Homology of labelled configuration spaces via spectral Lie algebra methods

Project Information

HLCSSLA

Grant agreement ID: 101150469

DOI

10.3030/101150469

Project closed

EC signature date 12 March 2024

Start date 1 April 2024

End date 31 March 2026

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 214 934,40

Coordinated by

KOBENHAVNS UNIVERSITET
Denmark

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

Reporting period: 2024-04-01 to 2026-03-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.