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1. Stable homotopy groups of spheres

   https://doi.org/10.1073/pnas.2012335117  

   Isaksen, Daniel C.; Wang, Guozhen; Xu, Zhouli (September 2020, Proceedings of the National Academy of Sciences)

   We discuss the current state of knowledge of stable homotopy groups of spheres. We describe a computational method using motivic homotopy theory, viewed as a deformation of classical homotopy theory. This yields a streamlined computation of the first 61 stable homotopy groups and gives information about the stable homotopy groups in dimensions 62 through 90. As an application, we determine the groups of homotopy spheres that classify smooth structures on spheres through dimension 90, except for dimension 4. The method relies more heavily on machine computations than previous methods and is therefore less prone to error. The main mathematical tool is the Adams spectral sequence. 

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2. Localization and nilpotent spaces in -homotopy theory

   https://doi.org/10.1112/S0010437X22007321  

   Asok, Aravind; Fasel, Jean; Hopkins, Michael J. (March 2022, Compositio Mathematica)

   Abstract For a subring $$R$$ of the rational numbers, we study $$R$$ -localization functors in the local homotopy theory of simplicial presheaves on a small site and then in $${\mathbb {A}}^1$$ -homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in $${\mathbb {A}}^1$$ -homotopy theory, paying attention to future applications for vector bundles. We show that $$R$$ -localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space $$BGL_n$$ is $${\mathbb {A}}^1$$ -nilpotent when $$n$$ is odd, and analyze the (more complicated) situation where $$n$$ is even as well. We establish analogs of various classical results about rationalization in the context of $${\mathbb {A}}^1$$ -homotopy theory: if $-1$ is a sum of squares in the base field, $${\mathbb {A}}^n \,{\setminus}\, 0$$ is rationally equivalent to a suitable motivic Eilenberg–Mac Lane space, and the special linear group decomposes as a product of motivic spheres. 

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3. K-theoretic Tate–Poitou duality and the fiber of the cyclotomic trace

   https://doi.org/10.1007/s00222-020-00952-z  

   Blumberg, Andrew J.; Mandell, Michael A. (February 2020, Inventiones mathematicae)

   Let p∈Z be an odd prime. We prove a spectral version of Tate–Poitou duality for the algebraic K-theory spectra of number rings with p inverted. This identifies the homotopy type of the fiber of the cyclotomic trace K(OF)∧p→TC(OF)∧p after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of Z. 

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4. A Trace Map on Higher Scissors Congruence Groups

   https://doi.org/10.1093/imrn/rnae153  

   Bohmann, Anna_Marie; Gerhardt, Teena; Malkiewich, Cary; Merling, Mona; Zakharevich, Inna (August 2024, International Mathematics Research Notices)

   Abstract Cut-and-paste $$K$$-theory has recently emerged as an important variant of higher algebraic $$K$$-theory. However, many of the powerful tools used to study classical higher algebraic $$K$$-theory do not yet have analogues in the cut-and-paste setting. In particular, there does not yet exist a sensible notion of the Dennis trace for cut-and-paste $$K$$-theory. In this paper we address the particular case of the $$K$$-theory of polyhedra, also called scissors congruence $$K$$-theory. We introduce an explicit, computable trace map from the higher scissors congruence groups to group homology, and use this trace to prove the existence of some nonzero classes in the higher scissors congruence groups. We also show that the $$K$$-theory of polyhedra is a homotopy orbit spectrum. This fits into Thomason’s general framework of $$K$$-theory commuting with homotopy colimits, but we give a self-contained proof. We then use this result to re-interpret the trace map as a partial inverse to the map that commutes homotopy orbits with algebraic $$K$$-theory. 

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5. Torus actions, Morse homology, and the Hilbert scheme of points on affine space

   https://doi.org/10.46298/epiga.2021.6792  

   Totaro, Burt (August 2021, Épijournal de Géométrie Algébrique)

   We formulate a conjecture on actions of the multiplicative group in motivichomotopy theory. In short, if the multiplicative group G_m acts on aquasi-projective scheme U such that U is attracted as t approaches 0 in G_m toa closed subset Y in U, then the inclusion from Y to U should be anA^1-homotopy equivalence. We prove several partial results. In particular, over the complex numbers,the inclusion is a homotopy equivalence on complex points. The proofs use ananalog of Morse theory for singular varieties. Application: the Hilbert schemeof points on affine n-space is homotopy equivalent to the subspace consistingof schemes supported at the origin. 

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