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Dennis Sullivan

Publications and source records attributed to Dennis Sullivan.

30 records · Page 2Linked to original sources
arXiv2008

Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs

We interpret mathematically the pair (master equation, solution of master equation) up to equivalence, as the pair (a presentation of a free triangular dga T over a combination operad O, dga map of T into C, a dga over O) up to homotopy equivalence of dgOa maps, see Definition 1. We sketch two general applications: I to the theory of the definition and homotopy theory of infinity versions of general algebraic structures including noncompact frobenius algebras and Lie bialgebras. Here the target C would be the total Hom complex between various tensor products of another chain complex B, C = HomB, O describes combinations of operations like composition and tensor product sufficient to describe the algebraic structure and one says that B has the algebraic structure in question. II to geometric systems of moduli spaces up to deformation like the moduli of J holomorphic curves. Here C is some geometric chain complex containing the fundamental classes of the moduli spaces of the geometric problem. We also discuss analogues of homotopy groups and Postnikov systems for maps and impediments to using them related to linear terms in the master equation called anomalies.

math.QA↗
arXiv2007

String Topology: Background and Present State

The data of a "2D field theory with a closed string compactification" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following assumptions. It is assumed the punctures are labeled and divided into nonempty sets of inputs and outputs. The inputs are marked by a tangent direction and the outputs are weighted by nonnegative real numbers adding to unity. It is assumed the gluing of inputs to outputs lands on the pseudomanifold boundary of the cell decomposition and the entire pseudomanifold boundary is decomposed into pieces by all such factorings. It is further assumed that the action is equivariant with respect to the toroidal action of rotating the markings. A main result of compactified string topology is the Theorem (closed strings): Each oriented smooth manifold has a 2D field theory with a closed string compactification on the equivariant chains of its free loop space mod constant loops. The sum over all surface types of the top pseudomanifold chain yields a chain X satisfying the master equation dX + X*X = 0 where * is the sum over all gluings. This structure is well defined up to homotopy. The genus zero parts yields an infinity Lie bialgebra on the equivariant chains of the free loop space mod constant loops. The higher genus terms provide further elements of algebraic structure called a "quantum Lie bialgebra" partially resolving the involutive identity. There is also a compactified discussion and a Theorem 2 for open strings as the first step to a more complete theory. We note a second step for knots.

math.GT↗
arXiv2007

Axiomatic Characterization of Ordinary Differential Cohomology

Cheeger-Simons differential characters, 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 desciption 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 a DeRham sequence) to form a commutative DNA-like array of functors called the Character Diagram. Theorem 1.1 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 above is unique up to unique natural equivalence. Theorem 1.2 shows the 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 Theorem 1.1 couples the naturality with results about approximating smooth singular cycles and homologies by embedded pseudomanifolds.

math.AT↗
arXiv2006

Infinity Structure of Poincare Duality Spaces

We show that the complex $C_\bullet X$ of rational simplicial chains on a compact and triangulated Poincaré duality space $X$ of dimension $d$ is an A$_\infty$ coalgebra with $\infty$ duality. This is the structure required for an A$_\infty$ version of the cyclic Deligne conjecture. One corollary is that the shifted Hochschild cohomology $HH^{\bullet+d} (C^\bullet X, C_\bullet X)$ of the cochain algebra $C^\bullet X$ with values in $C_\bullet X$ has a BV structure. This implies, if $X$ is moreover simply connected, that the shifted homology $H_{\bullet+d}LX$ of the free loop space admits a BV structure. An appendix by Dennis Sullivan gives a general local construction of $\infty$ structures.

math.AT↗
arXiv1999

String Topology

Consider two families of closed oriented curves in a d-manifold. At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an i-dimensional family and a j-dimensional family will produce an (i+j-d+2)-dimensional family. Our purpose is to describe mathematical structure behind such interactions.

math.GT↗
arXiv1996

Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichmüller Space

There exists on each Teichmüller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichmüller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q$ bundles to construct such an universal version. Our universal objects exist over the universal space, $T_\infty$, which is the direct limit of the $T_g$ as the genus varies over the tower of all unbranched coverings of any base surface. The bundles and the connecting isomorphisms are equivariant with respect to the natural action of the universal commensurability modular group.

alg-geom↗
arXiv1995

Topological conjugacy of circle diffeomorphisms

The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy. In 1985, one of us (Sullivan) gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than Sullivan's Zygmund condition and vice versa. This paper will give the third criterion which is implied by either of the above criteria.

math.DS↗
arXiv1994

Universal Hodge Bundle and Mumford Isomorphisms on the Abelian Inductive Limit of Teichmüller Spaces

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain ``solenoidal Riemann surface'', $H_{\infty,abel}$. The main result constructs these universal $DET$ and Hodge line bundles on this direct limit, equips them with their Quillen metrics, and proves that they are related by the appropriate Mumford isomorphisms. Our work can be viewed as a contribution towards a non-perturbative formulation of the bosonic string theory, since it provides an interpretation of the Polyakov measure in a genus-independent fashion.

alg-geom↗
arXiv1993

Teichmüller Theory and the Universal Period Mapping via Quantum Calculus and the $H^{1/2}$ Space on the Circle

The Universal Teichmüller Space, $T(1)$, is a universal parameter space for all Riemann surfaces. In earlier work of the first author it was shown that one can canonically associate infinite- dimensional period matrices to the coadjoint orbit manifold $Diff(S^1)/Mobius(S^1)$ -- which resides within $T(1)$ as the (Kirillov-Kostant) submanifold of ``smooth points'' of $T(1)$. We now extend that period mapping $Π$ to the entire Universal Teichmüller space utilizing the theory of the Sobolev space $H^{1/2}(S^1)$. $Π$ is an equivariant injective holomorphic immersion of $T(1)$ into Universal Siegel Space, and we describe the Schottky locus utilizing Connes' ``quantum calculus''. There are connections to string theory. Universal Teichmüller Space contains also the separable complex submanifold $T(H_\infty)$ -- the Teichmüller space of the universal hyperbolic lamination. Genus-independent constructions like the universal period mapping proceed naturally to live on this completion of the classical Teichmüller spaces. We show that $T(H_\infty)$ carries a natural convergent Weil-Petersson pairing.

alg-geom↗
arXiv1991

Expanding direction of the period doubling operator

We prove that the period doubling operator has an expanding direction at the fixed point. We use the induced operator, a ``Perron-Frobenius type operator'', to study the linearization of the period doubling operator at its fixed point. We then use a sequence of linear operators with finite ranks to study this induced operator. The proof is constructive. One can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point.

math.DS↗