Knots and their manifold stories

A confluence of several strong currents in mathematics has invigorated knot theory and ancillary areas of 3 and 4-dimensional manifolds.

This workshop will bring together a multidisciplinary community to investigate the connections between what at one time seemed disparate areas of mathematical research. The underlying root and impetus has been knot theory. Numerous questions, both classical and modern have arisen. Among these are the four-dimensional topological surgery conjecture, which has been long connected to classical knot theory, the knot concordance problem, the classification of classical knot groups, classification of four dimensional homology cobordism, computation of localization and completion in homotopy theory, the L-theory of localized rings, the classification of knots and three manifolds via finite type and quantum invariants, classical interpretations of these modern invariants, and numerous other classical problems in knots and manifold theory, high and low dimensional.

The first (and oldest) such current is that of knots and Higher-dimensional manifolds, including surgery and homotopy theory and especially localization and completion of groups, rings and modules. The second current is that of topological 4-manifolds, specifically including the topological techniques of M. Freedman and A. Casson that distinguish this field from higher-dimensional manifolds. The third current is that of combinatorial (quantum) knot theory which began with work of V. Jones, E. Witten, M. Konsevitch and V. Vassiliev. The fourth current is that of von Neumann algebras and L^2 homology. Some of the earliest and most exciting applications of surgery theory were classification results for higher-dimensional knots and links in the work of A. Haefliger, J. Levine and M. Kervaire. S. Cappell and J. Shaneson developed surgery with coefficients to approach codimension-two placement, tangential structures with coefficients were classified in part by Taylor and Williams, and localization was introduced by Vogel and Le Dimet. Their work on a theoretical classification of concordance classes of links underscored the necessity of considering noncommutative localization. But to a great extent the difficulty of the algebra of noncommutative rings and localizations of modules has, until recently, obstructed the transfer of these algebraic techniques to low-dimensional situations.

When applied to 4-manifolds and classical knot concordance this new toolbox was found to be inadequate. The inability to obtain the embedded 2-disks whose existence were predicted by homotopy theory presented seemingly insurmountable barriers. Moreover, in low-dimensions the fundamental groups of the relevant spaces were usually large and nonabelian and demanded closer attention. A. Casson and C. Gordon first proved the inadequacy of the higher-dimensional tools in the context of classical knot concordance. They introduced new tools, including the Atiyah-Singer G-signature theorem, and showed that it was necessary and profitable to consider nonabelian covering spaces. The appearance of the G-signature theorem hinted at possible further use of analytic invariants. This was underscored in more recent work of M. Farber and J. Levine on homology cobordism of manifolds using eta invariants associated to finite-dimensional unitary representations of the fundamental group.

In topological 4-manifolds, the higher-dimensional techniques were augmented by remarkable techniques of M. Freedman, A. Casson and F. Quinn, involving infinite (but convergent) topological constructions called Casson towers. They also approximated embedded 2-disks with towers of iterated embedded surfaces called gropes. These topological constructions have, until recently, remained largely outside the scope of algebraic or analytical understanding, (although gropes were seen from the outset to be a topological reflection of the algebra of commutator series of the fundamental group). Although both the Casson-Gordon invariants and the Casson-Freedman towers were intrinsic to topological four-manifolds the connections between these two phenomena remained obscure. On the other hand, the field of combinatorial/quantum knot theory began with work of V. Jones on von Neumann algebras. Moreover noncommutative algebra underlay the work of Drinfeld that in turn suggested the primary invariant of quantum knot theory, the Kontsevitch integral. Together with ideas of E. Witten from physics and analysis, these suggested the primary invariants of quantum 3-manifolds, those of N. Reshitikin and V. Turaev. The field seemed to turn increasingly combinatorial and 2-dimensional, with major advances by many researchers in understanding the algebra of knots and 3-manifolds via planar projections of links, and through the algebra of trivalent (Feynman) diagrams. Connections with 4-manifolds and with higher-dimensional techniques seemed particularly elusive. Moreover the field struggled to find topological interpretations of these powerful new invariants. It remains open whether or not these represent complete invariants for knots.

Recently the merging of these currents accelerated in exciting ways.

In the recent work of T. Cochran, K. Orr and P. Teichner on classical knot concordance, the high-dimensional, 4-dimensional and von Neumann currents come together. They defined new invariants using noncommutative localization of modules and $L^2$ techniques - specifically the von Neumann rho invariants of Atiyah-Cheeger-Gromov. Moreover the relationship between the topology of gropes and the algebra of the derived series of the fundamental group is underscored, and a connection is established between the algebraic/homotopy-theoretic techniques (including those of Casson-Gordon) and the grope and tower constructions. Recent work of L. Rozansky, S. Garoufalidis and A. Kricker has combined quantum knot theory with several of these other currents. They have shown that the Konsevitch integral for knots and for boundary links (which is defined over the rationals) satisfies a certain hidden ?integrality? property, only described through the use of the language of surgery and of noncommutative localization of group rings. This has enabled them to ?lift? the Kontsevitch integral to a potentially more powerful invariant. This invariant has already been shown by Rozansky, Garoufalidis and Teichner to give strong new results about Alexander polynomial one knots (for example) and to suggest an organization of the Kontsevitch invariant (and hence of all Vassiliev finite type invariants) that better reflects the topology of knots. Finally, using ideas of K. Habiro, J. Conant and P. Teichner show that filtering knots by ?the size of the gropes they bound in 3-space? provides a possible connection between the 3- and 4-dimensional worlds. Specifically they show one version of this filtration is respected by the Kontsevich integral and captures much of its topological content; whereas another version yields a filtration consistent with recent tower filtrations of the classical knot concordance group. Several other provocative connections remain unexplored. First, many of the above constructions can be indexed by families of trivalent (Feynman) diagrams. Work of R. Schneiderman shows that these diagrams also parametrize certain higher-order intersection data among 2-spheres in 4-manifolds, refining the classical intersection theory of higher-dimensional manifolds to a more subtle theory, more effective in dimension four. Second, earlier researchers such as Culler-Shalen, Casson, Levine and Farber were able to address noncommutativity through studying finite dimensional complex representations of the fundamental group. Some techniques of algebraic geometry were then used profitably. Is there a possible interaction between these techniques and the regular representations into unitary operators on (infinite-dimensional) Hilbert space used to obstruct slicing knots? Lastly, the families of higher-order Alexander polynomials for 3-manifolds M, defined by S. Harvey using noncommutative algebra, surprisingly obstruct the existence of symplectic structures on M x S^1 even when Seiberg-Witten invariants fail. This hints at a hierarchy of higher-order Seiberg-Witten invariants. Can the methods of Osvath and Zabo be refined (perhaps by looking at equivariant intersection theory) to a nonabelian world that will reflect the above algebraic invariants?

The common source is knot theory. In its broadest context, this workshop seeks insights across fields. Homotopy theory and localization, surgery theory, functional analysis, topological four manifolds, classical knot theory in high and low dimensions, combinatorial and geometric three manifold theory, representation theory, physics and quantum topology are all finding a common area for shared discourse rooted in knot theory and illustrated in the discussion above. Such connections demand deeper exploration only available through the collaborative enterprise and through the dissolution of the artificial barriers between fields. The workshop will reinforce this interdisciplinary emphasis and will encourage the interaction of researchers across these fields. We anticipate only five talks a day, with the first day of the workshop consisting of expository talks on each of the four topics mentioned above - higher dimensional manifolds and knots, topological four manifolds and tower constructions, quantum invariants of knots and three manifolds, and analytic invariants.

This workshop proposal is timed to benefit and complement the PIMS thematic program in knot theory and 3-manifolds at UBC in the summer of 2004, sponsored by PIMS and organized by Dale Rolfsen, who is a coorganizer of this proposed workshop. This benefit may prove especially effective if the workshop occurs either the week before or after the Vancouver program.

Confirmed Participants

Schedule (PDF file)

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