Dev P. Sinha

Operads and knot spaces

Let Em denote the space of embeddings of the interval I = [?1,1] in the cube Im with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot; see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I,Im) x i?lmm(l, Im). In [28], McClure and Smith define a cosimplicial object O* associated to an operad with multiplication 0, whose homotopy invariant totalization we denote Tot(?#); see Definition 2.17 and Definition 2.5 below. Let /Cm denote the rath Kontsevich operad, introduced in [22], whose entries are compactified configuration spaces and which is weakly equivalent to the little ra-disks operad [37]; see Definition 4.1 and Theorem 4.5.

The topology of spaces of knots: cosimplicial models

We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial. These models build on the calculus of isotopy functors and are weakly homotopy equivalent to knot spaces when the ambient dimension is greater than three. The mapping space model, and the evaluation map on which it builds, is suitable for analysis through differential topology. The cosimplicial model gives rise to spectral sequences which converge to ohomology and homotopy groups of spaces of knots when they are connected. We explicitly identify and establish vanishing lines in these spectral sequences.

A one-dimensional embedding complex

We give the 1rst explicit computations of rational homotopy groups of spaces of “long knots” in Euclidean spaces. We de1ne a spectral sequence which converges to these rational homotopy groups whose E 1 term is de1ned in terms of familiar Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E 1 term is zero, and make calculations of E 2 in a 1nite range. c 2002 Elsevier Science B.V. All

Lie coalgebras and rational homotopy theory II: Hopf invariants

We give a new, definitive resolution of the homotopy periods problem modulo torsion. This problem was addressed most famously by Sullivan, on the first page and then Section 11 of his seminal paper [24]. There Sullivan starts to give explicit integrals over a sphere which determine whether two maps from that sphere are equivalent in the quotient of the homotopy groups of a simply connected space modulo their torsion subgroups. These integrals are a direct generalization of Whitehead’s formula for Hopf invariant [25], and they encode “linking numbers.” More integrals along these lines were given by Haefliger [10] and some framework was developed by Novikov [17]. Before Sullivan’s work, Boardman and Steer [3] gave a set of linking number invariants defined for maps to a suspension. Finally, Hain in his thesis [11] uses Chen integrals as his model for the bar construction to give one resolution of the homotopy period question, but produces integrals over products of spheres with simplices rather than over spheres themselves and an auxiliary formalism is required to address redundancy. We give a complete set of “integrals” (we can use integer-valued cochains if we wish) over the sphere which distinguish maps modulo torsion, proceeding from a simple use of the bar construction. We also connect these integrals to Quillen functors in the framework of differential graded Lie coalgebras, as developed in [23]. On the formal side, we are then able to evaluate Hopf invariants on Whitehead products and are able to understand the naturality of these maps in the long exact sequence of a fibration. For applications, we can show for example that the homotopy groups of homogeneous spaces are controlled by classical linking numbers. We proceed in two steps, first using the classical bar complex to define integer-valued homotopy functionals which coincide with evaluation of the cohomology of ΩX on the looping of a map from S to X . One basic, apparently new, observation is that calculations in the classical bar complex yield the Hopf invariant formula of Whitehead (as well as those of Haefliger and Sullivan). We establish basic properties and give examples. In the second part we use the Harrison complex on commutative cochains, the standard cohomology theory for commutative algebras, and thus must switch to rational coefficients. Using our graph coalgebraic presentation, we show that a product-coproduct formula established geometrically in the bar complex descends to the duality predicted by Koszul-Moore theory. In summary, we find that rational homotopy groups are controlled by “linking phenomena,” and that this linking phenomena are best described in the language of Lie coalgebras. One direction we plan to pursue further is the use of Hopf invariants to realize Koszul-Moore duality isomorphisms in general. A second direction we plan to pursue is that of spaces which are not simply connected, where the graph coalgebra models seem relevant even for K(π, 1) spaces.

Embedding calculus knot invariants are of finite type

We show that the map on components from the space of classical long knots to the n-th stage of its Goodwillie-Weiss embedding calculus tower is a map of monoids whose target is an abelian group and which is invariant under clasper surgery. We deduce that this map on components is a finite type-(n-1) knot invariant. We also compute the second page in total degree zero for the spectral sequence converging to the components of this tower as Z-modules of primitive chord diagrams, providing evidence for the conjecture that the tower is a universal finite-type invariant over the integers. Key to these results is the development of a group structure on the tower compatible with connect-sum of knots, which in contrast with the corresponding results for the (weaker) homology tower requires novel techniques involving operad actions, evaluation maps, and cosimplicial and subcubical diagrams.

Real equivariant bordism and stable transversality obstructions for Z/2

In this paper we compute homotopical equivariant bordism for the group Z/2, namely MO Z/2 * , geometric equivariant bordism n Z/2 * , and their quotient as modules over geometric bordism. This quotient is a module of stable transversality obstructions. We construct these rings from knowledge of their localizations.

Algebraic Topology: Applications and New Directions

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the groups of de Rham-Witt forms of the ring k. At present, the validity of the formula depends on a conjecture that concerns the combinatorial structure of a new family of polytopes that we call stunted regular cyclic polytopes. The polytopes in question appear as the intersections of regular cyclic polytopes with (certain) linear subspaces. We verify low-dimensional cases of the conjecture. This leads to unconditional new results on K_2 and K_3 which extend earlier results by Krusemeyer for K_0 and K_1.