Smoothing maps into algebraic sets and spaces of flat connections

TL;DR

This paper proves that continuous maps from smooth manifolds to real algebraic sets can be smoothly approximated, and applies this to analyze characteristic classes and homotopy groups of flat connection spaces.

Contribution

It introduces a method to smooth maps into algebraic sets and explores implications for characteristic classes and homotopy groups of flat connection spaces.

Findings

Continuous maps into algebraic sets are homotopic to smooth maps.

Lower bounds are established for homotopy groups of flat connection spaces.

Applications to characteristic classes of vector bundles are demonstrated.

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.