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1. Finite quotients of 3-manifold groups

   https://doi.org/10.1007/s00222-024-01262-4  

   Sawin, Will; Wood, Melanie Matchett (July 2024, Inventiones mathematicae)

   For and finite groups, does there exist a 3-manifold group with as a quotient but no as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments. 

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2. Bridge trisections of knotted surfaces in 4-manifolds

   https://doi.org/10.1073/pnas.1717171115  

   Meier, Jeffrey; Zupan, Alexander (October 2018, Proceedings of the National Academy of Sciences)

   We prove that every smoothly embedded surface in a 4-manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4-manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a generalized bridge trisection, extends the authors’ definition of bridge trisections for surfaces in S 4 . Using this construction, we give diagrammatic representations called shadow diagrams for knotted surfaces in 4-manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside ℂ ℙ 2 . Using these examples, we prove that there exist exotic 4-manifolds with ( g , 0 ) —trisections for certain values of g. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams. 

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3. Naturality and Mapping Class Groups in Heegaard Floer Homology

   https://doi.org/10.1090/memo/1338  

   Juhász, András; Thurston, Dylan; Zemke, Ian (September 2021, Memoirs of the American Mathematical Society)

   We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds. 

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4. Trisections and link surgeries

   https://doi.org/10.53733/94  

   Kirby, Robion; Thompson, Abigail (September 2021, New Zealand Journal of Mathematics)

   We examine questions about surgery on links which arise naturally from the trisection decomposition of 4-manifolds developed by Gay and Kirby \cite{G-K3}. These links lie on Heegaard surfaces in $$\#^j S^1 \times S^2$$ and have surgeries yielding $$\#^k S^1 \times S^2$$. We describe families of links which have such surgeries. One can ask whether all links with such surgeries lie in these families; the answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer. 

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5. 3–Manifolds without any embedding insymplectic 4–manifolds

   https://doi.org/10.2140/gt.2024.28.3357  

   Daemi, Aliakbar; Lidman, Tye; Miller_Eismeier, Mike (November 2024, Geometry & Topology)

   We show that there exist infinitely many closed 3–manifolds that do not embed in closed symplectic 4–manifolds, disproving a conjecture of Etnyre–Min–Mukherjee. To do this, we construct L–spaces that cannot bound positive- or negative-definite manifolds. The arguments use Heegaard Floer correction terms and instanton moduli spaces. 

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