Publications (61)40.56 Total impact
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ABSTRACT: An obstruction theory for representing homotopy classes of surfaces in 4manifolds by immersions with pairwise disjoint images is developed, using the theory of nonrepeating Whitney towers. The accompanying higherorder intersection invariants provide a geometric generalization of Milnor's linkhomotopy invariants, and can give the complete obstruction to pulling apart 2spheres in certain families of 4manifolds. It is also shown that in an arbitrary simply connected 4manifold any number of parallel copies of an immersed surface with vanishing selfintersection number can be pulled apart, and that this is not always possible in the nonsimply connected setting. The order 1 intersection invariant is shown to be the complete obstruction to pulling apart 2spheres in any 4manifold after taking connected sums with finitely many copies of S^2\times S^2; and the order 2 intersection indeterminacies for quadruples of immersed 2spheres in a simply connected 4manifold are shown to lead to interesting number theoretic questions.  [Show abstract] [Hide abstract]
ABSTRACT: The first part of this paper exposits a simple geometric description of the KirbySiebenmann invariant of a 4manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higherorder intersection invariants of Whitney towers studied by the authors, particularly for the 4ball. In the second part of this paper, a general theory of quadratic forms is developed and then specialized from the noncommutative to the commutative to finally, the symmetric settings. The intersection invariant for twisted Whitney towers is shown to be the universal symmetric refinement of the framed intersection invariant. As a corollary we obtain a short exact sequence that has been essential in the understanding of Whitney towers in the 4ball.Geometry and Topology Monographs 06/2012; DOI:10.2140/gtm.2012.18.35 
Article: Cohomotopy sets of 4manifolds
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ABSTRACT: Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4manifold X to the 3sphere, and to enumerate the homotopy classes of maps from X to the 2sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4manifolds and provide a framework for the study of Morse 2functions on 4manifolds, a subject that has garnered considerable recent attention.Geometry and Topology Monographs 03/2012; DOI:10.2140/gtm.2012.18.161  [Show abstract] [Hide abstract]
ABSTRACT: This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a treevalued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higherorder SatoLevine and higherorder Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4ball bounded by a link in the 3sphere. Applications include computation of the grope filtration, and new geometric characterizations of Milnor's link invariants.Geometry & Topology 02/2012; DOI:10.2140/gt.2012.16.1419 · 0.82 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We show that the Artin representation on concordance classes of string links induces a welldefined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bingdoubles of any string knot with nonzero Arf invariant. We also continue J. Levine's work [20, 21, 22] comparing two filtrations of the group of homology cobordism classes of 3dimensional homology cylinders, one defined in terms of an Artintype representation (the Johnson filtration) and one defined using clasper surgery (the GoussarovHabiro filtration). In particular, the associated graded groups are completely classified up to an unknown 2torsion summand for the GoussarovHabiro filtration, for which we obtain an upper bound, in a precisely analogous fashion to the classification of the Whitney tower filtration of link concordance.  [Show abstract] [Hide abstract]
ABSTRACT: This survey discusses our results and conjectures concerning supersymmetric field theories and their relationship to cohomology theories. A careful definition of supersymmetric Euclidean field theories is given, refining Segal's axioms for conformal field theories. We state and give an outline of the proof of various results relating field theories to cohomology theories.  [Show abstract] [Hide abstract]
ABSTRACT: We show how to measure the failure of the Whitney move in dimension 4 by constructing higherorder intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4manifolds. For Whitney towers on immersed disks in the 4ball, we identify some of these new invariants with previously known link invariants such as Milnor, SatoLevine, and Arf invariants. We also define higherorder SatoLevine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higherorder invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4ball. A conjecture regarding the nontriviality of the higherorder Arf invariants is formulated, and related implications for filtrations of string links and 3dimensional homology cylinders are described.Proceedings of the National Academy of Sciences 05/2011; 108(20):81318. DOI:10.1073/pnas.1018581108 · 9.81 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: This paper describes the relationship between the first nonvanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2complex in the 4ball, built from immersed disks bounded by the given link in the 3sphere together with finitely many `layers' of Whitney disks. The intersection invariant is a higherorder generalization of the intersection number between two immersed disks in the 4ball, well known to give the linking number of the link on the boundary, which measures intersections among the Whitney disks and the disks bounding the given link, together with information that measures the twists (framing obstructions) of the Whitney disks. This interpretation of Milnor invariants as higherorder intersection invariants plays a key role in the classifications of both the framed and twisted Whitney tower filtrations on link concordance (as sketched in this paper). Here we show how to realize the higherorder Arf invariants, which also play a role in the classifications, and derive new geometric characterizations of links with vanishing Milnor invariants of length less than or equal to 2k.Journal of Topology 02/2011; DOI:10.1112/jtopol/jtt025 · 0.86 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The first part of this paper completes the classification of Whitney towers in the 4ball that was started in three related papers. We provide an algebraic framework allowing the computations of the graded groups associated to geometric filtrations of classical link concordance by order n (twisted) Whitney towers in the 4ball. Higherorder SatoLevine invariants and higherorder Arf invariants are defined and shown to be the obstructions to framing a twisted Whitney tower. In the second part of this paper, a general theory of quadratic forms is developed and then specialized from the noncommutative to the commutative to finally, the symmetric settings. The intersection invariant for twisted Whitney towers is shown to be the universal symmetric refinement of the framed intersection invariant. UPDATE: The results of the first six sections of this paper have been subsumed into the paper "Whitney tower concordance of classical links."  [Show abstract] [Hide abstract]
ABSTRACT: This paper describes grope and Whitney tower filtrations on the set of concordance classes of classical links in terms of class and order respectively. Using the treevalued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) welldefined connected sum operation. Twisted Whitney towers are also introduced, along with a corresponding quadratic enhancement of the intersection theory for framed Whitney towers that measures Whitneydisk framing obstructions. The obstruction theory in the framed setting is strengthened, and the relationships between the twisted and framed filtrations are described in terms of exact sequences which show how higherorder SatoLevine and higherorder Arf invariants are obstructions to framing a twisted Whitney tower. The results from this paper combine with those in \cite{CST2,CST3,CST4} to give a classifications of the filtrations; see our survey \cite{CST0} as well as the end of the introduction. UPDATE: This paper has been completely subsumed into the paper "Whitney tower concordance of classical links" \cite{WTCCL}.  [Show abstract] [Hide abstract]
ABSTRACT: The authors review the notion of field theories as functors, pioneered by Atiya, Kontsevich in topological case and Segal in the conformal case. They give precise meaning to the notion of smoothness for such functors by introducing family versions of the relevant bordism categories. The authors generalize these notions to supersymmetric field theories, also valid in higher dimensions, which they then continue to study in the simplest case that of dimension zero. The main contribution of the paper is to make all new mathematical notions regarding supersymmetric field theories precise. The authors observe that concordance gives an equivalence relation which can be defined for geometric objects over manifolds for which “pullbacks” and “isomorphisms” make sense. By Stokes’ theorem two closed nforms on X are concordant if and only if they represent the same de Rham cohomology class; two vector bundles with connections are concordant if and only if they are isomorphic as vector bundles. Passing from a Euclidean field theory (EFT) over X to its concordance class forgets the geometric information while retaining the topological information: differential forms of a specific degree n arise by forgetting the Euclidean geometry (on superpoints) and working with topological field theories (TFTs) instead. As a consequence, concordance classes of such field theories are shown to represent de Rham cohomology. The authors’ results are consistent with the formal group point of view towards (complex oriented) cohomology theories, where the additive formal group gives an ordinary rational cohomology, the multiplicative group gives Ktheory and the formal groups associated to elliptic curves lead to elliptic cohomology.01/2011; 2(1). DOI:10.4171/QT/12  [Show abstract] [Hide abstract]
ABSTRACT: In his study of the group of homology cylinders, J. Levine made the conjecture that a certain homomorphism eta': T > D' is an isomorphism. Here T is an abelian group on labeled oriented trees, and D' is the kernel of a bracketing map on a quasiLie algebra. Both T and D' have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory, and the homology of the group of automorphisms of the free group. In this paper, we confirm Levine's conjecture. This is a central step in classifying the structure of links up to grope and Whitney tower concordance, as explained in other papers of this series. We also confirm and improve upon Levine's conjectured relation between two filtrations of the group of homology cylinders.Geometry & Topology 12/2010; DOI:10.2140/gt.2012.16.555 · 0.82 Impact Factor 
Article: Traces in monoidal categories
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ABSTRACT: The main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category. This construction is a common generalization of the trace for endomorphisms of dualizable ob jects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property.Transactions of the American Mathematical Society 10/2010; DOI:10.1090/S000299472012056157 · 1.10 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The authors use a combination of bordism theory and surgery theory to classify up to scobordism the closed oriented topological 4manifolds with geometrically twodimensional fundamental groups satisfying property (WAA): (i) Whitehead group Wh(π) vanishes; (ii) The assembly map A 5 :H 5 (π;𝕃 0 )→L 5 (ℤπ) is surjective; (iii) The assembly map A 4 :H 4 (π;𝕃 0 )→L 4 (ℤπ) is injective. In particular, these properties are shown to be satisfied by the family of solvable BaumslagSolitar groups B(k)={a,b∣aba 1 =b k ,k∈ℤ}· Since the groups B(k) are solvable, Freedman’s scobordism theorem may to applied to complete the homeomorphism classification. Theorem A: For closed oriented 4manifolds with solvable BaumslagSolitar fundamental groups, and given type and KirbySiebenmann invariant, any isometry between equivariant intersection forms can be realized by a homeomorphism. Theorem B: For any closed oriented 4manifold M with fundamental group π=B(k), the quotient π 2 (M) † =π 2 (M)/R(s M ) is a finitely generated, stablyfree ℤπmodule and the induced form (s M ) † is nonsingular. Conversely, (i) Any nonsingular ℤπhermitian form on a finitely generated, stablyfree ℤπmodule is realized as (s M ) † by a closed oriented 4manifold M with fundamental group B(k)· (ii) Up to homeomorphism, there are exactly two such manifolds for odd forms, distinguished by the KirbySiebenmann invariant. If k is even, an even form determines a manifold of type (II) uniquely; type (III) does not occur. For k odd, there is exactly one 4manifold with a given even intersection form in each type (II) or (III). More generally, the authors show: Theorem C: For closed oriented 4manifolds with geometrically twodimensional fundamental groups satisfying properties (WAA), and given KirbySiebenmann invariant, any isometry between equivariant intersection forms inducing an isomorphism of w 2 types can be realized by an scobordism. In the proofs, the modified surgery approach to classification developed by the second author in [Ann. Math. (2) 149, No. 3, 707–754 (1999; Zbl 0935.57039)] plays a key role. In particular, the methods differ from the classical surgery approach in that they do not need to understand the homotopy classification first. This leads to the reduction of the scobordism classification to an easier bordism question.Journal of Topology and Analysis 06/2009; DOI:10.1142/S1793525309000084 · 0.34 Impact Factor 
Article: Strings, Fields and Topology
01/2009; DOI:10.4171/OWR/2009/28  [Show abstract] [Hide abstract]
ABSTRACT: Introduction: In recent years, the interplay between traditional geometric topology and theoretical physics, in particular quantum field theory, has played a significant role in the work of many researchers. The idea of this workshop was to bring these people together so that the fields will be able to grow together in the future. Most of the talks were related to various flavors of field theories and differential cohomology theories.01/2009; 6(2).  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we answer a question of Mike Freedman, regarding the efficiency of positive topological field theories as invariants of smooth manifolds in dimensions greater than 4. We show that simply connected closed 5manifolds can be distinguished by such invariants. Using Barden's classification, this follows from our result which says that homology groups and the vanishing of cohomology operations with finite coefficients are detected by positive topological field theories. Moreover, we prove that in the nonsimply connected case, as well as in all dimensions d> 5, the universal manifold pairing (and in particular, ddimensional positive topological field theories) are not sufficient to distinguish compact dmanifolds with boundary S 3 × S n , n> 1, and S 4 k −1 , k> 1. The latter case is equivalent to the same statement for closed 4kmanifolds.Journal of Topology 07/2008; DOI:10.1112/jtopol/jtn016 · 0.86 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Closed oriented 4manifolds with the same geometrically 2dimensional fundamental group (satisfying certain properties) are classified up to $s$cobordism by their $w_2$type, equivariant intersection form and the KirbySiebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed oriented 4manifolds with solvable BaumslagSolitar fundamental groups, including a precise realization result. 
Article: New Constructions of Slice Links
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ABSTRACT: We use techniques of Freedman and Teichner to prove that, under certain circumstances, the multiinfection of a slice link is again slice (not necessarily smoothly slice). We provide a general context for proving links are slice that includes many of the previously known results. Comment: 19 pagesCommentarii Mathematici Helvetici 09/2007; DOI:10.4171/CMH/175 · 0.90 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this paper we answer a question of Mike Freedman, regarding the efficiency of positive topological field theories as invariants of smooth manifolds in dimensions > 4. We show that simply connected closed 5manifolds can be distinguished by such invariants. Using Barden's classification, this follows from our observation that homology groups and the vanishing of cohomology operations with finite coefficients are detected by positive topological field theories. Moreover, we prove that in the nonsimply connected case, as well as in all dimensions d > 5, the universal manifold pairing (and in particular, ddimensional positive topological field theories) are not sufficient to distinguish compact dmanifolds with boundary S 3 × S n , n > 1 and S 4k−1 , k > 1. The latter case is equivalent to the same statement for closed 4kmanifolds.
Publication Stats
758  Citations  
40.56  Total Impact Points  
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Institutions

2006–2011

University of California, Berkeley
 Department of Mathematics
Berkeley, CA, United States


1994–2003

University of California, San Diego
 Department of Mathematics
San Diego, California, United States


1992–1995

Johannes GutenbergUniversität Mainz
Mayence, RheinlandPfalz, Germany
