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

Smoothing maps into algebraic sets and spaces of flat connections