The Goodwillie-Weiss Tower and Knot Theory - Past and Present
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1 The Goodwillie-Weiss Tower and Knot Theory - Past and Present Dev P. Sinha University of Oregon June 24, 2014 Goodwillie Conference, Dubrovnik, Croatia New work joint with Budney, Conant and Koytcheff
2 Late(?) 1980 s Vassiliev studies the algebraic topology of spaces of knots (and its complement ), and from that insight defines finite-type knot invariants. They are soon after found to have significant connections with the Jones polynomial, Lie theory and physics through Chern-Simons theory. It is still not known how fine they are as an invariant. There are extremely interesting open conjectures about their relationship to hyperbolic volume of a knot complement. The Lie theoretic invariants categorify in interesting ways.
3 Vassiliev, Birman-Lin, and Bar Natan show that a type-n knot invariant gives rise to a functional on A n, a combinatorially defined module, namely as trivalent graphs including cyclic orderings of edges incident to vertices and a spine. modulo an anti-symmetry relation and STU: = -
4 Kontsevich produces two sets of integrals, one which bears his name and one which are called Bott-Taubes integrals (first case studied by Bar Natan in his thesis with Witten). These integrals take any complex(/real/ rational(!)) functional on A n and produce a type-n knot invariant. Integrals like these establish formality of Euclidean configuration spaces, and are also used in Kontsevich s seminal work on deformation quantization.
5 1996(?) Tom gives a talk at the Stanford Topology Conference, conjecturing that his and Michael Weiss s calculus of isotopy functors, when applied to classical knots, is related to the theory of finite-type invariants. He bases this conjecture on the Bousfield-Kan cohomology spectral sequence for a cosimplicial model he sees for the tower for knots, seeing an alternate (and more common) presentation for functionals on A n, namely as chord diagrams modulo a one-term and a four-term relation.
6 2002(+) I develop an alternate cosimplicial model, along with a mapping space model, for the Goodwillie-Weiss tower for knots. The cosimplicial model gives rise to Goodwillie s spectral sequence (which Turchin showed is isomorphic to Vassiliev s spectral sequence before knowing about Goodwillie s spectral sequence).
7 2002(+) The mapping space model is a multi-relative mapping space, conveniently given as a homotopy limit: holim Conf 0 Conf 1 Conf 2 Conf 1 Conf 0 Conf 1 Conf 0, where Conf i = Conf i M,.
8 2002(+) The map from the knot space to the mapping space model ev n : K AM n sends an embedding f to the evaluation map - that is, the induced map on configuration spaces or Gauss map- Conf n f : Conf n I Conf n M. (This is not your typical way to map to a homotopy limit!)
9 2003 Budney, Conant, Scannell and I show that the map from the knot space to AM 3, on components, gives rise to the unique type-two knot invariant. The third component has π 0 = Z. It has as associated graded π 3 (fiber Conf 3 (I 3 ) Conf 2 (I 3 )) = π 3 (fiber S 2 S 2 S 2 S 2 ). The Hopf invariant which detects this Whitehead product gives rise to a knot invariant defined by the tower. Evaluating this Hopf invariant explicitly leads to a new formula for the invariant by counting (with coefficients of ±1 and 0) quadrisecants on a knot.
10 z y 2003 The flow of information goes: knot/ isotopy evaluation map/ homotopy (relative) linking of preimages of sub manifolds Conf 3 f(t ) 2 f(t ) 1 f(t ) 3 x The mapping space model makes this analysis possible.
11 2004 Volic, in his thesis, considers the homology tower, associated to the functor Emb HR, where all models are obtained by smashing object-wise with HR. He shows that the 2nth stage recovers all finite-type n invariants over R, resolving Goodwillie s question over R! He does this by showing that the Bott-Taubes integrals, which come from Chern-Simons theory and are used to produce all type-n invariants over R, extend to produce invariants on the tower. So this says more new about the tower than it does about knots.
12 2006 Conant shows that the zero line of the homotopy spectral sequence for Emb Q has as in degree ( n, n) the module A I n 1 defined as before but now demanding the diagrams be non-separated. Primitive type-(n 1) invariants are determined by functionals on these modules. The presentation given in the spectral sequence had not been considered before, and is non-trivial to compare to the standard presentation.
13 2008 Motivated by the linking/hopf invariant used by Budney, Conant, Scannell and S-, Ben Walter and I show how cocycles in Bar C (X ) give rise to explicit, geometric Hopf invariants, elements of Hom (π (X ), Z). When X is simply connected with finitely generated homotopy groups, these Hopf invariants span a full-rank submodule. (The cokernel of the map from the Bar construction to Hom (π (X ), Z) is the Hopf invariant one module of X.)
14 2008 For example, if ω 1 and ω 2 are two-cocycles on S 2 which indicate distinct points, so that ω 1 ω 2 is a bar cocycle then the corresponding Hopf invariant is calculated as a linking number. In general, these Hopf invariants are defined through linking with correction.
15 2005; 2010 A basic issue with the standard tower is that there is no group structure. We like group structures, and the theory of finite type invariants demands one. S-, using McClure-Smith results: π 0 of the limit of the tower is a monoid. Turchin, extending McClure-Smith theory: Every space in the tower is a two-fold loop space, and the maps between them are two-fold loop maps. Turchin and Dwyer-Hess (current work) also investigate de-loopings.
16 Now, with Budney, Conant, Koytcheff We show that the map on components from the knot space to AM n, the nth stage in the Goodwillie-Weiss tower, is a type-(n 1) invariant. We also show that the homotopy spectral sequence is a spectral sequence of abelian groups (including total degrees 0 and 1), and that collapse of this spectral sequence in total degree zero plus a (relatively easy?) conjecture of Goodwillie would show that all weight systems give rise to finite-type invariants. (This would be a major result, especially considering that Chern-Simons theory gives the theory over the reals. By the way, what about embedded string topology?)
17 Showing the tower gives abelian group valued invariant First significant step: showing this is abelian group valued invariant! We reproduce Turchin s result that the tower itself is a tower of 2-cubes spaces, but in a framed setting. (Recall/ retroactively see Robin s talk to see how.) More critically, we show that the evaluation map ev n : K AM n preserves 1-cubes actions.
18 Showing the tower gives abelian group valued invariant We show that π 0 is a group inductively (if 0 H G K 0 exact sequence of monoids; H and K groups implies G is). For this we need that the fiber sequences for the layers in the tower Ω n X AM n AM n 1 are maps of monoids, with the latter surjective. In fact it seems to preserve overlapping 1-cubes (that is, a 2-cubes) actions, which would establish that Shubert s commutativity of connect sum of knots from 1947 is compatible with Steenrod s formula for 1 from 1948.
19 Establishing finite type Conant had around 2008 produced an argument that if π 0 (ev n ) is abelian-group valued and preserves connect sum, then it is type n 1. His argument uses Habiro moves, an alternate (non-trivial) approach to finite-type theory. The map from the knot space to this tower factors the map to the HR tower Volic studied. Group structure was part of the definition there. One gets finite-type through a more elementary approach, and sees type n precisely at stage 2n (no sooner).
20 The spectral sequence The spectral sequence which converges to π (AM n ) coming from the cosimplicial model has E 1 given by homotopy groups of configuration spaces. The reduced E 1 can be calculated as the homotopy of the total fiber of a cube made from the forgetting (codegeneracy) maps (a nice application of cubical diagrams to simplicial topology). Following Goodwillie-Weiss, we use the standard fiber sequences n 1 S 2 Conf n (R 3 ) Conf n 1 (R 3 ) to for example start with the four-cube with Conf 4 (R 3 ) as initial and obtain its total fiber as that of the following.
21 The spectral sequence S 2 S 2 S 2 S 2 S 2 S 2 S 2 S 2 S 2 S 2 S 2 S 2 pt.
22 The spectral sequence Looping down once we see terms of Ω k S 2. Hilton showed that this is equivalent to the product x B ΩS x +1, where B is a choice of basis for the free Lie algebra on k generators and x is its degree. (One takes the Whitehead product maps S x +1 k S 2, loops them down, and then uses H-space structure to get a map from the product. Then compute map on homology to see result, by Whitehead.) We may choose these bases for different wedge facts consistently so that on the looped-down version we are projecting in the products to factors in which not all generators appear.
23 The spectral sequence In the end, the reduced E 1 p,q is B π qs x +1 - unstable homotopy groups of spheres! The zero-line is torsion-free and the one-line has both free summands and two-torsion (lots of η). Conant s calculation only required integral coefficients, so we can see A I n 1 as the E 2 n,n.
24 The spectral sequence Remarkably, collapse of the spectral sequence (potentially a purely algebro-topological question) + ev n being surjective on π 0 compatibly with the E 2 isomorphism would show that all weight systems integrate to finite type invariants (over the integers)! The π 0 surjectivity (without compatibility with E 2 isomorphism) was Brian and Ismar s warm-up thesis problem, as this is what is predicted by the Goodwillie-Klein estimates in higher dimensions). It seems approachable through elementary methods. If this integration were to happen through homotopy theory, geometric topologists would still want to understand how the homotopy classes of a knots evaluation map could be measured geometrically.
25 Putting it together through the Hopf invariant approach While the zero-line of the E 2 term is known, it is an open question for example whether the A I n have any torsion, in particular any two-torsion. The one-line is not known, beyond the observation that the copies of η everywhere survive to E 2. If the spectral sequence does collapse at E 2 (citing Dundas?), then we can show that the linear dual of π 0 (AM n ) would be given by (relative) Hopf invariants. Thus, we are led to study relative Hopf invariants directly to see how they give knot invariants. We can consider a tri-graded object with bar constructions on cochains of configuration spaces, along with the cosimplicial/ AM n differential.
26 Putting it together through the Hopf invariant approach We have just started this work, but it seems that if we use cochains in Conf n (R 3 ) then some of the Hopf invariants give rise to Polyak-Viro arrow diagram counts. These are known to give finite-type invariants, but it is not known how many. If we can connect: functional on A I n some bar cocycle/ relative Hopf invariant an arrow diagram count which realizes the functional on A I n, then Polyak-Viro invariants would give all finite-type invariants. Maybe formality in the appropriate range (formality over the integers is false by work of Salvatore) or emerging connections with factorization homology could come into play as well.
27 Thanks, Tom! You ve helped a lot of people keep themselves occupied.
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