Peter Teichner

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

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Publications (61)40.56 Total impact

  • Rob Schneiderman, Peter Teichner
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    ABSTRACT: An obstruction theory for representing homotopy classes of surfaces in 4-manifolds by immersions with pairwise disjoint images is developed, using the theory of non-repeating Whitney towers. The accompanying higher-order intersection invariants provide a geometric generalization of Milnor's link-homotopy invariants, and can give the complete obstruction to pulling apart 2-spheres in certain families of 4-manifolds. It is also shown that in an arbitrary simply connected 4-manifold any number of parallel copies of an immersed surface with vanishing self-intersection number can be pulled apart, and that this is not always possible in the non-simply connected setting. The order 1 intersection invariant is shown to be the complete obstruction to pulling apart 2-spheres in any 4-manifold after taking connected sums with finitely many copies of S^2\times S^2; and the order 2 intersection indeterminacies for quadruples of immersed 2-spheres in a simply connected 4-manifold are shown to lead to interesting number theoretic questions.
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    James Conant, Rob Schneiderman, Peter Teichner
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    ABSTRACT: The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--ball. In the second part of this paper, a general theory of quadratic forms is developed and then specialized from the non-commutative 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 4--ball.
    Geometry and Topology Monographs 06/2012; DOI:10.2140/gtm.2012.18.35
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    Robion Kirby, Paul Melvin, Peter Teichner
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    ABSTRACT: Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. 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 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.
    Geometry and Topology Monographs 03/2012; DOI:10.2140/gtm.2012.18.161
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    James Conant, Rob Schneiderman, Peter Teichner
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    ABSTRACT: This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued 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 higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. 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
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    James Conant, Rob Schneiderman, Peter Teichner
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    ABSTRACT: We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles 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 3-dimensional homology cylinders, one defined in terms of an Artin-type representation (the Johnson filtration) and one defined using clasper surgery (the Goussarov-Habiro filtration). In particular, the associated graded groups are completely classified up to an unknown 2-torsion summand for the Goussarov-Habiro filtration, for which we obtain an upper bound, in a precisely analogous fashion to the classification of the Whitney tower filtration of link concordance.
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    Stephan Stolz, Peter Teichner
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    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.
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    Jim Conant, Rob Schneiderman, Peter Teichner
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    ABSTRACT: We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previously known link invariants such as Milnor, Sato-Levine, and Arf invariants. We also define higher-order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney tower. Together with Milnor invariants, these higher-order invariants are shown to classify the existence of (twisted) Whitney towers of increasing order in the 4-ball. A conjecture regarding the nontriviality of the higher-order Arf invariants is formulated, and related implications for filtrations of string links and 3-dimensional homology cylinders are described.
    Proceedings of the National Academy of Sciences 05/2011; 108(20):8131-8. DOI:10.1073/pnas.1018581108 · 9.81 Impact Factor
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    James Conant, Rob Schneiderman, Peter Teichner
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    ABSTRACT: This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Whitney disks. The intersection invariant is a higher-order generalization of the intersection number between two immersed disks in the 4-ball, 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 higher-order 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 higher-order 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
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    James Conant, Rob Schneiderman, Peter Teichner
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    ABSTRACT: The first part of this paper completes the classification of Whitney towers in the 4-ball 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 4-ball. Higher-order Sato-Levine invariants and higher-order 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 non-commutative 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."
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    James Conant, Rob Schneiderman, Peter Teichner
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    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 tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) well-defined 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 Whitney-disk 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 higher-order Sato-Levine and higher-order 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}.
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    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 n-forms 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 K-theory and the formal groups associated to elliptic curves lead to elliptic cohomology.
    01/2011; 2(1). DOI:10.4171/QT/12
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    James Conant, Rob Schneiderman, Peter Teichner
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    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 quasi-Lie 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
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    Stephan Stolz, Peter Teichner
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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/S0002-9947-2012-05615-7 · 1.10 Impact Factor
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    Peter Teichner, Matthias Kreck, Ian Hambelton
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    ABSTRACT: The authors use a combination of bordism theory and surgery theory to classify up to s-cobordism the closed oriented topological 4-manifolds with geometrically two-dimensional fundamental groups satisfying property (W-AA): (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 Baumslag-Solitar groups B(k)={a,b∣aba -1 =b k ,k∈ℤ}· Since the groups B(k) are solvable, Freedman’s s-cobordism theorem may to applied to complete the homeomorphism classification. Theorem A: For closed oriented 4-manifolds with solvable Baumslag-Solitar fundamental groups, and given type and Kirby-Siebenmann invariant, any isometry between equivariant intersection forms can be realized by a homeomorphism. Theorem B: For any closed oriented 4-manifold M with fundamental group π=B(k), the quotient π 2 (M) † =π 2 (M)/R(s M ) is a finitely generated, stably-free ℤπ-module and the induced form (s M ) † is non-singular. Conversely, (i) Any non-singular ℤπ-hermitian form on a finitely generated, stably-free ℤπ-module is realized as (s M ) † by a closed oriented 4-manifold M with fundamental group B(k)· (ii) Up to homeomorphism, there are exactly two such manifolds for odd forms, distinguished by the Kirby-Siebenmann 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 4-manifold with a given even intersection form in each type (II) or (III). More generally, the authors show: Theorem C: For closed oriented 4-manifolds with geometrically two-dimensional fundamental groups satisfying properties (W-AA), and given Kirby-Siebenmann invariant, any isometry between equivariant intersection forms inducing an isomorphism of w 2 -types can be realized by an s-cobordism. 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 s-cobordism classification to an easier bordism question.
    Journal of Topology and Analysis 06/2009; DOI:10.1142/S1793525309000084 · 0.34 Impact Factor
  • Dennis Sullivan, Stephan Stolz, Peter Teichner
    01/2009; DOI:10.4171/OWR/2009/28
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    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).
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    Matthias Kreck, Peter Teichner
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    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 5-manifolds 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 non-simply connected case, as well as in all dimensions d> 5, the universal manifold pairing (and in particular, d-dimensional positive topological field theories) are not sufficient to distinguish compact d-manifolds 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 4k-manifolds.
    Journal of Topology 07/2008; DOI:10.1112/jtopol/jtn016 · 0.86 Impact Factor
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    Ian Hambleton, Matthias Kreck, Peter Teichner
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    ABSTRACT: Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to $s$-cobordism by their $w_2$-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed oriented 4-manifolds with solvable Baumslag-Solitar fundamental groups, including a precise realization result.
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    Tim D. Cochran, Stefan Friedl, Peter Teichner
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    ABSTRACT: We use techniques of Freedman and Teichner to prove that, under certain circumstances, the multi-infection 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 pages
    Commentarii Mathematici Helvetici 09/2007; DOI:10.4171/CMH/175 · 0.90 Impact Factor
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    Matthias Kreck, Peter Teichner
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    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 5-manifolds 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 non-simply connected case, as well as in all dimensions d > 5, the universal manifold pairing (and in particular, d-dimensional positive topological field theories) are not sufficient to distinguish compact d-manifolds 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 4k-manifolds.

Publication Stats

758 Citations
40.56 Total Impact Points

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 Gutenberg-Universität Mainz
      Mayence, Rheinland-Pfalz, Germany