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1. Integral models for spaces via the higher Frobenius

   https://doi.org/10.1090/jams/998  

   Yuan, Allen (January 2023, Journal of the American Mathematical Society)

   We give a fully faithful integral model for simply connected finite complexes in terms of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$is the data of its Spanier-Whitehead dual, as an ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring, together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s $Q$-construction acts on the $\mathrm{\infty <\#comment/>}$-category of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic $K$-theory which we callpartial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of ${\mathbb{F}}_p$up to $p$-completion. 

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2. Property G and the 4-genus

   https://doi.org/10.1090/btran/153  

   Ni, Yi (January 2024, Transactions of the American Mathematical Society, Series B)

   We say a null-homologous knot $K$in a $3$-manifold $Y$has Property G, if the Thurston norm and fiberedness of the complement of $K$is preserved under the zero surgery on $K$. In this paper, we will show that, if the smooth $4$-genus of $K\times <\#comment/>\left\{0\right\}$(in a certain homology class) in $(Y\times <\#comment/>[0,1\left]\right)\mathrm{\#<\#comment/>}N\stackrel{¯<\#comment/>}{\mathbb{C}{P}^2}$, where $Y$is a rational homology sphere, is smaller than the Seifert genus of $K$, then $K$has Property G. When the smooth $4$-genus is $0$, $Y$can be taken to be any closed, oriented $3$-manifold. 

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3. Fillable contact structures from positive surgery

   https://doi.org/10.1090/btran/200  

   Mark, Thomas; Tosun, Bülent (August 2024, Transactions of the American Mathematical Society, Series B)

   We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot $K$in a contact 3-manifold $Y$, gives rise to a weakly fillable contact structure. We show that this happens if and only if $Y$itself is weakly fillable, and $K$is isotopic to the boundary of a properly embedded symplectic disk inside a filling of $Y$. Moreover, if $Y^{\prime }$is a contact manifold arising from positive contact surgery along $K$, then any filling of $Y^{\prime }$is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of $Y$. Using this result we deduce several necessary conditions for a knot in the standard 3-sphere to admit a fillable positive surgery, such as quasipositivity and equality between the slice genus and the 4-dimensional clasp number, and we give a characterization of such knots in terms of a quasipositive braid expression. We show that knots arising as the closure of a positive braid always admit a fillable positive surgery, as do knots that have lens space surgeries, and suitable satellites of such knots. In fact the majority of quasipositive knots with up to 10 crossings admit a fillable positive surgery. On the other hand, in general, (strong) quasipositivity, positivity, or Lagrangian fillability need not imply a knot admits a fillable positive contact surgery. 

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4. Rigidity of nonpositively curved manifolds with convex boundary

   https://doi.org/10.1090/proc/16475  

   Ghomi, Mohammad; Spruck, Joel (November 2023, Proceedings of the American Mathematical Society)

   We show that a compact Riemannian $3$-manifold $M$with strictly convex simply connected boundary and sectional curvature $K\le <\#comment/>a\le <\#comment/>0$is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv <\#comment/>a$on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for $K\ge <\#comment/>a\ge <\#comment/>0$. 

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5. Equivariant Infinite Loop Space Theory: The Space Level Story

   https://doi.org/10.1090/memo/1540  

   May, J; Merling, Mona; Osorno, Angélica (January 2025, Memoirs of the American Mathematical Society)

   We rework and generalize equivariant infinite loop space theory, which shows how to construct $G$-spectra from $G$-spaces with suitable structure. There is a classical version which gives classical $\mathrm{\Omega <\#comment/>}$- $G$-spectra for any topological group $G$, but our focus is on the construction of genuine $\mathrm{\Omega <\#comment/>}$- $G$-spectra when $G$is finite. We also show what is and is not true when $G$is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical $G$-spectra for general $G$but fails for genuine $G$-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature. 

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