ANR-FNS - Appel à projets générique 2018 - FNS

Operads, Calculus and Homotopy theory methods in Topology – OCHoTop

Operads, Calculus, and Homotopy theory methods in Topology

This proposal addresses problems of topology, in the field of fundamental mathematics. The general purpose of our project is to develop applications of operads (an algebraic device) for the definition of invariants associated to links, manifolds and stratified spaces (classical objects of topology).

Topological invariants through operads

In recent years, considerable effort has been devoted to the elaboration and computation of topological invariants in the setting of rational homotopy theory, which provides information about the homotopy of spaces modulo torsion. Indeed, research in operad theory has led, as a notable outcome,to a description of the rational homotopy of embedding spaces of manifolds of arbitrary dimensions in terms of the homology of graph complexes. This result leads to a generalization of the definition of the classical Vassiliev invariants of links, which correspond to graph homology classes associated to embeddings of the 1-dimensional circle into the 3-dimensional sphere. These applications rely on topological methods known as the Goodwillie-Weiss calculus, and one has to assume that the codimension of the embeddings is large enough in order to ensure the convergence of the method.<br />The main idea of our project is to explore combinatorial and topological operations underlying the definition of our operads, as well as the existence of algebro-geometric structures underlying our objects, in order to obtain new invariants for links, manifolds, and stratified spaces, which should enable us to go beyond the limitations of the models of rational homotopy theory. In the 3-dimensional case, various invariants have been defined by using methods of quantum topology. The idea is that the theory of operads could again be used in combination with other methods of topology in order to produce higher dimensional versions of these invariants. The ultimate goal is to produce computable invariants, which may reflect fine torsion information, and which should enable us to handle relative topological structures of low codimension.

The main examples of operads that we consider are the operads of little disks which model homotopy commutative algebra structures. Our project relies on the results of recent researches which use the little disks operads to build homotopical models of embedding spaces, as well as new homology theories for manifolds, such as the factorization homology. We intend to study applications of graph complexes for the computation of the homotopy of these operadic models of embedding spaces and for the construction of classes in these new operadic homology theories. We also intend to study the homotopy of objects, called modular operads, which are modelled on the structures of graph complexes, and to explore the applications of these objects for the study of wiring diagrams in neurosciences.

- Definition of analogues of the Eilenberg-MacLane spaces for intersection cohomology et solution of a problem stated by Goresky-MacPherson in 1984 : the representability of intersection cohomology.
- Description in terms of graph complexes of the rational homotopy of mapping spaces on the little disk operads and of the rational homotopy of the spaces of embeddings with compact support of a manifold in a Euclidean space.
- Invariance result for the coHochschild homology of differential graded coalgebra (this homology agrees with a homology theory defined in terms.of categories).
- Comparison results on the constructions by surgery and by sum of states of homotopical quantum fields theories (HQFT) of dimension 3 with an aspherical target : they are connected by the graded center of monoidal categories graded by group.

This main and most immediate impact, which one may expect from our project, will be in the field of fundamental mathematics.
The renewal of the theory of operads, initiated in the 90’s, has already had deep impact on mathematics. Let us mention the work of Kontsevich and Tamarkin, on the deformation-quantization of Poisson manifolds. This renewal was in part motivated by Kontsevich’s ideas on the applications of graph complexes for the study of the cohomology of moduli spaces and of outer spaces. The Koszul duality of operads, discovered by Ginzburg and Kapranov, moreover provides e?cient
methods to carry out effective computations by using operads.
We expect that our proposal will enable us to extend the scope of these applications of operads in order to obtain effective combinatorial descriptions of invariants, generalizing the kind of (cohomological) invariants studied by Kontsevich, which occur in topology and in algebraic geometry.
To be specific, let us mention the relationship between operadic mapping spaces and embedding spaces, which provides a generalization of the Vassiliev approach to the study of knot spaces in low dimensional topology, as well as the applications of operads for the definition of factorization homology of manifolds, and the applications of rational homotopy theory methods for the study of stratified spaces. The idea is that one could use these connections to develop new applications, in these research subjects, of the effective methods of the theory of operads and of homotopy theory that have been introduced since this renewal of the subject in the 90’s.

- David Chataur (with Daniel Tanré), Natural operations in Intersection Cohomology. Preprint arXiv:2005.04960 (2020).
- Sylvain Douteau, Homotopy theory of stratified spaces. Algebr. Geom. Topol. 21 (2021), no. 1, pp. 507–541.
- Benoit Fresse (with Victor Turchin and Thomas Willwacher), On the rational homotopy type of embedding spaces of manifolds in Rn. Preprint arXiv:2008.08146 (2020).
- Benoit Fresse, Lorenzo Guerra, On a notion of homotopy Segal E-infinity-Hopf cooperad. Preprint arXiv:2011.11333 (2020).
- Kathryn Hess (with Brooke Shipley), Invariance properties of co{H}ochschild homology. J. Pure Applied Algebra 225 (2021), no. 2.
- Jens Kjaer, Unstable v1-Periodic Homotopy of Simply Connected, Finite H-Spaces, using Goodwillie Calculus. Preprint arXiv:1905.05269 (2019).
- Alexis Virelizier (with Vladimir Turaev), On 3-dimensional Homotopy Quantum Field Theory III: Comparison of two approaches. Int. J. Math. 31 (2020), pp. 1-57.

This proposal addresses problems of topology, in the field of fundamental mathematics. The general purpose of our project is to develop applications of operads (an algebraic device) for the definition of invariants associated to links, manifolds and stratified spaces (classical objects of topology).

In recent years, considerable effort has been devoted to the elaboration and computation of such invariants in the setting of rational homotopy theory, which provides information about the homotopy of spaces modulo torsion. Indeed, research in operad theory has led, as a notable outcome, to a description of the rational homotopy of embedding spaces of manifolds of arbitrary dimensions in terms of the homology of graph complexes. This result leads to a generalization of the definition of the classical Vassiliev invariants of links, which correspond to graph homology classes associated to embeddings of the 1-dimensional circle into the 3-dimensional sphere. These applications rely on topological methods known as the Goodwillie-Weiss calculus,and one has to assume that the codimension of the embeddings is large enough in order to ensure the convergence of the method.

The main idea of our project is to explore combinatorial and topological operations underlying the definition of our operads, as well as the existence of algebro-geometric structures underlying our objects, in order to obtain new invariants for links, manifolds, and stratified spaces, which should enable us to go beyond the limitations of the models of rational homotopy theory. In the 3-dimensional case, various invariants have been
defined by using methods of quantum topology. The idea is that the theory of operads could again be used in combination with other methods of topology in order to produce higher dimensional versions of these invariants.

The ultimate goal is to produce computable invariants, which may reflect fine torsion information, and which should enable us to handle relative topological structures of low codimension. In addition, we intend to explore the potential applications of generalized operad structures to the modelling of wiring diagrams in neuroscience and computer science.

Roughly, an operad is an object, consisting of collections of operations, which governs a category of algebras. The classical categories of associative algebras, of commutative algebras, and of Lie algebras, for instance, can be described in terms of operads. The main examples of operads considered in this project are the operads of little discs (the E_n-operads), which are used to model hierarchies of homotopy commutative structures, and some variants of the little discs operads that naturally occur in problems of algebra and topology.

Project coordination

Benoit Fresse (Laboratoire Paul Painlevé)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.

Partner

LPP Laboratoire Paul Painlevé
Laboratoire pour la topologie et les neurosciences

Help of the ANR 247,320 euros
Beginning and duration of the scientific project: December 2018 - 48 Months

Useful links

Explorez notre base de projets financés

 

 

ANR makes available its datasets on funded projects, click here to find more.

Sign up for the latest news:
Subscribe to our newsletter