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1. Almost non-negatively curved 4-manifolds with torus symmetry

   https://doi.org/10.1090/proc/15093  

   Harvey, John; Searle, Catherine (January 2020, Proceedings of the American Mathematical Society)

   Wei, Guofang (Ed.)

   We prove that if a closed, smooth, simply-connected 4-manifold with a circle action admits an almost non-negatively curved sequence of invariant Riemannian metrics, then it also admits a non-negatively curved Riemannian metric invariant with respect to the same action. The same is shown for torus actions of higher rank, giving a classification of closed, smooth, simply-connected 4-manifolds of almost non-negative curvature under the assumption of torus symmetry. 

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2. Wall's Stable Realization for Diffeomorphisms of Definite 4-manifolds

   Ruberman, Daniel; Strle, Sašo (April 2025, Indiana University mathematics journal)

   Demeter, Ciprian; Jolly, Michael; Judge, Chris; Le, Nam; Levenberg, Norm; Mandell, Michael; Pilgrim, Kevin; Sternberg, Peter; Strauch, Matthias; Wang, Shouhong (Ed.)

   ABSTRACT. Let X be a smooth simply connected closed 4- manifold with definite intersection form. We show that any automorphism of the intersection form of X is realized by a dif- feomorphism of X#(S2×S2). This extends and completes Wall’s foundational result from 1964. 

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3. SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

   https://doi.org/10.1017/fms.2019.11  

   LEVINE, ADAM SIMON; LIDMAN, TYE (January 2019, Forum of Mathematics, Sigma)

   We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to $$S^{2}$$ but do not admit a spine (that is, a piecewise linear embedding of $$S^{2}$$ that realizes the homotopy equivalence). This is the remaining case in the existence problem for codimension-2 spines in simply connected manifolds. The obstruction comes from the Heegaard Floer $$d$$ invariants. 

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4. Special Lagrangian submanifolds of log Calabi-Yau manifolds

   Collins, Tristan C.; Jacob, A.; Lin, Y.-S. (January 2021, Duke mathematical journal)

   null (Ed.)

   We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D is a smooth divisor in the linear system of K_Y with D^2=d, then X=Y/D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface. 

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5. Contact surgery numbers

   https://doi.org/10.4310/JSG.2023.v21.n6.a4  

   Etnyre, John; Kegel, Marc; Onaran, Sinem (January 2023, Journal of Symplectic Geometry)

   It is known that any contact $$3$$-manifold can be obtained by rationally contact Dehn surgery along a Legendrian link $$L$$ in the standard tight contact $$3$$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $$L$$ describing a given contact $$3$$-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the $$3$$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $$S^1\times S^2$$, the Poincar\'e homology sphere and the Brieskorn sphere $$\Sigma(2,3,7)$$. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $$3$$-sphere. We further obtain results for the $$3$$-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest. 

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