2Partially supported by NSF Grant DMS9626330

Documenta Mathematica · Extra Volume ICM 1998 · II · 443–452

2. The knot surgery construction

Let X be a simply connected oriented smooth closed 4-manifold. Its most basic invariant is its intersection form

Q_{X} : H_{2}(X; Z) ⊗ H_{2}(X; Z) → Z

defined by counting signed transverse intersections of embedded oriented surfaces representing given homology classes. It is a famous theorem of M. Freedman [10] that Q_{X} determines the homeomorphism type of X, and an equally renowned theorem of S.K. Donaldson [1] that Q_{X} is not sufficient to determine the diffeomorphism type of X. In this section we shall discuss geometric operations on a given smooth 4-manifold which preserve the underlying topological structure and alter its smooth structure. In particular, we shall consider the following construction: Let X be a simply connected smooth 4-manifold which contains a smoothly embedded torus T of self-intersection 0. Given a knot K in S^{3}, we replace a tubular neighborhood of T with S^{1} × (S^{3} \ K) to obtain the knot surgery manifold X_{K}.

More formally, this procedure is accomplished by performing 0-framed surgery on K to obtain the 3-manifold M_{K}. The meridian m of K can be viewed as a circle in M_{K}; so in S^{1}×M_{K} we have the smooth torus T_{m} = S^{1}×m of self-intersection 0. Since a neighborhood of m has a canonical framing in M_{K}, a neighborhood of the torus T_{m} in S^{1} × M_{K} has a canonical identification with T_{m} × D^{2}. The knot surgery manifold X_{K} is given by the fiber sum

X_{K} = X#_{T =T_{m}}S^{1} × M_{K} = (X \ T × D^{2}) ∪ (S^{1} × M_{K} \ T_{m} × D^{2})

where the two pieces are glued together so as to preserve the homology class [pt × ∂D^{2}]. This latter condition does not, in general, completely determine the isotopy type of the gluing, and X_{K} is taken to be any manifold constructed in this fashion.

Because S^{1} ×(S^{3} \K) has the same homology as a tubular neighborhood of T in X (and because the gluing preserves [pt × ∂D^{2}]) the homology and intersection form of X_{K} will agree with that of X. If it is also assumed that X \ T is simply connected, then π_{1}(X_{K}) = 1; so X_{K} will be homeomorphic to X.

In order to distinguish the diffeomorphism types of the X_{K}, we rely on Seiberg-Witten invariants. We view the Seiberg-Witten invariant of a smooth 4-manifold as a multivariable (Laurent) polynomial. To do this, recall that the Seiberg-Witten invariant of a smooth closed oriented 4-manifold X with b_{2}^{+}(X) > 1 is an integer-valued function which is defined on the set of spin^{c} structures over X (cf. [28]). In case H_{1}(X, Z) has no 2-torsion (for example, as here where X is simply connected) there is a natural identification of the spin^{c} structures of X with the characteristic elements of H_{2}(X, Z) (i.e. those elements k whose Poincaré duals k̂ reduce mod 2 to w_{2}(X)). In this case we view the Seiberg-Witten invariant as

SW_{X} : {k ∈ H_{2}(X, Z)|k̂ ≡ w_{2}(T X) (mod 2))} → Z.

The sign of SW_{X} depends on an orientation of H^{0}(X, R) ⊗ det H_{+}^{2}(X, R) ⊗ det H^{1}(X, R). If SW_{X}(β) ≠ 0, then β is called a basic class of X. It is a fundamental fact that the set of basic classes is finite. Furthermore, if β is a basic class, then so is −β with SW_{X}(−β) = (−1)^{(e+sign)(X)/4} SW_{X}(β) where e(X) is the Euler number and sign(X) is the signature of X.

Now let {±β_{1}, …, ±β_{n}} be the set of nonzero basic classes for X. Consider variables t_{β} = exp(β) for each β ∈ H^{2}(X; Z) which satisfy the relations t_{α+β} = t_{α}t_{β}. We may then view the Seiberg-Witten invariant of X as the Laurent polynomial

SW_{X} = SW_{X}(0) + ∑_{j=1}^{n} SW_{X}(β_{j}) · (t_{β_{j}} + (−1)^{(e+sign)(X)/4} t_{β_{j}}^{−1}).

As an example of this notational device, consider the simply connected minimally elliptic surface E(n) with holomorphic Euler characteristic n and no multiple fibers. Its Seiberg-Witten invariant is SW_{E(n)} = (t − t^{−1})^{n−2} where t = t_{F} for F the fiber class. Thus, SW_{E(n)}((n−2m)F) = (−1)^{m−1}\binom{n−2}{m−1} for m = 1, …, n−1 and SW_{E(n)}(α) = 0 for any other α. When b^{+}(X) > 1, the Laurent polynomial SW_{X} is a diffeomorphism invariant of X.

For our theorem, we need to place a mild hypothesis on the embedded torus T. We say that a smoothly embedded torus representing a nontrivial homology class [T] is c-embedded if there is a neighborhood N of T in X and a diffeomorphism φ : N → U where U is a neighborhood of a cusp fiber in an elliptic surface and φ(T) is a smooth elliptic fiber in U. Equivalently, T is c-embedded if it contains two simple closed curves which generate π_{1}(T) and which bound vanishing cycles in X. Note that a c-embedded torus has self-intersection 0.

Theorem 2.1 ([6]). Let X be a simply connected oriented smooth 4-manifold with b^{+} > 1. Suppose that X contains a c-embedded torus T with π_{1}(X \ T) = 1, and let K be any knot in S^{3}. Then the knot surgery manifold X_{K} is homeomorphic to X and has Seiberg-Witten invariant

SW_{X_{K}} = SW_{X} · Δ_{K}(t)

where Δ_{K}(t) is the symmetrized Alexander polynomial of K and t = exp(2[T]).

For example, the theorem applies to the K3-surface E(2) where T is a smooth elliptic fiber, and since SW_{E(2)} = 1, we have SW_{E(2)_{K}} = Δ_{K}(t). It is a theorem

of Seifert that any Laurent polynomial of the form P (t) = a_{0} + ∑_{j=1}^{n} a_{j}(t^{j} + t^{−j})

with coefficient sum P (1) = ±1 is the Alexander polynomial of some knot in S^{3}. Call such a Laurent polynomial an A-polynomial. It follows that if (X, T) satisfies the hypothesis of Theorem 2.1, then for any A-polynomial P (t), there is a smooth simply connected 4-manifold X_{P} which is homeomorphic to X and has Seiberg-Witten invariant SW_{X_{P}} = SW_{X} · P (t) where t = exp(2[T]). In particular, for each A-polynomial P (t), there is a manifold homeomorphic to the K3-surface with SW = P (t).

The relationship between Seiberg-Witten type invariants and the Alexander polynomial was first discovered by Meng and Taubes. In [17] they showed that the 3-manifold Seiberg-Witten invariant is related to Milnor torsion.

If one starts with a fibered knot K, then S^{1} × M_{K} is a surface bundle over a torus and thus carries a symplectic structure [27] for which T_{m} is a symplectic submanifold. Thus if X is a symplectic 4-manifold containing a c-embedded symplectic torus T, then X_{K} = X#_{T =T_{m}}S^{1} × M_{K} is also symplectic [11, 16]. In a fashion similar to the treatment of the Seiberg-Witten invariant as a Laurent polynomial, one can view the Gromov invariant of a symplectic 4-manifold X as a polynomial Gr_{X} = ∑ Gr_{X}(β) t_{β} where Gr_{X}(β) is the usual Gromov invariant of β. Let A_{K}(t) = t^{d}Δ_{K}(t) denote the normalized Alexander polynomial, where d is the degree of Δ_{K}(t). As a corollary to Theorem 2.1 and the theorems of Taubes relating the Seiberg-Witten and Gromov invariants of a symplectic 4−manifold [25, 26] we have:

Corollary 2.2 ([6]). Let X be a symplectic 4-manifold with b^{+} > 1 containing a symplectic c-embedded torus T. If K is a fibered knot, then X_{K} is a symplectic 4-manifold whose Gromov invariant is Gr_{X_{K}} = Gr_{X} · A_{K}(τ) where τ = exp([T]).

This last calculation can also be made purely within the realm of symplectic topology [12, 15]. Our interest is directed more to the opposite situation. The Alexander polynomial of a fibered knot is monic; i.e. its top coefficient is ±1. On the other hand:

Corollary 2.3 ([6]). If Δ_{K}(t) is not monic, then X_{K} does not admit a symplectic structure. Furthermore, if X contains a homologically nontrivial surface Σ_{g} of genus g disjoint from T with [Σ_{g}]^{2} < 2 − 2g if g > 0 or [Σ_{g}]^{2} < 0 if g = 0, then X_{K} with the opposite orientation does not admit a symplectic structure.

Until the summer of 1996, it was still a plausible conjecture (sometimes called the ‘minimal conjecture’) that each irreducible simply connected 4-manifold should admit a symplectic structure with one of its orientations. The first counterexamples to this conjecture were constructed by Z. Szabo [22]. The knot surgery construction gives a multitude of examples of simply connected irreducible ‘nonsymplectic’ 4-manifolds. In fact, if X is simply connected with SW_{X} ≠ 0 and if X contains a c-embedded torus T with π_{1}(X \ T) = 1, then Theorem 2.1 and Corollary 2.3 imply that there are infinitely many distinct nonsymplectic smooth 4-manifolds X_{K} homeomorphic to X.

If K_{1} and K_{2} have the same Alexander polynomial, Seiberg-Witten invariants are not able to distinguish X_{K_{1}} from X_{K_{2}}. For example, take X = E(2). Then X_{K} has a self-intersection 0 homology class σ satisfying σ ·[T] = 1 which is represented by an embedded surface of genus g(K) + 1 where g(K) is the genus of K. One might hope that these classes could be used to distinguish X_{K_{1}} from X_{K_{2}} when g(K_{1}) ≠ g(K_{2}).

Conjecture . For X = E(2), the manifolds X_{K_{1}} and X_{K_{2}} are diffeomorphic if and only if K_{1} and K_{2} are equivalent knots.

The proof of Theorem 2.1 proceeds by successively simplifying the manifold X_{K} in a fashion which mimics the calculation of the Alexander polynomial of K via skein relations. Recall that Δ_{K}(t) can be calculated via the relation

Δ_{K_{+}}(t) = Δ_{K_{−}}(t) + (t^{1/2} − t^{−1/2}) · Δ_{K_{0}}(t) (1)

where K_{+} is an oriented knot or link, K_{−} is the result of changing a single oriented positive (right-handed) crossing in K_{+} to a negative (left-handed) crossing, and K_{0} is the result of resolving the crossing as shown in Figure 1.

[figure: K_{+}; K_{−}; K_{0}; Figure 1]

The point of using (1) to calculate Δ_{K} is that K can be simplified to an unknot via a sequence of crossing changes. One builds a ‘resolution tree’ starting from K and at each stage adding the bifurcation of Figure 2, where each K_{+}, K_{−}, K_{0} is a knot or 2-component link, and so that at the bottom of the tree, there are only unknots, and split links. Then, because the Alexander polynomial of an unknot is 1, and is 0 for a split link (of more than one component) one can can work backwards using (1) to calculate Δ_{K}(t).

[figure: K_{+}; K_{−}; K_{0}; Figure 2]

The manifold X_{K_{+}} can be obtained from X_{K_{−}} by means of a (+1)-log transform on a nullhomologous torus in X_{K_{−}}, and then the gluing theorems of [18] show that SW_{X_{K_{+}}} can be computed in terms of the Seiberg-Witten invariants of X_{K_{−}} and a manifold X_{K_{−},0} obtained by a 0-log transform on X_{K_{−}}. With some work, this leads to a related resolution diagram of 4-manifolds where each knot K′ corresponds to X_{K′}, and this diagram can be used to prove Theorem 2.1.

We conclude this section by pointing out that the knot surgery construction can be generalized to manifolds with b^{+} = 1 and to links in S^{3} of more than one component in a more-or-less obvious way. One glues the complements of cembedded tori in 4-manifolds to the product of S^{1} with the link complement. See [6] for details. For example, if to each boundary component of S^{1} ×(S^{3} \N(L)) we glue E(1) minus the neighborhood of a smooth elliptic fiber, we obtain a manifold with SW = Δ_{L}(t_{1}, …, t_{n}), the multivariable Alexander polynomial of the link. Szabo’s examples in [22] can be obtained from this construction.

3. Embeddings of surfaces in 4-manifolds

Knot surgery can also be used to change the embedding of a surface in a fixed 4-manifold. To motivate the construction, note that one can tie a knot in the core {0}×I of a cylinder D^{2}×I by removing a tubular neighborhood of a meridian circle and replacing it with a knot complement S^{3} \K. We shall perform a parametrized version of this construction in the 4-manifold setting. Consider an oriented surface Σ of genus g > 0 which is smoothly embedded in a simply connected 4-manifold X. Let α be a simple closed curve on Σ which is part of a symplectic basis, and let α × I be an annular neighborhood of α in Σ. In X we see the neighborhood D^{2} × α × I. For a fixed knot K in S^{3}, we parametrize the above construction so as to perform it on each of the cylinders D^{2} × {y} × I, y ∈ α, to obtain an embedded surface Σ_{K}. This is equivalent to performing knot surgery on the (nullhomologous) rim torus R = ∂D^{2} × α. We call this operation rim surgery.

Theorem 3.1 ([7]). Let X be a simply connected smooth 4-manifold with an embedded surface Σ of positive genus. Suppose that π_{1}(X \ Σ) = 1. Then for each knot K in S^{3}, rim surgery produces a surface Σ_{K}, and there is a homeomorphism (X, Σ) ≅ (X, Σ_{K}).

The Seiberg-Witten invariant can be used to study these embeddings, but first, an auxilliary construction is needed. For each positive integer g, let Y_{g} be the union of the Milnor fiber of the (2, 2g + 1, 4g + 1) Brieskorn singularity and a generalized nucleus consisting of the 4-manifold obtained as the trace of the 0framed surgery on (2, 2g + 1) torus knot in ∂B^{4} and a −1 surgery on a meridian. Then Y_{g} is a Kahler surface and admits a holomorphic fibration over CP^{1} with generic fiber a surface S_{g} of genus g.

Let (X, Σ) be as in Theorem 3.1, and suppose that the self-intersection Σ^{2} = 0. We call (X, Σ) an SW-pair if satisfies the property that SW_{X#_{Σ=Sg} Y_{g}} ≠ 0. (In general, if Σ^{2} = n > 0, one makes this definition by first blowing up n times.) For example, if X is symplectic and Σ is a symplectic submanifold (of square 0), then X#_{Σ=S_{g}}Y_{g} is symplectic, and it follows that (X, Σ) is an SW-pair. In X#_{Σ=S_{g}}Y_{g}, the rim torus R becomes homologically essential and is c-embedded. We can use Theorem 2.1 to calculate Seiberg-Witten invariants:

SW_{X#_{Σ_{K} =Sg} Y_{g}} = SW_{(X#_{Σ=Sg} Y_{g})_{K}} = SW_{X#_{Σ=Sg} Y_{g}} · Δ_{K}(r)

where r = exp(2[R]), viewing [R] as a class in the fiber sum. We have:

Theorem 3.2 ([7]). Consider any SW-pair (X, Σ) with Σ^{2} ≥ 0. If K_{1} and K_{2} are two knots in S^{3} and if there is a diffeomorphism of pairs (X, Σ_{K_{1}}) ≅ (X, Σ_{K_{2}}), then Δ_{K_{1}}(t) = Δ_{K_{2}}(t).

As a special case:

Theorem 3.3 ([7]). Let X be a simply connected symplectic 4-manifold and Σ a symplectically embedded surface of positive genus and nonnegative self-intersection. Assume also that π_{1}(X \ Σ) = 1. If K_{1} and K_{2} are knots in S^{3} and if (X, Σ_{K_{1}}) ≅ (X, Σ_{K_{2}}), then Δ_{K_{1}}(t) = Δ_{K_{2}}(t). Furthermore, if Δ_{K}(t) ≠ 1, then Σ_{K} is not smoothly ambient isotopic to a symplectic submanifold of X.

The second part of the theorem holds because if Σ_{K} were symplectic, X#_{Σ_{K}=S_{g}}Y_{g} would be a symplectic manifold. The symplectic form ω on this manifold is inherited from the forms on X and Y_{g}; so ⟨ω, R⟩ = 0. But SW_{X#_{Σ_{K} =Sg} Y_{g}} = SW_{X#_{Σ=Sg} Y_{g}} · Δ_{K}(r), and it follows that the among the basic classes k of X#_{Σ_{K}=S_{g}}Y_{g}, more than one has ⟨ω, k⟩ maximal. This contradicts the fact that, for a symplectic manifold, the maximality of ⟨ω, K⟩ characterizes the canonical class among all basic classes [24].

4. Fiber sums of holomorphic Lefschetz fibrations

In this section we shall construct for every integer g ≥ 3 a pair (X_{g}, X′_{g}) of simply connected complex surfaces carrying holomorphic genus g Lefschetz fibrations with the property that their fiber sum (along a regular fiber) is a symplectic 4-manifold Z_{g} which supports no complex structure; in fact Z_{g} is not even homeomorphic to a complex manifold.

Let T (p, q) denote the (p, q) torus knot in S^{3} and let N(p, q) denote the 4-manifold obtained by attaching a 2-handle to the 4-ball along T (p, q) with 0framing. It is well known that N(p, q) is a Lefschetz fibration over D^{2} with generic fiber a Riemann surface of genus g(p, q) = (p − 1)(q − 1)/2. Let W (p, q) denote the canonical resolution of the Brieskorn singularity Σ(p, q, pq), the Seifert-fibered 3-manifold with three exceptional fibers of order p, q, and pq, and with H_{1} = Z. It is known that W (p, q) also supports the structure of a genus g(p, q) Lefschetz fibration over D^{2} with a singular fiber over 0 which is a sequence of 2-spheres plumbed according to the resolution diagram of Σ(p, q, pq). Finally, let

Z(p, q) = W (p, q) ∪ N(p, q).

The manifold Z(p, q) is a rational surface which is diffeomorphic to the connected sum of CP^{2} and r(p, q) copies of \overline{CP}^{2} for some computable integer r(p, q). Furthermore, Z(p, q) supports the structure of a holomorphic Lefschetz fibration whose fiber has genus g(p, q).

Now consider nontrivial torus knots T (p, q) and T (p′, q′) with the property that g(p, q) = g(p′, q′). (This is possible for every g(p, q) ≥ 3.) Let F (p, q; p′, q′) denote the fiber sum along a regular fiber of Z(p, q) with Z(p′, q′). Then F (p, q; p′, q′) is a simply connected symplectic 4-manifold with

c_{1}^{2} = 10 + 8g(p, q) − r(p, q) − r(p′, q′), χ = (b^{+} + 1)/2 = 1 + g(p, q).

Furthermore, F (p, q; p′, q′) supports the structure of a Lefschetz fibration with fiber of genus g(p, q). A computation of the Seiberg-Witten invariants of F (p, q; p′, q′) shows that, up to sign, there is a unique Seiberg-Witten basic class. It follows that F (p, q; p′, q′) is minimal.

Conjecture . F (p, q; p′, q′) supports the structure of a complex 4-manifold if and only if {p, q} = {p′, q′}.

As evidence, consider the pairs (2, 2n + 1) and (3, n + 1), n ≢ 2 mod 3. For F (2, 2n + 1; 3, n) one can show that r(2, 2n + 1) = 4n + 4 and r(3, n + 1) = 3n + 7 so that

c_{1}^{2} = n − 2, χ = n + 1.

Thus, c_{1}^{2} = χ − 3, which violates the Noether inequality c_{1}^{2} ≥ 2χ − 6. This means that F (2, 2n + 1; 3, n) is a minimal symplectic 4-manifold that is not even homotopy equivalent to a complex manifold. In fact, it can be shown that the fiber sum of Z(2, 2n + 1) with itself is the elliptic surface E(n + 1) and that the fiber sum of Z(3, n + 1) with itself is a Horikawa surface with χ = n + 1. Furthermore F (2, 2n + 1; 3, n) can be obtained from E(n + 1) by removing from Z(2, 2n + 1) \ F ⊂ E(n + 1), F a regular fiber, the regular neighborhood of the configuration of (n − 2) 2-spheres:

•  •  …  • −(n + 1) −2  −2

whose boundary is the lens space L((n−1)^{2}, −n) and replacing it with the rational ball that this lens space bounds. (See [3] for all the details concerning this rational blowdown procedure.) Thus F (2, 2n + 1; 3, n) is the manifold Y (n) constructed in Lemma 7.5 of [3].

5. Homeomorphic but non-diffeomorphic 4-manifolds with the same Seiberg-Witten invariants

In this section we construct examples of a pair (X_{1}, X_{2}) of symplectic 4manifolds with X_{1} homeomorphic to X_{2}, SW_{X_{1}} = SW_{X_{2}}, but X_{1} is not diffeomorphic to X_{2}. To do this choose a pair of fibered 2-bridge knots K(α, β_{1}) and K(α, β_{2}) with the same Alexander polynomials; for example K_{1} = K(105, 64) and K_{2} = K(105, 76) with Alexander polynomial

Δ_{K}(t) = t^{−4} − 5t^{−3} + 13t^{−2} − 21t^{−1} + 25 − 21t + 13t^{2} − 5t^{3} + t^{4}.

Although these knots have the same Alexander polynomial, they can be distinguished by the fact that their branch covers are the lens spaces L(α, β_{1}) and L(α, β_{2}) which are distinct; in our specific case L(105, 64) is not diffeomorphic to L(105, 76). These knots are also distinguished by their dihedral linking numbers; let S_{K_{1}} and S_{K_{2}} denote the 2-fold covers of S^{3} branched over K_{1} and K_{2}, with lifted branched loci K̃_{1} and K̃_{2}, respectively. Thus we have knots K̃_{i} in S_{K_{i}} = L(α, β_{i}). Take the α-fold covers of these lens spaces to obtain links L_{i} = {K_{1}^{(i)}, …, K_{α}^{(i)}} which are the lifts of the branch loci K̃_{i}. The linking numbers of the links L_{1} and L_{2} are known as the ‘dihedral linking numbers’ of the 2-bridge knot K(α, β).

Now perform the knot surgery construction of §2 on the K3 surface, replacing T^{2} × D^{2} with S^{1} × (S_{K_{j}} \ K̃_{j}). The resulting 4-manifolds are the manifolds X_{i}. Either by adapting the arguments of [6] or by using [12] or [15], it can be shown that SW_{X} = SW_{Y} = Δ_{K}(t) · Δ_{K}(−t). Unfortunately, the X_{i} are not simply connected (but are homeomorphic). In particular, π_{1}(X_{1}) = π_{1}(X_{2}) = Z_{α}, and the α-fold covers X̃_{1} and X̃_{2} of X_{1} and X_{2} are not diffeomorphic. To see this, observe that X̃_{i} is obtained as our link construction in [6] (cf. § 2) by gluing one copy of E(2) minus a neighborhood of a smooth elliptic fiber to every boundary component of S^{1} × (S^{3} \ L_{i}). It follows from [6] that

SWX̃_{i} = Δ_{L_{i}}(t_{1}, …, t_{α}) · ∏_{j=1}^{α}(t_{j}^{1/2} − t_{j}^{−1/2})

Since the linking numbers of the links L_{1} and L_{2} are different, it can be shown the Hosokawa polynomials of the links L_{1} and L_{2}, when evaluated at 1 are distinct [9]. Thus their Alexander polynomials are different and X̃_{1} is not diffeomorphic to X̃_{2}.

There is a lesson to be learned from these examples. One must consider the Seiberg-Witten invariants of a 4-manifold X together with those of all of its covers as the appropriate invariant for X.

6. Nonsymplectic 4-manifolds with one basic class

Recall from § 2, that if k is a basic class of X, so is −k. Because of this, we say that X has n basic classes if the set {k | SW_{X}(k) ≠ 0}/{±1} consists of n elements. There are abundant examples of 4-manifolds with one basic class. Minimal nonsingular algebraic surfaces of general type have one basic class (the canonical class) [28]. The authors and others have constructed many examples of minimal symplectic manifolds with one basic class and χ − 3 ≤ c_{1}^{2} < 2χ − 6. (These manifolds cannot admit complex structures due to the geography of complex surfaces.) However, the examples described here are the first nonsymplectic manifolds with one basic class.

Let X = E(2) and T a smooth elliptic fiber. For a knot K of genus g form the knot surgery construction to obtain X_{K}. In X_{K} there is a surface Σ of genus g + 1 with [Σ]^{2} = 0 and [Σ] · [T] = 1. Let M be the 3-manifold obtained from 0-surgery on the trefoil knot. Then S^{1} × M is a T^{2}-fiber bundle over T^{2}. The fiber sum of g + 1 copies of the fiber bundle gives a 4-manifold Y which is an F = T^{2}-bundle over a surface of genus g + 1, and it is easily seen that there is a section C. Furthermore, Y is a symplectic 4-manifold with c_{1}(Y) = −2g[F]. Our example, corresponding to the genus g knot K is Z_{K} = X_{K}#_{Σ=C}Y. We perform this fiber sum so that Z_{K} is a spin 4-manifold [11]. It can be seen to be simply connected.

Write the symmetrized Alexander polynomial of K as Δ_{K}(t) = a_{0} +

∑_{n=1}^{d} a_{n}(t^{n} + t^{−n}), and call d the degree of Δ_{K}(t). Since the genus of K is g,

we have d ≤ g. If K is an alternating knot, for example, then d = g. Say that the Alexander polynomial of K has maximal degree if d = g. Using techniques of [20] we calculate:

Theorem 6.1 ([8]). Let K be a knot in S^{3} whose Alexander polynomial has maximal degree. Then Z_{K} has one basic class, k, with |SW_{Z_{K}} (k)| = a_{d}, the top coefficient of Δ_{K}(t). When |a_{d}| > 1, Z_{K} is nonsymplectic.

References

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Ronald Fintushel

Department of Mathematics Michigan State University East Lansing, Michigan 48824 U.S.A. ronfint@math.msu.edu Ronald J. Stern Department of Mathematics University of California Irvine, California 92697 U.S.A. rstern@uci.edu