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1. Infinite homotopy stable class for 4-manifolds with boundary

   Conway, Anthony; Crowley Diarmuid, Powell Mark. (August 2023, Pacific journal of mathematics)

   We show that for every odd prime $$q$$, there exists an infinite family~$$\{M_i\}_{i=1}^{\infty}$$ of topological 4-manifolds that are all stably homeomorphic to one another, all the manifolds $$M_i$$ have isometric rank one equivariant intersection pairings and boundary $$L(2q, 1) \# (S^1 \times S^2)$$, but they are pairwise not homotopy equivalent via any homotopy equivalence that restricts to a homotopy equivalence of the boundary. 

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2. Bounds on Cheeger–Gromov invariants and simplicial complexity of triangulated manifolds

   https://doi.org/10.1515/Crelle-2024-0003  

   Lim, Geunho; Weinberger, Shmuel (February 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   .We show the existence of linear bounds on Wall 𝜌-invariants of PL manifolds, employing a new combinatorial concept of 𝐺-colored polyhedra. As an application, we show how the number of h-cobordism classes of manifolds simple homotopy equivalent to a lens space with 𝑉 simplices and the fundamental group of Z n grows in 𝑉. Furthermore, we count the number of homotopy lens spaces with bounded geometry in 𝑉. Similarly, we give new linear bounds on Cheeger–Gromov 𝜌-invariants of PL manifolds endowed with a faithful representation also. A key idea is to construct a cobordism with a linear complexity whose boundary is π 1 -injectively embedded, using relative hyperbolization. As an application, we study the complexity theory of high-dimensional lens spaces. Lastly, we show the density of 𝜌-invariants over manifolds homotopy equivalent to a given manifold for certain fundamental groups. This implies that the structure set is not finitely generated. 

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3. Ricci flow and diffeomorphism groups of 3-manifolds

   https://doi.org/10.1090/jams/1003  

   Bamler, Richard; Kleiner, Bruce (August 2022, Journal of the American Mathematical Society)

   We complete the proof of the Generalized Smale Conjecture, apart from the case of R P 3 RP^3 , and give a new proof of Gabai’s theorem for hyperbolic 3 3 -manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms, except S 3 S^3 and R P 3 RP^3 , as well as hyperbolic manifolds, to prove that the space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3 3 -manifold X X , the inclusion Isom ⁡ ( X , g ) → Diff ⁡ ( X ) \operatorname {Isom}(X,g)\rightarrow \operatorname {Diff}(X) is a homotopy equivalence for any Riemannian metric g g of constant sectional curvature. 

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4. Graphing, homotopy groups of spheres, and spaces of long links and knots

   https://doi.org/10.1017/fms.2024.114  

   Koytcheff, Robin (January 2025, Forum of Mathematics, Sigma)

   We study homotopy groups of spaces of long links in Euclidean space of codimension at least three. With multiple components, they admit split injections from homotopy groups of spheres. We show that, up to knotting, these account for all the homotopy groups in a range which depends on the dimensions of the source manifolds and target manifold and which roughly generalizes the triple-point-free range for isotopy classes. Just beyond this range, joining components sends both a parametrized long Borromean rings class and a Hopf fibration to a generator of the first nontrivial homotopy group of the space of long knots. For spaces of equidimensional long links of most source dimensions, we describe generators for the homotopy group in this degree in terms of these Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map which increases source and target dimensions by one. 

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5. On the universal pairing for 2‐complexes

   https://doi.org/10.1112/blms.70130  

   Khovanov, Mikhail; Krushkal, Vyacheslav; Nicholson, John (June 2025, Bulletin of the London Mathematical Society)

   Abstract The universal pairing for manifolds was defined and shown to lack positivity in dimension 4 in [Freedman, Kitaev, Nayak, Slingerland, Walker, and Wang, J. Geom. Topol.9(2005), 2303–2317]. We prove an analogous result for 2‐complexes, and show that the universal pairing does not detect the difference between simple homotopy equivalence and 3‐deformations. The question of whether these two equivalence relations are different for 2‐complexes is the subject of the Andrews–Curtis conjecture. We also discuss the universal pairing for higher dimensional complexes and show that it is not positive. 

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