[Paper Review] Smooth approximation in algebraic sets and the topological Atiyah-Segal map

Thomas John Baird, Daniel A. Ramras|arXiv (Cornell University)|Jun 14, 2012

Homotopy and Cohomology in Algebraic Topology20 references3 citations

TL;DR

This paper establishes that continuous maps from a smooth, closed manifold M to a real algebraic set X in R^n are homotopic to smooth maps within X, enabling a smooth approximation theory for algebraic sets. The result is applied to characterize characteristic classes of vector bundles from continuous families of complex group representations, clarifying phenomena in deformation K-theory and yielding new insights into spaces of flat connections over aspherical manifolds.

ABSTRACT

Let X be a real algebraic set in R^n. We show that all continuous maps from M to X, with M a smooth, closed manifold, are homotopic (in X) to smooth maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations. This leads to an explanation for certain phenomena in deformation K-theory, and has applications to spaces of flat connections over aspherical manifolds.

Motivation & Objective

• To establish a smooth approximation theorem for continuous maps from smooth manifolds into real algebraic sets.
• To understand the topological implications of continuous families of complex group representations on vector bundles.
• To explain observed phenomena in deformation K-theory through the lens of smooth approximation and characteristic classes.
• To analyze spaces of flat connections over aspherical manifolds using the refined topological structure provided by the Atiyah-Segal map.

Proposed method

• Utilize the structure of real algebraic sets in R^n to construct homotopies between continuous maps and smooth maps with values in the same set.
• Apply techniques from differential topology and algebraic geometry to ensure the homotopy remains within the algebraic set X.
• Relate continuous families of complex group representations to vector bundles and their characteristic classes via the topological Atiyah-Segal map.
• Use the smooth approximation result to lift topological invariants to smooth settings, enabling analysis via differential methods.
• Analyze the cohomological structure of spaces of flat connections on aspherical manifolds using the refined characteristic classes derived from the approximation.

Experimental results

Research questions

• RQ1 Can every continuous map from a smooth, closed manifold to a real algebraic set be homotoped to a smooth map within the same set?
• RQ2 How do continuous families of complex group representations give rise to characteristic classes of vector bundles?
• RQ3 What topological obstructions or structures emerge in deformation K-theory due to the smooth approximation of such families?
• RQ4 How does the Atiyah-Segal map refine the understanding of spaces of flat connections over aspherical manifolds?

Key findings

• Continuous maps from smooth, closed manifolds to real algebraic sets in R^n are homotopic to smooth maps within the same set, establishing a smooth approximation result.
• The smooth approximation enables a coherent treatment of characteristic classes associated to continuous families of complex group representations.
• The framework explains previously observed phenomena in deformation K-theory through the interplay of topology and algebraic structure.
• The results provide new cohomological invariants for spaces of flat connections over aspherical manifolds, derived from the refined Atiyah-Segal map.