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1. Topological coHochschild homology and the homology of free loop spaces

   https://doi.org/10.1007/s00209-021-02879-4  

   Bohmann, Anna Marie ; Gerhardt, Teena ; Shipley, Brooke ( May 2022 , Mathematische Zeitschrift)
2. Spatiotemporal Persistent Homology for Dynamic Metric Spaces

   https://doi.org/10.1007/s00454-019-00168-w  

   Kim, Woojin ; Mémoli, Facundo ( February 2020 , Discrete & Computational Geometry)

   Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). In this paper we extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter “spatiotemporal” filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov–Hausdorff distance which permits metrization of the collection of all DMSs. On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well-known interleaving distance. In either case, the metric d can be computed in polynomial time. 

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3. On the quotient of the homology cobordism group by Seifert spaces

   https://doi.org/10.1090/btran/110  

   Hendricks, Kristen ; Hom, Jennifer ; Stoffregen, Matthew ; Zemke, Ian ( September 2022 , Transactions of the American Mathematical Society, Series B)

   We prove that the quotient of the integer homology cobordism group by the subgroup generated by the Seifert fibered spaces is infinitely generated. 

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4. Equivariant localization and completion in cyclic homology and derived loop spaces

   https://doi.org/10.1016/j.aim.2020.107005  

   Chen, Harrison ( April 2020 , Advances in Mathematics)
5. Representation Homology of Topological Spaces

   https://doi.org/10.1093/imrn/rnaa023  

   Berest, Yuri ; Ramadoss, Ajay C ; Yeung, Wai-Kit ( February 2020 , International Mathematics Research Notices)

   null (Ed.)

   Abstract In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday–Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other homology theories associated with spaces (such as Pontryagin algebras, ${{\mathbb{S}}}^1$-equivariant homology of the free loop space, and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces, and some 3-dimensional manifolds, such as link complements in ${\mathbb{R}}^3$ and the lens spaces $ L(p,q) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in ${\mathbb{R}}^3$. 

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