Smoothing maps into algebraic sets and spaces of flat connections

Abstract

Let X be a real algebraic subset of R^n and M a smooth, closed manifold. We show that all continuous maps from M to X are homotopic (in X) to C^\infty maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower bounds on the ranks of the homotopy groups of spaces of flat connections over aspherical manifolds.

15 pages. To appear in Geometriae Dedicata. Changes in v4: Removed material on the topological Atiyah-Segal map, which will appear in a separate article. Various other small changes to the exposition

References in corpus (2)

• A finite dimensional approach to the strong Novikov conjecture
• K-theoretic descent and a motivic Atiyah-Segal theorem