Peter Teichner

Max Planck Institute for Mathematics, Bonn, North Rhine-Westphalia, Germany

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Publications (62)43.19 Total impact

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    Daniel Kasprowski · Markus Land · Mark Powell · Peter Teichner
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    ABSTRACT: We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant intersection forms are stably isometric. We also find explicit algebraic invariants that determine the stable classification for spin manifolds in this class.
    Full-text · Article · Nov 2015
  • 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.
    No preview · Article · Oct 2012 · Documenta mathematica Journal der Deutschen Mathematiker-Vereinigung
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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.
    Preview · Article · Jun 2012 · Geometry and Topology Monographs
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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.
    Preview · Article · Mar 2012 · Geometry and Topology Monographs
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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.
    Preview · Article · Feb 2012 · Geometry & Topology
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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.
    Preview · Article · Feb 2012 · Quantum Topology
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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.
    Full-text · Article · Jul 2011
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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.
    Preview · Article · May 2011 · Proceedings of the National Academy of Sciences
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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 our classifications [J. Conant, R. Schneiderman and P. Teichner, ‘Higher-order intersections in low-dimensional topology’, Proc. Natl Acad. Sci. USA 108 (2011) 8131–8138; J. Conant, R. Schneiderman and P. Teichner, ‘Whitney tower concordance of classical links’, Geom. Topol. 16 (2012) 1419–1479] of both the framed and twisted Whitney tower filtrations on link concordance. 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 length at most 2k Milnor invariants.
    Preview · Article · Feb 2011 · Journal of Topology
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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."
    Preview · Article · Jan 2011
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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}.
    Preview · Article · Jan 2011
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    Full-text · Article · Jan 2011 · Quantum Topology
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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.
    Preview · Article · Dec 2010 · Geometry & Topology
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    Stephan Stolz · Peter Teichner
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    ABSTRACT: This paper contains the construction, examples and properties of a trace and a trace pairing for certain morphisms in a monoidal category with switching isomorphisms. Our construction of the categorical trace is a common generalization of the trace for endomorphisms of dualizable objects in a balanced monoidal category and the trace of nuclear operators on a topological vector space with the approximation property. In a forthcoming paper, applications to the partition function of super-symmetric field theories will be given.
    Full-text · Article · Oct 2010 · Transactions of the American Mathematical Society
  • Henning Hohnhold · Stephan Stolz · Peter Teichner
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    ABSTRACT: In memory of Raoul Bott, friend and mentor. Abstract. There are many models for the K-theory spectrum known today, each one having its own history and applications. The purpose of this note is to give an elementary description of eight such models (and certain completions of them) and to relate all of them by canonical maps, some of which are homeomorphisms (rather than just homotopy equivalences). Our survey begins with Raoul Bott’s iterated spaces of minimal geodesics in orthogonal groups, whichheusedtoprovehisfamousperiodicity theorem, and includes Milnor’s spaces of Clifford module structures as well as the Atyiah – Singer spaces of Fredholm operators. From these classical descriptions we move via spaces of unbounded operators and super-semigroups of operators to our most recent model, which is given by certain spaces of supersymmetric (1|1)-dimensional field theories. These spaces were introduced by the second two authors for the purpose of generalizing them to spaces of certain supersymmetric (2|1)dimensional Euclidean field theories that are conjectured to be related to the Hopkins – Miller spectrum TMF of topological modular forms.
    No preview · Article · Oct 2010
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    Peter Teichner · Matthias Kreck · Ian Hambelton
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    ABSTRACT: Closed oriented 4-manifolds with the same geometrically two-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.
    Full-text · Article · Jun 2009 · Journal of Topology and Analysis
  • Dennis Sullivan · Stephan Stolz · Peter Teichner

    No preview · Article · Jan 2009
  • Stephan A. Stolz · Dennis Sullivan · Peter Teichner

    No preview · Article · Jan 2009
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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 S3 × Sn, n > 1, and S4k−1, k > 1. The latter case is equivalent to the same statement for closed 4k-manifolds.
    Full-text · Article · Jul 2008 · Journal of Topology
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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.
    Full-text · Article · Feb 2008

Publication Stats

879 Citations
43.19 Total Impact Points

Institutions

  • 2012
    • Max Planck Institute for Mathematics
      Bonn, North Rhine-Westphalia, Germany
  • 2002-2011
    • University of California, Berkeley
      • Department of Mathematics
      Berkeley, California, United States
  • 2005
    • Rice University
      • Department of Mathematics
      Houston, Texas, 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