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  1. ; ; (, International Mathematics Research Notices)
    Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $$X$$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the $${\mathbb P}^{1}$$ suspension of $$X$$; we also analyze a host of variations on this observation. Our approach yields many examples of $${\mathbb A}^{1}$$-$$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine $${\mathbb A}^{1}$$-contractible smooth schemes. 
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  2. ; ; (, 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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