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1. Robertson’s conjecture and universal finite generation in the homology of graph braid groups

   https://doi.org/10.1007/s00029-024-00971-1  

   Knudsen, Ben; Ramos, Eric (November 2024, Selecta Mathematica)

   Abstract We formulate a categorification of Robertson’s conjecture analogous to the categorical graph minor conjecture of Miyata–Proudfoot–Ramos. We show that these conjectures imply the existence of a finite list of atomic graphs generating the homology of configuration spaces of graphs—in fixed degree, with a fixed number of particles, under topological embeddings. We explain how the simplest case of our conjecture follows from work of Barter and Proudfoot–Ramos, implying that the category of cographs is Noetherian, a result of potential independent interest. 

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2. Cyclic branched covers of Seifert links and properties related to the ADE$ADE$ link conjecture

   https://doi.org/10.1112/jlms.70178  

   Boyer, Steven; Gordon, Cameron_McA; Hu, Ying (May 2025, Journal of the London Mathematical Society)

   Abstract In this article, we show that all cyclic branched covers of a Seifert link have left‐orderable fundamental groups, and therefore admit co‐oriented taut foliations and are not ‐spaces, if and only if it is not an link up to orientation. This leads to a proof of the link conjecture for Seifert links. When is an link up to orientation, we determine which of its canonical ‐fold cyclic branched covers have nonleft‐orderable fundamental groups. In addition, we give a topological proof of Ishikawa's classification of strongly quasi‐positive Seifert links and we determine the Seifert links that are definite, resp., have genus zero, resp. have genus equal to its smooth 4‐ball genus, among others. In the last section, we provide a comprehensive survey of the current knowledge and results concerning the link conjecture. 

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3. Cohomological and motivic inclusion–exclusion

   https://doi.org/10.1112/S0010437X24007292  

   Das, Ronno; Howe, Sean (September 2024, Compositio Mathematica)

   We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods. 

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4. Endoscopic decompositions and the Hausel–Thaddeus conjecture

   https://doi.org/10.1017/fmp.2021.7  

   Maulik, Davesh; Shen, Junliang (January 2021, Forum of Mathematics, Pi)

   Abstract We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $$\mathrm {SL}_n$$ - and $$\mathrm {PGL}_n$$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $$\mathrm {SL}_n$$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p -adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler. 

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5. Robertson's Conjecture in algebraic topology

   Knudsen, Ben; Ramos, Eric (January 2023, Séminaire lotharingien de combinatoire)

   One of the most famous results in graph theory is that of Kuratowski’s theorem, which states that a graph $$G$$ is non-planar if and only if it contains one of $$K_{3,3}$$ or $$K_5$$ as a topological minor. That is, if some subdivision of either $$K_{3,3}$$ or $$K_5$$ appears as a subgraph of $$G$$. In this case we say that the question of planarity is determined by a finite set of forbidden (topological) minors. A conjecture of Robertson, whose proof was recently announced by Liu and Thomas, characterizes the kinds of graph theoretic properties that can be determined by finitely many forbidden minors. In this extended abstract we will present a categorical version of Robertson’s conjecture, which we have proven in certain cases. We will then illustrate how this categorification, if proven in all cases, would imply many non-trivial statements in the topology of graph configuration spaces. 

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