My research interests have evolved over the years, and mainly center around low-dimensional topology and knot theory. To me, low-dimensional topology means the topology of 3 and 4-dimensional manifolds, and the interesting interactions between these subjects. An important development in the last twenty years in these subjects has been the growth of analytical/geometric methods, especially those of gauge theory, and this has been central to my research. Knots appear naturally in this work as a source of interesting examples, and as a way of describing 3-manifolds and 4-manifolds. My oldest work is concerned with Casson-Gordon invariants, and also with high-dimensional knot theory. My published papers are in my publication list and I also have a list of available preprints. (This latter is out of date; for recent papers you'd do better on the math arxiv.) What follows is a description of some of my work, roughly in reverse chronological order. I haven't discussed every single paper, or gone all the way back to my first papers, but hope to do that some day.
The mysterious nature of topology in dimension 4 is a theme that runs through my work, from the beginning to the present day. This section is about some of the most recent aspects of my research on the more topological side; the one that follows (describing work with Saveliev and Mrowka) is a little more analysis-based.
Absolutely exotic 4-manifolds: An ongoing theme in four-dimensional topology is the search for exotic smooth structures on 4-manifolds. Such structures are known to exist for most simply-connected manifolds, but the quest gets a lot harder as the size (as measured by the second Betti number) goes down. The Holy Grail would be an exotic CP2 or even an exotic 4-sphere (a counterexample to the Poincaré conjecture). Some years ago, Selman Akbulut (Michigan State University) showed how to find smooth structures on contractible manifolds that are exotic relative to the boundary; this is the phenomenon now known as a cork. Selman and I gave a technique for converting relatively exotic manifolds into absolutely exotic ones. Our technique uses some 3-manifold topology related to my much older work on invertible homology cobordisms.
Non-orientable surfaces in 4-manifolds: The study of embeddings of surfaces in 4-manifolds is absolutely fundamental, and goes back to the earliest days. For instance, questions about existence and intersections of embedded spheres are tied to the failure (in the smooth category) and partial success (in the topological setting) of programs to adapt surgery techniques that work in high dimensions. The most famous instance is the proof of the Thom conjecture, which gives a lower bound on the genus of an embedded surface carrying a given integral homology class in many 4-manifolds, most famously CP2. With Adam Levine (now at Princeton University), Saso Strle (University of Ljubljana, Slovenia) and my Brandeis colleague Ira Gessel, I established lower bounds for non-orientable surfaces carrying a given Z2 homology class. The techniques come from the Floer homology proof of the Thom conjecture, and require some strenuous calculation of the Floer groups (inluding the `correction terms') for non-orientable circle bundles. Adam and I further extended the scope of the correction terms (using additional algebraic structures in Floer theory) and have applied them to link cobordism.
Stabilization phenomena: A classical theorem of Wall, in some sense the second theorem in 4-manifold theory (the first being Rohlin's theorem) says that homotopy equivalent simply-connected 4-manifolds become diffeomorphic after stabilization. In this setting, stabilization means connected sum with S2 x S2 or CP2 # CP2. Wall's proof gave no indication of the number of stabilizations, but recent work of many authors shows that just one stabilization suffices in all known cases. For many years, I wondered whether this same result would hold for other exotic behavior in 4-manifolds, in particular whether the non-isotopic diffeomorphisms I constructed some years ago (see below) would become isotopic after a single stabilization. This would imply the same 1-stable equivalence for the resulting metrics of positive scalar curvature. A joint project with Dave Auckly (Kansas State University), Hee Jung Kim (Seoul National University), and Paul Melvin (Bryn Mawr) shows that this is indeed the case.
For more than a decade, I have been working with Nikolai Saveliev (University of Miami) on a project relating gauge theory in dimension 4 and Rohlin's invariant of 3-manifolds. We have been studying 4-manifolds that look homologically like a circle cross a homology 3-sphere; such a manifold is called a homology S1 x S3. Such manifolds have Rohlin invariants (just like homology 3-spheres) and an important conjecture states that the Rohlin invariant of a homotopy S1 x S3 must vanish.
There is a Donaldson-type invariant defined for a homology S1 x S3, and we have been working on showing that this invariant reduces, modulo 2, to the Rohlin invariant. This would imply the conjecture stated above. We have made some progress on this, and have shown it in a number of cases, in particular the case of mapping tori of diffeomorphisms of homology 3-spheres. We have also investigated similar questions, relating to homology tori, and have a new proof of Casson's result relating the Casson and Rohlin invariants in dimension 3. There are (so far) three papers coming from this work, plus a survey paper. The result on homology tori is related to a paper I wrote with Saso Strle in which we evaluated the Seiberg-Witten invariant, modulo 2, of a homology 4-torus.
My current work with Nikolai in this area focuses on using Taubes' periodic-end technology to study the Dirac operator on the infinite cyclic cover of a homology S1 x S3. We have used this circle of ideas to give another analytical interpretation of the Rohlin invariant, and applied this interpretation to show that certain non-orientable 4-manifolds do not admit metrics of positive scalar curvature. More recently, we worked with Tom Mrowka (MIT) to define a Seiberg-Witten invariant of a homology S1 x S3. The invariant is a combination of the count of irreducible solutions to the Seiberg-Witten equations and the index of an end-periodic Dirac operator. This inspired a second paper, establishing an index theorem for end-periodic operators analogous to the famous Atiyah-Patodi Singer index theorem. We found a surprising application of this circle of ideas, showing that the `end-periodic η-invariant' that arises in the index theorem can be used to detect components in the moduli space of positive scalar curvature metrics on some even-dimensional manifolds. We have made some calculations of the end-periodic η-invariant' and of the Seiberg-Witten invariant of the famous Inoue surface.
The remarkable Heegaard-Floer theory of Ozsvath and Szabo turns out to be extremely useful in the study of smooth knot and link concordance. Numerical invariants such as τ(K) and the d-invariants of surgeries and branched covers give lots of delicate information about the surfaces that knots and links bound in 4-space. I have worked on several aspects of this theory, outlined below.
Knots of polynomial one: One of the remarkable results of the 1980's arose as a conjunction of Donaldson's theorem (on intersections of 4-manifolds) and Freedman's surgery theory for 4-manifolds. Together, these showed that knots with Alexander polynomial = 1 are topologically slice, but not necessarily smoothly slice. Chuck Livingston (Indiana University), Matt Hedden (Michigan State) used Heegaard-Floer theory (d-invariants of branched covers) to show that not all topologically slice knots are smoothly concordant to knots of polynomial one.
Link concordance: The main theme here is the distinction between concordance properties of a link, and those of its individual components. For instance, Jae Choon Cha (POSTECH, Korea), Taehee Kim (Konkuk, Korea), Saso Strle (University of Ljubljana, Slovenia), and I found examples of 2-component links with Alexander polynomial 1, are topologically concordant to the Hopf link, have unknotted components, but which are not smoothly concordant to the Hopf link. Cha and I found examples of topologically slice links whose individual components are smoothly concordant to the unknot, but which are not smoothly concordant to any link with unknotted components. Most recently, Strle and I explored the `parallel link' construction of A. Kawauchi (two copies of a knot, drawn parallel to each other, with linking number 0; also known as the (2,0) cable, and showed that the d-invariants of a knot K give obstructions to the (2,0) cable of K being concordant to a split link.
Heegaard-Floer invariants of knots in branched covers: One of the more useful Heegaard-Floer invariants, due to Ozsvath and Szabo is an integer-valued knot invariant τ(K). In fact, τ(K) is an invariant of smooth knot concordance, and is definitely different from such classical concordance invariants such as the knot signature. Eli Grigsby (Boston College), Saso Strle, and I have extended the τ-invariant to give a large collection of concordance invariants. Our idea was motivated by an analogy with the Casson-Gordon invariants, which are signatures associated to a cyclic branched cover Y of S3 branched over a knot K. Briefly, for every surjection φ:H1(Y) -> Zp, there are invariants σφ(K) defined in terms of equivariant signatures. Roughly speaking, for a slice knot K, there is a subgroup G of H1(Y) such that |G|2 = |H1(Y)| with the property that σφ(K) = 0 whenever φ(G) =0. We showed that something very similar is true for the τ-invariants of the inverse image of K in the branched cover Y. There's a τ-invariant for each Spinc structure on Y; these are parameterized by H1(Y), and we showed that the τ-invariants associated to the subgroup G must all vanish. This turns out to give lots of new information on smooth knot concordance. Right now, our results are all for 2-bridge knots, but we are learning how to calculate these τ-invariants for covers of more complicated knots.
One of the hardest (and most important) questions about smooth knot concordance is whether a knot K whose Whitehead double Wh(K) is slice must in fact have been slice in the first place. (The converse is easily seen to be true.) More recently, the same question has been asked about the Bing double B(K), which is similar to the Whitehead double but produces a link of two components. Chuck Livingston, Jae Choon Cha and I showed that if B(K) is (topologically) slice, then K is algebraically slice. In particular, the Arf invariant of K must vanish; this result has some relevance to 4-manifold theory.
In 1998, I used 1-parameter Donaldson theory to give the first examples of diffeomorphisms of 4-manifolds that are homotopic, but not isotopic, to the identity. The idea of parameterized gauge theory is that if one has a manifold X for which the formal dimension of some moduli space has negative dimension (-k), then one can define a 0-dimensional moduli space whenever one has a k-dimensional family (ie a bundle with fiber X, and with k-dimensional base). In subsequent papers, I refined the method to give lots of information about the diffeomorphism group, for instance that the kernel of the map from isotopy to homotopy is infinitely generated. (This is opposite from what happens in the topological case, where this map is basically an isomorphism.) Using this technique in the context of Seiberg-Witten theory, I showed that the space of positive scalar curvature metrics on a 4-manifold can be disconnected. (This is related to the work described in the section on periodic index theory.) I'm currently working on higher-parameter generalizations of these results.
In the early 80's, Simon Donaldson had the remarkable insight that the study of the Yang-Mills moduli space held the key to all sorts of previously inaccessible problems about the topology of 4-manifolds. My research in this area centered around the fundamental idea (originating in Donaldson's work) that one should try to understand the moduli space of a manifold cut into two pieces meeting along a common boundary, in terms of the moduli space of the pieces. It turned out that to study gauge theory on manifolds with boundary, one proceeded by adding an infinite collar to the boundary, and studied gauge theory on the resulting non-compact manifold. Several years work in this direction went into a book with Tom Mrowka and John Morgan (Columbia) that lays the analytical foundations for this study in some generality. Technically, we made no assumptions of non-degeneracy of the critical points of the Chern-Simons functional along the boundary; this level of generality is useful in many applications.
The book was concerned with the Yang-Mills equations, but much of the analysis adapts quite directly to the Seiberg-Witten equations on manifolds with boundary. I wrote a number of papers using these ideas (and some with less analysis) to study problems about representing homology classes by surfaces of low genus. This included results about spheres and tori of negative self-intersection. The case of spheres turned out to be a key ingredient in the proof by Fintushel and Stern of the blowup formula for Donaldson invariants. Of course most of this was superseded by the solution of the Thom conjecture by Kronheimer-Mrowka and Morgan-Szabo-Taubes, and the extension to negative self-intersections of Oszvath-Szabo.
Parallel to my work on gauge theory, I studied the analytical/differential geometric invariants of 3-manifolds known as η-invariants. These arise as part of the formula for the dimension of the Yang-Mills moduli space, and as interesting Riemannian invariants in their own right. I used the relation between gauge theory and η-invariants to prove some results about homology cobordisms; this has applications to knot cobordism that should be worked out more fully. Rob Meyerhoff (Boston College) and I got interested in understanding how the η-invariants of hyperbolic manifolds (and the related Chern-Simons invariants) behaved under various cut/paste operations, such as mutation.
With Paul Kirk (Indiana University) and Eric Klassen (Florida State University) I wrote a paper giving topological techniques to compute spectral flows and therefore some η-invariants. My first work in this general area was a proof that mutation preserves the property of having a hyperbolic metric and doesn't change Gromov's norm. The proof relies, among other things, on extending the Freedman-Hass-Scott results about compact least area surfaces to the case of least-area properly embedded surfaces in finite-volume hyperbolic 3-manifolds. I've come back to mutations a few times over the years, most recently to show that mutation doesn't change the Floer homology of a homology 3-sphere.
A spinoff of my interest in η-invariants and Chern-Simons invariants was the observation that these invariants behave slightly differently when taking finite covers. I used this to give a new obstruction to an abstract group being the fundamental group of a closed 3-manifold.
I combined my interests in 4-dimensional knot theory and hyperbolic geometry by investigating a version of Gromov's invariant for knots in the 4-sphere. Using this, I could show that some knots can't have Seifert-fibered Seifert surfaces (same Seifert, different usages, though!). Buried in this paper is another result that has turned out to be useful in many settings: any 3-manifold is invertibly homology cobordant to a hyperbolic 3-manifold. This idea (with modifications) showed up 25 years later in my work with Akbulut on exotic smooth structures on 4-manifolds!
I was always interested in the Kummer construction, in which one quotients the 4-torus by an involution (and then resolves the singularities) to obtain a K3 surface (aka a quartic surface). One day I came across an old paper of Nikulin that showed, using binary linear codes and algebraic geometry, that the maximal number of nodal singularities that a (singular) K3 surface can have is 16, exactly the number coming from the Kummer construction. Contemplating this, I eventually realized that the fact that the torus is aspherical meant that Nikulin's argument about codes was really about the topology of 2-fold covers, and his algebraic geometry argument could be rephrased in purely topological terms. So I showed that the maximal number of disjoint topological 2-spheres that could be embedded in a K3 surface is 16. Not long after I wrote this, a preprint by David Jaffe appeared that discussed the much harder problem of the maximum number of nodes on a sextic surface (the case of the quintic having been done in 1979 by Beauville). I wrote to Jaffe, and we combined some of my topological ideas with his existing work in algebraic geometry and coding theory, to establish the upper bound of 65 for the number of nodes. By an amazing coincidence, W. Barth had just at that moment constructed a nodal sextic with 65 nodes. The case of surfaces with degree greater than 6 remains tantalizingly out of reach. There are some great pictures of (the real parts of) these surfaces by Stefan Endrass, and by Oliver Labs on his page AlgebraicSurface.net.
I plan to write about some of my other research topics (doubly slice knots, embedding problems, assorted other papers) one of these days.