Abstract
Let \(X\subset \mathbb {R}^n\) be a real algebraic set and \(M\) a smooth, closed manifold. We show that all continuous maps \(M\rightarrow X\) 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.
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Notes
We use Euclidean topology throughout, except that “irreducible algebraic set” means irreducible in the Zariski topology. We will identify all real algebraic sets with their real points throughout this section.
One may establish the existence of such a diagram using the theory of regular neighborhoods, as presented in Hudson–Zeeman [13], instead of the theory of rug functions.
To construct such a metric on a principal \(G\)-bundle \(P\) over a manifold \(N\), where \(G\) is a Lie group, note that if \(P\) is trivial over \(U\subset N\) then \(T(P|_{U}) \cong T(U) \times T_e (G) \times G,\) and any choice of metric on \(T(U) \oplus ( T_e (G)\times U)\) gives rise, under translation by \(G\), to a \(G\)-invariant inner product on \(T(P|_{U})\). These inner products can be pasted together using a partition of unity on \(N\).
References
Bochnak, J., Kucharz, W.: Real algebraic morphisms represent few homotopy classes. Math. Ann. 337(4), 909–921 (2007)
Dummit, D.S., Foote, R.M.: Abstract Algebra, 3rd edn. Wiley, Hoboken (2004)
Durfee, A.H.: Neighborhoods of algebraic sets. Trans. Am. Math. Soc. 276(2), 517–530 (1983)
Fine, B., Kirk, P., Klassen, E.: A local analytic splitting of the holonomy map on flat connections. Math. Ann. 299(1), 171–189 (1994)
Gaĭfullin, A.A.: The manifold of isospectral symmetric tridiagonal matrices and the realization of cycles by aspherical manifolds. Tr. Mat. Inst. Steklova, 263(Geometriya, Topologiya i Matematicheskaya Fizika. I):44–63, 2008. Translation in Proc. Steklov Inst. Math. 263 (2008), no. 1, 38–56
Gaĭfullin, A.A.: Realization of cycles by aspherical manifolds. Uspekhi Mat. Nauk, 63(3(381)):157–158, 2008. Translation in Russian Math. Surveys 63 (2008), no. 3, 562–564
Ghiloni, R.: Second order homological obstructions on real algebraic manifolds. Topol. Appl. 154(17), 3090–3094 (2007)
Gottlieb, D.H.: Applications of bundle map theory. Trans. Am. Math. Soc. 171, 23–50 (1972)
Grothendieck, A.: Classes de Chern et représentations linéaires des groupes discrets. In: Dix Exposés sur la Cohomologie des Schémas, pp. 215–305. North-Holland, Amsterdam (1968)
Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall Inc., Englewood Cliffs (1974)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hirschhorn, P.S.: Model Categories and Their Localizations, Volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2003)
Hudson, J.F.P., Zeeman, E.C.: On regular neighbourhoods. Proc. Lond. Math. Soc. 3(14), 719–745 (1964)
Kollár, J.: Lectures on Resolution of Singularities, Volume 166 of Annals of Mathematics Studies. Princeton University Press, Princeton (2007)
Kucharz, W., Maciejewski, Ł.: Complexification and homotopy. Homol. Homotopy Appl. 16(1), 159–165 (2014)
Mitter, P.K., Viallet, C.-M.: On the bundle of connections and the gauge orbit manifold in Yang-Mills theory. Commun. Math. Phys. 79(4), 457–472 (1981)
Moise, E.E.: Geometric topology in dimensions \(2\) and \(3\). In: Graduate Texts in Mathematics, Vol. 47. Springer, New York (1977)
Ramras, D.A.: Excision for deformation \(K\)-theory of free products. Algebr. Geom. Topol. 7, 2239–2270 (2007)
Ramras, D.A.: Yang-Mills theory over surfaces and the Atiyah-Segal theorem. Algebr. Geom. Topol. 8(4), 2209–2251 (2008)
Ramras, D.A.: The stable moduli space of flat connections over a surface. Trans. Am. Math. Soc. 363(2), 1061–1100 (2011)
Spanier, E.H.: Algebraic Topology. Springer, New York (1981). Corrected reprint
Thom, R.: Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28, 17–86 (1954)
Acknowledgments
The first author thanks Juan Souto and Frances Kirwan for helpful conversations. The second author thanks Ben Wieland. In addition, the suggestions of the anonymous referees helped to improve the exposition.
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The first author was partly supported by an NSERC discovery Grant. The second author was partially supported by NSF Grants DMS-0804553 and DMS-0968766 and a Collaboration Grant from the Simons Foundation.
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Baird, T., Ramras, D.A. Smoothing maps into algebraic sets and spaces of flat connections. Geom Dedicata 174, 359–374 (2015). https://doi.org/10.1007/s10711-014-0022-z
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DOI: https://doi.org/10.1007/s10711-014-0022-z