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RESEARCH STATEMENT Four-Dimensional Topology Is RESEARCH STATEMENT LUKE WILLIAMS Four-dimensional topology is interesting because it is the only dimension admitting infinitely many smoothings of the same topological manifold. My desire to understand this peculiarity has steered my research to problems in 4-manifolds. To address these problems, I utilize tools from knot theory, differential and symplectic geometry, as well as algebraic topology. The large number of possible smoothings of the same underlying space makes this subject chal- lenging. The primary problem is one of classification: Given two 4-manifolds, how do we distin- guish or identify them? In many cases, Freedman has a complete answer for determining when two 4-manifolds are homeomorphic. Far less is known when we investigate 4-manifolds up to diffeomorphism. One of my own projects encounters this subtly head-on. I provide previously unknown diffeomor- phisms between families of 4-manifolds. As Freedman's techniques failed to apply, these families were not even known to be homeomorphic, let alone diffeomorphic. Other projects of mine deal with the types of geometric structures supported by \small" 4-manifolds { those with minimal topology (e.g. rational balls). As we remove topology from the equation, it becomes harder to obstruct the possible types of supported geometry. I have also had the opportunity to work with undergraduate students applying techniques coming from the 4-dimensional theory to other areas of topology such as knot theory. 1. Previous Work In general, finding diffeomorphisms can be quite difficult. Dimension four is the last holdout for the Poincar´econjecture which, in this case, asserts that the 4-sphere supports a unique smooth structure. This is the simplest closed 4-manifold and yet we do not know if it supports a non- standard smooth structure! Nevertheless, we can distinguish certain smooth structures on many closed 4-manifolds. Gauge theoretical quantities such as the Seiberg-Witten invariants, which count solutions of particular PDE's on a smooth manifold, have played a key role in differentiating between smooth structures. In general, these invariants are computationally difficult. To aid in calculation, we can start with a manifold with known SW-invariants, and attempt to collapse some of its topology via a surgical operation, tracing the effect on the invariants along the way. The blow-up operation from algebraic geometry is an example.1 Its reverse, known as a blow-down, reduces the topology by killing the exceptional sphere, a generator of second homology. Identifying Rational Balls in the Rational Blow-Down. The blow-down can be generalized to the rational blow-down where we remove a neighborhood of a configuration of spheres bounding a lens space (a cyclic quotient of S3) from a 4-manifold and glue a rational homology ball in its place. This kills each sphere in the configuration at the possible expense of introducing new nontrivial elements into the fundamental group. Fintushel and Stern, as well as Park, have investigated how the SW-invariants change under a rational blow-down. The rational balls used in this procedure are not known to be unique. In fact, Kadokami and Yamada note that there are two 2-parameter families of rational balls, Am;n and Bp;q, in the 1Blowing-up a point in a complex surface consists of replacing that point by a sphere which parameterizes the complex lines passing through it { exchanging the neighborhood of the point with that of the complex projective line (the exceptional sphere) inside complex projective space. 1 2 LUKE WILLIAMS literature that could be used. It is reasonable to wonder if the result of performing a rational blow- down depends on which family one chooses to work from. I address this problem by settling the question of whether or not these two families coincide smoothly. To that end, I explicitly construct the following diffeomorphisms Theorem 1 ([3]). There exist diffeomorphisms between the boundaries of Bp;q and Am;n that extend to diffeomorphisms between the respective spaces. I do not have to rely on building diffeomorphisms to be able to identify these smoothings. Often, geometric information can greatly curtail the number of possibilities. For instance, the diffeomorphism types of 4-manifolds that symplectically fill lens spaces equipped with specific contact structures are completely enumerated by Lisca. Therefore, if a 4-manifold with lens space boundary (as is the case in the aforementioned rational balls) admits a Stein structure appropriately compatible with the lens space, then one only has to look to Lisca's list to determine the 4-manifold in question { a simpler problem. It was previously known that each rational ball Bp;q supports such a Stein structure and thus falls under Lisca's list. I prove the same in the case of Am;n: Theorem 2 ([3]). Each member of the family Am;n supports a Stein structure. Lisca's classification immediately implies that the two families coincide without ever appealing to specific diffeomorphisms. This result is an example of how the differential geometry of a space can be used to determine that space. Non-Stein rational balls. Differences in smooth structures can, in a sense, be localized. There- fore, understanding small exotic four manifolds gets at the heart of understanding exoticity in general. For instance, it is known that any pair of homeomorphic, non-diffeomorphic, closed, sim- ply connected, 4-manifolds are related by a \cork-twist." That is, the diffeomorphism type of one manifold can be changed to the other by locating a contractible manifold, known as a cork, remov- ing it and regluing it by a diffeomorphism of its boundary which extends as a homeomorphism, but not as a diffeomorphism, over the whole cork. As Stein structures provide considerable control on allowable smoothings, it is reasonable to wonder what small manifolds fail to support Stein structures? In the previous project, I found that both rational balls supported Stein structures. In fact, this is a defining property for them! However, one can ask are there non-Stein rational balls? It turn out that such phenomena abound; in an ongoing project, I prove Theorem 3 ([4]). Associated to each finitely presented group G with finite nontrivial abelianization, there exists a smooth rational 4-ball XG, whose fundamental group coincides with G; such that XG fails to support a Stein structure. Furthermore, XG can be chosen so that the boundary is irreducible and so that no Stein structure is supported in either orientation. Such examples can often be built from a thickened Cayley complex of the desired fundamental group. The choices involved in thickening amounts to choosing attaching maps and framings for 4-dimensional 2-handles (one for each relation in a presentation of the given group). If the fundamental group admits a nontrivial finite index subgroup, then we can look to the associated cover and investigate whether or not such a cover supports a Stein structure. The aforementioned choices allow the creation of surfaces in the cover violating the adjunction inequality. Obstructing Smooth Sliceness. The existence of small 4-manifolds with particular boundaries can play a useful role in answering questions in other areas of low-dimensional topology. A knot K is smoothly slice if it bounds a properly embedded smooth 2-disk in the 4-ball. Given a slice disk for a knot, we can consider the rational ball obtained from the 2-fold cover of B4 branched over the slice disk. This ball then bounds the 2-fold cover of S3 branched over the knot. If the cover of S3 branched over K also bounds any other negative definite 4-manifold, then one can glue the two to obtain a closed, negative definite manifold. Donaldson's celebrated diagonalization theorem ensures RESEARCH STATEMENT 3 that the resulting intersection form must be standard. This provides a powerful obstruction to K being smoothly slice first noticed by Casson and Akbulut. I have used this approach successfully to show that no member of a 5-parameter family of Montesinos knots could be smoothly slice [2]. 2. Ongoing and Future Work The Structure of the Homology Cobordism Group. In recent years, there has been consider- able study devoted to the structure of the 3-dimensional homology cobordism group Θ3 { the group Z of homology 3-spheres under connect sum up to the equivalence relation that two such spheres are equivalent if their is a 4-dimensional cobordism between them that looks like a product cobordism at the level of homology. Even determining which homology 3-spheres bound acyclic 4-manifolds is important and difficult. One such class of examples are the Brieskorn spheres Σ(2; 3; 12n + 1) for 2 3 12n+1 3 n > 2 (the link of the singularity z1 + z2 + z3 = 0 in C ). It is well known that Σ(2; 3; 13) and Σ(2; 3; 25) are trivial in Θ3 and that each Σ(2; 3; 12n + 1) bounds a negative definite plumbing Γ . Z n One can apply Donaldson's diagonalization theorem to Γn in an attempt to obstruct the sphere Σ(2; 3; 12n + 1) from being trivial in the homology cobordism group. Such an obstruction fails as the intersection form associated to Γn embeds in the standard lattice. I believe that this failure can provide a route to building the desired homology cobordism from Σ(2; 3; 12n+1) to S3. Indeed, by investigating the embedding into the standard lattice, one obtains a change of basis from that lattice to the intersection form associated to the negative definite plumbing. By tracing this change of basis at the handle level, I hope to embed the plumbing in a 2 connect sum of CP 's. Taking the complement of that embedding then gives the desired cobordism.
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