HLCSSLA · Homology of labelled configuration spaces via spectral Lie algebra methods

Homology of labelled configuration spaces via spectral Lie algebra methods

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.

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