Dennis Sullivan

Closed String Operators in Topology Leading to Lie Bialgebras and Higher String Algebra

Imagine a collection of closed oriented curves depending on parameters in a smooth d-manifold M. Along a certain locus of configurations strands of the curves may intersect at certain sites in M. At these sites in M the curves may be cut and reconnected in some way. One obtains operators on the set of parametrized collections of closed curves in M. By making the coincidences transversal and compactifying, the operators can be made to act in the algebraic topology of the free loop space of M when M is oriented. The process reveals collapsing sub graph combinatorics like that for removing infinities from Feynman graphs.

Axiomatic characterization of ordinary differential cohomology

The Cheeger-Simons differential characters, the Deligne cohomology in the smooth category, the Hopkins-Singer construction of ordinary differential cohomology, and the recent Harvey-Lawson constructions are each in two distinct ways abelian group extensions of known functors. In one description, these objects are extensions of integral cohomology by the quotient space of all differential forms by the subspace of closed forms with integral periods. In the other, they are extensions of closed differential forms with integral periods by the cohomology with coefficients in the circle. These two series of short-exact sequences mesh with two interlocking long-exact sequences (the Bockstein sequence and the de Rham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Our first theorem shows that on the category of smooth manifolds and smooth maps, any package consisting of a functor into graded abelian groups together with four natural transformations that fit together so as to form a Character Diagram as mentioned earlier is unique up to a unique natural equivalence. Our second theorem shows that natural product structure on differential characters is uniquely characterized by its compatibility with the product structures on the known functors in the Character Diagram. The proof of our first theorem couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

The homotopy invariance of the string topology loop product and string bracket

Let M n be a closed, oriented, n-manifold, and LM its free loop space. In [4] a commutative algebra structure in homology, H�(LM), and a Lie algebra structure in equivariant homology H S 1 � (LM), were defined. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M1 ! M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equivalence, Lf : LM1 ! LM2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds

Transverse string topology and the cord algebra

We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.

String Topology in Dimensions Two and Three

Let V denote the vector space with basis the conjugacy classes in the fundamental 4 group of an oriented surface S. In 1986 Goldman [1] constructed a Lie bracket [,] on V. If a and b are conjugacy classes, the bracket [a; b] is defined as the signed sum over intersection points of the conjugacy classes represented by the loop products taken at the intersection points. In 1998 the authors constructed a bracket on higher dimensional manifolds which is part of String Topology [2]. This happened by accident while working on a problem posed by Turaev [3], which was not solved at the time. The problem consisted in characterizing algebraically which conjugacy classes on the surface S are represented by simple closed curves. Turaev was motivated by a theorem of Jaco and Stallings [4,5] that gave a group theoretical statement equivalent to the three dimensional Poincare conjecture. This statement involved simple conjugacy classes. Recently a number of results have been achieved which illuminate the area around Turaev’s problem. Now that the conjecture of Poincar`e has been solved, the statement about groups of Jaco and Stallings is true and one may hope to find a Group Theory proof. Perhaps the results to be described here could play a role in such a proof. See Sect. 3 for some first steps in this direction.

Memories of Raoul Bott

In fall ’69 I arrived in the Boston area driving across the country from Berkeley to take up a fellowship at MIT. A big attraction was to learn more about differential geometry mathematics from Raoul Bott. Right away Raoul tried to help me find an apartment in Cambridge.