The paper deals with rational maps between real algebraic sets. We are interested in the rational maps which extend to continuous maps defined on the entire source space. In particular, we prove that every continuous map between unit spheres is homotopic to a rational map of such a type. We also establish connections with algebraic cycles and vector bundles.
Investigated are continuous rational maps of nonsingular real algebraic varieties into spheres. In some cases, necessary and sufficient conditions are given for a continuous map to be approximable by continuous rational maps. In particular, each continuous map between unit spheres can be approximated by continuous rational maps.
AbstractWe investigate stratified-algebraic vector bundles on a real algebraic
variety X. A stratification of X is a finite collection of pairwise
disjoint, Zariski locally closed subvarieties whose union is X. A
topological vector bundle ξ on X is called a stratified-algebraic
vector bundle if, roughly speaking, there exists a stratification {\mathcal{S}}
of X such that the restriction of ξ to each stratum S in {\mathcal{S}}
is an algebraic vector bundle on S. In particular, every algebraic
vector bundle on X is stratified-algebraic. It turns out that
stratified-algebraic vector bundles have many surprising properties,
which distinguish them from algebraic and topological vector bundles.
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