7 papers
Algebraic vs. analytic: a survey
Aravind Asok, Tom Bachmann, Michael J. Hopkins
Given a pair of complex algebraic varieties, we can ask: does every analytic map admit an algebraic representative? Such questions are most interesting when the source and target a…
Efficient generation of projective modules: a motivic view
Aravind Asok, Morgan Opie, Brian Shin +1
Assume is a field and is a smooth -algebra of dimension . If is a projective module of rank , then it is well-known that can be generated by -elements…
On P^1-stabilization in unstable motivic homotopy theory
Aravind Asok, Tom Bachmann, Michael J. Hopkins
We analyze stabilization with respect to in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectiv…
Splitting vector bundles over real algebraic varieties
Aravind Asok, Jean Fasel, Samuel Lerbet
Suppose is a smooth affine real variety and is a vector bundle over . We analyze the problem of splitting off a free rank one summand from in cor…
Algebraic vector bundles and -local A^1-homotopy theory
Aravind Asok, Jean Fasel, Michael J. Hopkins
Using techniques of A^1-homotopy theory, we produce motivic lifts of elements in classical homotopy groups of spheres; these lifts provide polynomial maps of spheres and allow us t…
Unstable motivic and real-étale homotopy theory
Aravind Asok, Tom Bachmann, Elden Elmanto +1
We prove that for any base scheme , real étale motivic (unstable) homotopy theory over coincides with unstable semialgebraic topology over (that is, sheaves of spaces o…