Stable etale realization and etale cobordism

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an \'etale topological realization of the stable motivic homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero. On the other hand we get a natural setting for \'etale cohomology theories. In particular, we define and discuss an \'etale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting from \'etale cohomology. Finally, we construct maps from algebraic to \'etale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element.