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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. Exotic Mazur manifolds and knot trace invariants

   https://doi.org/10.1016/j.aim.2021.107994  

   Hayden, Kyle; Mark, Thomas E; Piccirillo, Lisa (November 2021, Advances in Mathematics)

   From a handle-theoretic perspective, the simplest contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic. Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant ν is an invariant of the trace of a knot, i.e. the smooth 4-manifold obtained by attaching a 2-handle to the 4-ball along K. This provides a computable, integer-valued diffeomorphism invariant that is effective at distinguishing exotic smooth structures on knot traces and other simple 4-manifolds, including when other adjunction-type obstructions are ineffective. We also show that the concordance invariants τ and ϵ are not knot trace invariants. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct surgeries to $$S^1 \times S^2$$, resolving a question from Problem 1.16 in Kirby's list. 

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3. From zero surgeries to candidates for exotic definite 4‐manifolds

   https://doi.org/10.1112/jlms.12800  

   Manolescu, Ciprian; Piccirillo, Lisa (August 2023, Journal of the London Mathematical Society)

   Abstract One strategy for distinguishing smooth structures on closed 4‐manifolds is to produce a knot in that is slice in one smooth filling of but not slice in some homeomorphic smooth filling . In this paper, we explore how 0‐surgery homeomorphisms can be used to potentially construct exotic pairs of this form. To systematically generate a plethora of candidates for exotic pairs, we give a fully general construction of pairs of knots with the same zero surgeries. By computer experimentation, we find five topologically slice knots such that, if any of them were slice, we would obtain an exotic 4‐sphere. We also investigate the possibility of constructing exotic smooth structures on in a similar fashion. 

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4. Topological Realizations of Line Arrangements

   https://doi.org/10.1093/imrn/rnx190  

   Ruberman, Daniel; Starkston, Laura (August 2017, International Mathematics Research Notices)

   Abstract A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for lines in a projective plane over a field. An important variation allows for pseudolines: embedded circles (isotopic to $$\mathbb R\rm{P}^1$$) in the real projective plane. In this article we investigate whether a configuration is realized by a collection of 2-spheres embedded, in symplectic, smooth, and topological categories, in the complex projective plane. We find obstructions to the existence of topologically locally flat spheres realizing a configuration, and show for instance that the combinatorial configuration corresponding to the projective plane over any finite field is not realized. Such obstructions are used to show that a particular contact structure on certain graph manifolds is not (strongly) symplectically fillable. We also show that a configuration of real pseudolines can be complexified to give a configuration of smooth, indeed symplectically embedded, 2-spheres. 

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5. Non‐isotopic splitting spheres for a split link in S4$S^4$

   https://doi.org/10.1112/plms.70038  

   Hughes, Mark; Kim, Seungwon; Miller, Maggie (April 2025, Proceedings of the London Mathematical Society)

   Abstract We show that there exist split, orientable, 2‐component surface‐links in with non‐isotopic splitting spheres in their complements. In particular, for non‐negative integers with , the unlink consisting of one component of genus and one component of genus contains in its complement two smooth splitting spheres that are not topologically isotopic in . This contrasts with link theory in the classical dimension, as any two splitting spheres in the complement of a 2‐component split link are isotopic in . 

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