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1. 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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2. Harmonic spinors on the Davis hyperbolic 4-manifold

   https://doi.org/10.1142/S1793525320500247  

   Ratcliffe, John G.; Ruberman, Daniel; Tschantz, Steven T. (September 2021, Journal of Topology and Analysis)

   In this paper, we use the [Formula: see text]-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic [Formula: see text]-manifold that admits harmonic spinors. We also explicitly describe the spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the [Formula: see text]-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold. 

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3. Almost Symplectic Structures on Spin(7)−Manifolds

   Salur, Sema; Yalcinkaya, Eyup (July 2020, 12th ISAAC Congress, Portugal, Geometries Defined by Differential Forms Proceedings, 2020.)

   null (Ed.)

   A non-degenerate differential 2-form on an even dimensional manifold M^2n is called an almost-symplectic structure. A necessary condition for the existence of an almost-symplectic structure is that all odd-dimensional Stiefel-Whitney classes of M should vanish. In this paper, we prove that all odd-dimensional Stiefel-Whitney classes of a smooth, closed, connected, orientable 8-manifold with spin structure vanish. We also study the almost-symplectic structures on certain classes of Spin(7)-manifolds. 

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4. Embedding spheres in knot traces

   https://doi.org/10.1112/S0010437X21007508  

   Feller, Peter; Miller, Allison N.; Nagel, Matthias; Orson, Patrick; Powell, Mark; Ray, Arunima (October 2021, Compositio Mathematica)

   Abstract The trace of the $$n$$ -framed surgery on a knot in $$S^{3}$$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $$2$$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $$3$$ -dimensional knot invariants. For each $$n$$ , this provides conditions that imply a knot is topologically $$n$$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice. 

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5. OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

   https://doi.org/10.1017/fms.2018.16  

   ASOK, A.; FASEL, J.; HOPKINS, M. (March 2019, Forum of mathematics)

   Suppose is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension , it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general. 

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