Abstract. We show how the new invariants of 4-manifolds resulting from the Seiberg-Witten monopole equation lead quickly to a proof of the ‘Thom conjecture’. 1. Statement of the result The genus of a smooth algebraic curve of degree d in CP 2 is given by the formula g =(d−1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2-manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4andford = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,8,10]. In this note we confirm the conjecture. Theorem 1. Let Σ be an oriented 2-manifold smoothly embedded in CP 2 so as to represent the same homology class as an algebraic curve of degree d. Then the genus g of Σ satisfies g ≥ (d − 1)(d − 2)/2. Very recently, Seiberg and Witten [11,12,13] introduced new invariants of 4manifolds, closely related to Donaldson’s polynomial invariants [2], but in many respects much simpler to work with. The new techniques have led to more elementary proofs of many theorems in the area. Given the monopole equation and the vanishing theorem which holds when the scalar curvature is positive (something which was pointed out by Witten), the rest of the argument presented here is not hard to come by. A slightly different proof of the Theorem, based on the same techniques, has been found by Morgan, Szabo and Taubes. It is also possible to prove a version of Theorem 1 for other complex surfaces, without much additional work. This and various other applications will be treated in a later paper, with joint authors. 2. The monopole equation and the Seiberg-Witten invariants Let X be an oriented, closed Riemannian 4-manifold. Let a spin c structure on X be given. We write c for the spin c structure and write W + = W + c and W − = W − c for the associated spin c bundles. Thus W + is a U(2) bundle and Clifford multiplication

(i) Topology of embedded surfaces. Let X be a smooth, simply-connected 4-manifold, and ξ a 2-dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly

Abstract. For an oriented link L ⊂ S 3 = ∂D 4, let χs(L) be the greatest Euler characteristic χ(F) of an oriented 2-manifold F (without closed components) smoothly embedded in D 4 with boundary L. A knot K is slice if χs(K) = 1. Realize D 4 in C 2 as {(z, w) : |z | 2 + |w | 2 ≤ 1}. It has been conjectured that, if V is a nonsingular complex plane curve transverse to S 3, then χs(V ∩ S 3) = χ(V ∩ D 4). Kronheimer and Mrowka have proved this conjecture in the case that V ∩ D 4 is the Milnor fiber of a singularity. I explain how this seemingly special case implies both the general case and the “slice-Bennequin inequality ” for braids. As applications, I show that various knots are not slice (e.g., pretzel knots like P(−3, 5, 7); all knots obtained from a positive trefoil O{2,3} by iterated untwisted positive doubling). As a sidelight, I give an optimal counterexample to the “topologically locally-flat Thom conjecture”. 1. A brief history of sliceness A link is a compact 1-manifold without boundary L (i.e., finite union of simple closed curves) smoothly embedded in the 3-sphere S 3; a knot is a link with one component. If S 3 is realized in R 4 as, say, the unit sphere, then a natural way to construct links is to intersect suitable two-dimensional subsets X ⊂ R 4 with S 3; one may then ask how constraints on X are reflected in constraints on the link X ∩ S 3. For instance, Fox and Milnor (c. 1960) considered, in effect, the case that X is a smooth 2-sphere intersecting S 3 transversally; at Moise’s suggestion, Fox [5] adopted the adjective slice to describe the knots and links X ∩ S 3 so constructed. Fox and Milnor [6] gave a criterion for a knot K to be slice: its Alexander polynomial ∆K(t) ∈ Z[t, t −1] must have the form F(t)F(t −1). This shows that, for instance, the two trefoil knots O{2, ±3} are not slice (since ∆ O{2,±3} = t −1 −1+t is not of the form F(t)F(t −1)), but it says nothing about the two granny knots O{2, 3} O{2, 3}, O{2, −3} O{2, −3} (indeed, both granny knots share the Alexander polynomial

We show that the tree-level part of a theory of finite type invariants of 3-manifolds (based on surgery on objects called claspers, Y-graphs or clovers) is essentially given by classical algebraic topology in terms of the Johnson homomorphism and Massey products, for arbitrary 3-manifolds. A key role of our proof is played by the notion of a homology cylinder, viewed as an enlargement of the mapping class group, and an apparently new Lie algebra of graphs colored by H1(Σ) of a closed surface Σ, closely related to deformation quantization on a surface [AMR1, AMR2, Ko3] as well as to a Lie algebra that encodes the symmetries of Massey products and the Johnson homomorphism. In addition, we give a realization theorem for Massey products and the Johnson homomorphism by homology cylinders.

[FM], along with that of Murasugi [M] and Levine [Le1, Le2], revealed fundamental aspects of the structure of C. Since then there has been tremendous progress in 3- and 4-dimensional geometric topology, yet nothing more is now known about the underlying group structure of C than was known in 1969. In this paper we will describe new and unexpected classes of order 2 in C. What is known about C is quickly summarized. It is a countable abelian group. According to [Le2] there is a surjective homomorphism of φ: C → Z ∞ ⊕Z ∞ 2 ⊕Z ∞ 4. The results of [FM] quickly yield an infinite set of elements of order 2 in C, all of which are mapped to elements of order 2 by φ. The results just stated, and their algebraic consequences, present all that is known concerning the purely algebraic structure of C in either the smooth or topological locally flat category. For instance, one can conclude that elements of order 2 detected by homomorphisms to Z2, such as the Fox-Milnor examples, are not evenly divisible, but it remains possible that any given countable abelian group is a subgroup of C, including such groups as the infinite direct sum of copies of Q and Q/Z. Most succinctly, we know that C is isomorphic to a direct sum Z ∞ ⊕Z ∞ 2 ⊕ G, but all that is known about G is that it is countable and abelian. In particular, Fox and Milnor’s original question on the existence of torsion of order other than 2 remains completely open. Other basic questions regarding the structure of C appear in [Go, K1, K2].

Abstract. In knot concordance three genera arise naturally, g(K), g4(K), and gc(K): these are the classical genus, the 4–ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 ≤ g4(K) ≤ gc(K) ≤ g(K). Casson and Nakanishi gave examples to show that g4(K) need not equal gc(K). We begin by reviewing and extending their results. For knots representing elements in A, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon’s result that A is nontrivial implies that g4(K) can be nonzero for knots in A. Gilmer proved that g4(K) can be arbitrarily large for knots in A. We will prove that there are knots K in A with g4(K) = 1 and gc(K) arbitrarily large. Finally, we tabulate gc for all knots with 10 or fewer crossings. This requires the development of further obstructions and the description of previously unnoticed concordances. 1.

In 1926 Artin [3] described the construction of knotted 2–spheres in R 4. The intersection of each of these knots with the standard R 3 ⊂ R 4 is a nontrivial knot in R 3. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible that every knot is such a slice knot and it wasn’t until the early 1960s that Murasugi [84] and Fox and Milnor [24, 25] succeeded at proving that some knots are not slice. Slice knots can be used to define an equivalence relation on the set of knots in S 3: knots K and J are equivalent if K # − J is slice. With this equivalence the set of knots becomes a group, the concordance group of knots. Much progress has been made in studying slice knots and the concordance group, yet some of the most easily asked questions remain untouched. There are two related theories of concordance, one in the smooth category and the other topological. Our focus will be on the smooth setting, though the distinctions and main results in the topological setting will be included. Related topics must be excluded, in particular the study of link concordance. Our focus lies entirely in the classical setting; higher dimensional concordance theory is only mentioned when needed to understand the classical setting. 1.

We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two. The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for n> 1 there is an isomorphism φ from the concordance group Cn of knotted (2n−1)–spheres in S2n+1 to an algebraically defined group G±; furthermore, G± is isomorphic to the infinite direct sum Z ∞ ⊕Z ∞ 2 ⊕Z∞4. It was a startling consequence of the work of Casson and Gordon that in the classical case the kernel of φ on C1 is infinitely generated. Beyond this, little has been discovered about the pair (C1,φ). In this paper we present a new approach to studying C1 by introducing a group, M, defined as the quotient of the set of knots by the equivalence relation generated by concordance and positive mutation, with group operation induced by connected sum. We prove there is a factorization of φ, C φ1 φ2 −→M−→G−. Our main result is that both maps have infinitely generated kernels.

Abstract. In this paper we present a series of examples of new phenomena in the classical knot concordance group. First we show that for (almost) every Seifert form there is an infinite family of knots, distinct in concordance, having that form. Next we demonstrate that a number of results that are known to hold in higher dimensional concordance fail in the classical case. These include: (1) examples of knots with Seifert forms that split as direct sums of Seifert forms but the knots are not concordant to corresponding connected sums, and (2) knots with Alexander polynomials that factor as products of Alexander polynomials (with resultant 1) but the knots are not concordant to corresponding connected sums. We also provide examples showing that: (3) for almost every metabolic Seifert form M and for every Seifert form V, there are knots with Seifert form V ⊕ M which are not concordant to knots with Seifert form V, and (4) there are pairs of irreducible algebraically concordant Seifert forms V and W such that there are knots with Seifert form V that are not concordant to any knot with Seifert form W.

An n-dimensional µ-component boundary link is a codimension 2 embedding of spheres L = � S n ⊂ S n+2 µ such that there exist µ disjoint oriented embedded (n + 1)-manifolds which span the components of L. An Fµ-link is a boundary link together with a cobordism class of such spanning manifolds. The Fµ-link cobordism group Cn(Fµ) is known to be trivial when n is even but not finitely generated when n is odd. Our main result is an algorithm to decide whether two odd-dimensional Fµ-links represent the same cobordism class in C2q−1(Fµ) assum-ing q> 1. We proceed to compute the isomorphism class of C2q−1(Fµ), generalizing Levine’s computation of the knot cobordism group C2q−1(F1). Our starting point is the algebraic formulation of Levine, Ko and Mio who identify C2q−1(Fµ) with a surgery obstruction group, the Witt group G (−1)q,µ (Z) of µ-component