Homology of labelled configuration spaces via spectral Lie algebra methods

Información del proyecto

HLCSSLA

Identificador del acuerdo de subvención: 101150469

DOI

10.3030/101150469

Proyecto cerrado

Fecha de la firma de la CE 12 Marzo 2024

Fecha de inicio 1 Abril 2024

Fecha de finalización 31 Marzo 2026

Financiado con arreglo a

• Marie Skłodowska-Curie Actions (MSCA)

Coste total

Sin datos

Aportación de la UE

€ 214 934,40

Coordinado por

KOBENHAVNS UNIVERSITET
Denmark

Periodic Reporting for period 1 - HLCSSLA (Homology of labelled configuration spaces via spectral Lie algebra methods)

Período documentado: 2024-04-01 hasta 2026-03-31

Resumen del contexto y de los objetivos generales del proyecto

The project aims to develop a local-to-global strategy in two different areas that lie at the intersection of algebraic and geometric topology. More precisely, we investigate to what extent the local information of a certain structure or invariant determine the global information via gluing disks to form a manifold.

The first context is labelled configuration spaces. For M a manifold and k a natural number, the ordered configuration space is the collection of k distinct, ordered points in M equipped with a certain topology. Taking the quotient by the free action of the symmetric group that permutes the k points yields the unordered configuration space of k points in M. As a generalization, we can consider the labelled configuration space where we label each point by an element in another pointed space. The study of labelled configuration spaces dates back to as early as the 70s', pioneered by G. Segal and D. McDuff. It is of theoretical importance in mathematical subfields like homotopy theory, higher category theory, and iterated loop space theory, and has applications in embedding calculus of manifolds, modui space of Riemann surfaces, knot theory, number theory, mathematical physics, and even in motion planning. To measure the complexity of these spaces, we can compute algebraic invariants like the homology groups with coefficients in a finite field. This questions has been a major challenge in understanding unordered and labelled configuration spaces since the onset, and very few computations have been done. The goal of this project is to attack this classical problem with newly developed tools in algebraic topology and homotopy theory.

The second context is topological field theory. An nD-TFT is a symmetric monoidal functor out of the n-dimensional bordism category, which is roughly a coherent assignment of n-cobordisms between (n−1)-dimensional manifolds to maps between objects in a category. A major long-standing problem in the study of TFTs: the classification of nD-TFTs valued in a general∞-category V in terms of algebraic structures on objects in V. Since the open TFTs, i.e. restricting to disks and graph-like cobordisms between them, are known to be classified algebraically, we propose a to study whether they canonically extend to more complicated manifolds and cobordisms.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

For the first part of the project concerning labelled configuration spaces, substantial revisions were implemented for the two papers by the PI that lay the techincal foundation for the project. In particular, the algebraic filtrations leading to the iterative spectral sequences computing the mod p homology of labeled configuration spaces were clarified and made amenable to interact with the theory of synthetic spectra. Both papers were published in top subject-wise journals. Building upon these revisions, we were able to lay out a precise strategy to inductively compute the universal differentials in the bar spectral sequence against the spectral Lie operad. We are still in the process of implementing the strategy, since the technicality of the computation way exceeds the original expectation and requires developing further theories that address Koszul duality of monads that are obtained via quotienting a composite monad by a relation.

For the second part on TFTs, in dimension n=2, the PI, together with collaborators Jan Steinebrunner and Shaul Barkan, show that any open 2-dimensional topological field theory valued in a symmetric monoidal ∞-category (with suitable colimits) extends canonically to an open-closed field theory whose value at the circle is the Hochschild homology object of its value at the disk. As a corollary, we obtain an action of the moduli spaces of surfaces on the Hochschild homology object of E1-Frobenius algebras. In particular, we exhibits the open-closed 2-bordism category as the initial symmetric monoidal extension of the open 2-bordism category that admits Hochschild homology. In dimension infinity, an analogous result was established by the PI and Andrea Bianchi, where the open category is modeled by the ∞-category of graphs and the open-closed category by the ∞-category of graph cobordisms. In particular, we show that GrCob parametrizes natural operations among factorization homoliges of E_∞-Frobenius algebras.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

The main results are recorded in the previous section as main achievements. Potential impact of the results in the second part is a more refined understanding of string operations, i.e. natural operations on the cohomology of free loop spaces of manifolds.