Smoothing maps into algebraic sets and spaces of flat connections
Let X ⊂ R n be a real algebraic set and M a smooth, closed manifold. We show that all continuous maps M → X are homotopic (in X ) to C ∞ maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish l...
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| Published in | Geometriae dedicata Vol. 174; no. 1; pp. 359 - 374 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.02.2015
|
| Subjects | |
| Online Access | Get full text |
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| Abstract | Let
X
⊂
R
n
be a real algebraic set and
M
a smooth, closed manifold. We show that all continuous maps
M
→
X
are homotopic (in
X
) to
C
∞
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. |
|---|---|
| AbstractList | Let
X
⊂
R
n
be a real algebraic set and
M
a smooth, closed manifold. We show that all continuous maps
M
→
X
are homotopic (in
X
) to
C
∞
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. |
| Author | Baird, Thomas Ramras, Daniel A. |
| Author_xml | – sequence: 1 givenname: Thomas surname: Baird fullname: Baird, Thomas organization: Memorial University of Newfoundland – sequence: 2 givenname: Daniel A. surname: Ramras fullname: Ramras, Daniel A. email: dramras@iupui.edu organization: IUPUI |
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| Cites_doi | 10.1007/s00208-006-0061-3 10.2140/agt.2008.8.2209 10.1090/S0002-9947-1983-0688959-3 10.4310/HHA.2014.v16.n1.a9 10.1090/S0002-9947-2010-05218-3 10.1016/j.topol.2007.07.003 10.1112/plms/s3-14.4.719 10.1007/BF02566923 10.1007/BF01209307 10.1090/S0002-9947-1972-0309111-X 10.2140/agt.2007.7.2239 10.1007/BF01459778 10.1007/978-1-4684-9322-1 10.1007/978-1-4612-9906-6 |
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| Keywords | Chern class 55R37 Secondary: 57R20 Maps between classifying spaces 53C05 Real algebraic set Flat connection Primary: 14P05 Chern-Weil theory |
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| References | Mitter, Viallet (CR16) 1981; 79 Kollár (CR14) 2007 Bochnak, Kucharz (CR1) 2007; 337 Durfee (CR3) 1983; 276 Ramras (CR20) 2011; 363 CR6 CR5 Kucharz, Maciejewski (CR15) 2014; 16 Ramras (CR19) 2008; 8 Ghiloni (CR7) 2007; 154 CR17 CR9 Thom (CR22) 1954; 28 Gottlieb (CR8) 1972; 171 CR21 Hirschhorn (CR12) 2003 Guillemin, Pollack (CR10) 1974 Hatcher (CR11) 2002 Dummit, Foote (CR2) 2004 Ramras (CR18) 2007; 7 Fine, Kirk, Klassen (CR4) 1994; 299 Hudson, Zeeman (CR13) 1964; 3 PS Hirschhorn (22_CR12) 2003 PK Mitter (22_CR16) 1981; 79 DS Dummit (22_CR2) 2004 22_CR21 R Thom (22_CR22) 1954; 28 B Fine (22_CR4) 1994; 299 22_CR17 V Guillemin (22_CR10) 1974 A Hatcher (22_CR11) 2002 JFP Hudson (22_CR13) 1964; 3 22_CR9 DH Gottlieb (22_CR8) 1972; 171 DA Ramras (22_CR18) 2007; 7 22_CR6 W Kucharz (22_CR15) 2014; 16 J Bochnak (22_CR1) 2007; 337 22_CR5 DA Ramras (22_CR20) 2011; 363 J Kollár (22_CR14) 2007 AH Durfee (22_CR3) 1983; 276 R Ghiloni (22_CR7) 2007; 154 DA Ramras (22_CR19) 2008; 8 |
| References_xml | – volume: 337 start-page: 909 issue: 4 year: 2007 end-page: 921 ident: CR1 article-title: Real algebraic morphisms represent few homotopy classes publication-title: Math. Ann. doi: 10.1007/s00208-006-0061-3 contributor: fullname: Kucharz – ident: CR21 – year: 1974 ident: CR10 publication-title: Differential Topology contributor: fullname: Pollack – volume: 8 start-page: 2209 issue: 4 year: 2008 end-page: 2251 ident: CR19 article-title: Yang-Mills theory over surfaces and the Atiyah-Segal theorem publication-title: Algebr. Geom. Topol. doi: 10.2140/agt.2008.8.2209 contributor: fullname: Ramras – volume: 276 start-page: 517 issue: 2 year: 1983 end-page: 530 ident: CR3 article-title: Neighborhoods of algebraic sets publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1983-0688959-3 contributor: fullname: Durfee – year: 2003 ident: CR12 publication-title: Model Categories and Their Localizations, Volume 99 of Mathematical Surveys and Monographs contributor: fullname: Hirschhorn – volume: 16 start-page: 159 issue: 1 year: 2014 end-page: 165 ident: CR15 article-title: Complexification and homotopy publication-title: Homol. Homotopy Appl. doi: 10.4310/HHA.2014.v16.n1.a9 contributor: fullname: Maciejewski – volume: 363 start-page: 1061 issue: 2 year: 2011 end-page: 1100 ident: CR20 article-title: The stable moduli space of flat connections over a surface publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-2010-05218-3 contributor: fullname: Ramras – volume: 154 start-page: 3090 issue: 17 year: 2007 end-page: 3094 ident: CR7 article-title: Second order homological obstructions on real algebraic manifolds publication-title: Topol. Appl. doi: 10.1016/j.topol.2007.07.003 contributor: fullname: Ghiloni – ident: CR17 – ident: CR9 – volume: 3 start-page: 719 issue: 14 year: 1964 end-page: 745 ident: CR13 article-title: On regular neighbourhoods publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/s3-14.4.719 contributor: fullname: Zeeman – volume: 28 start-page: 17 year: 1954 end-page: 86 ident: CR22 article-title: Quelques propriétés globales des variétés différentiables publication-title: Comment. Math. Helv. doi: 10.1007/BF02566923 contributor: fullname: Thom – volume: 79 start-page: 457 issue: 4 year: 1981 end-page: 472 ident: CR16 article-title: On the bundle of connections and the gauge orbit manifold in Yang-Mills theory publication-title: Commun. Math. Phys. doi: 10.1007/BF01209307 contributor: fullname: Viallet – ident: CR6 – year: 2004 ident: CR2 publication-title: Abstract Algebra contributor: fullname: Foote – ident: CR5 – year: 2002 ident: CR11 publication-title: Algebraic Topology contributor: fullname: Hatcher – volume: 171 start-page: 23 year: 1972 end-page: 50 ident: CR8 article-title: Applications of bundle map theory publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1972-0309111-X contributor: fullname: Gottlieb – volume: 7 start-page: 2239 year: 2007 end-page: 2270 ident: CR18 article-title: Excision for deformation -theory of free products publication-title: Algebr. Geom. Topol. doi: 10.2140/agt.2007.7.2239 contributor: fullname: Ramras – year: 2007 ident: CR14 publication-title: Lectures on Resolution of Singularities, Volume 166 of Annals of Mathematics Studies contributor: fullname: Kollár – volume: 299 start-page: 171 issue: 1 year: 1994 end-page: 189 ident: CR4 article-title: A local analytic splitting of the holonomy map on flat connections publication-title: Math. Ann. doi: 10.1007/BF01459778 contributor: fullname: Klassen – ident: 22_CR21 doi: 10.1007/978-1-4684-9322-1 – volume: 16 start-page: 159 issue: 1 year: 2014 ident: 22_CR15 publication-title: Homol. Homotopy Appl. doi: 10.4310/HHA.2014.v16.n1.a9 contributor: fullname: W Kucharz – volume: 337 start-page: 909 issue: 4 year: 2007 ident: 22_CR1 publication-title: Math. Ann. doi: 10.1007/s00208-006-0061-3 contributor: fullname: J Bochnak – volume: 154 start-page: 3090 issue: 17 year: 2007 ident: 22_CR7 publication-title: Topol. Appl. doi: 10.1016/j.topol.2007.07.003 contributor: fullname: R Ghiloni – volume-title: Lectures on Resolution of Singularities, Volume 166 of Annals of Mathematics Studies year: 2007 ident: 22_CR14 contributor: fullname: J Kollár – volume-title: Abstract Algebra year: 2004 ident: 22_CR2 contributor: fullname: DS Dummit – volume: 363 start-page: 1061 issue: 2 year: 2011 ident: 22_CR20 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-2010-05218-3 contributor: fullname: DA Ramras – volume: 276 start-page: 517 issue: 2 year: 1983 ident: 22_CR3 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1983-0688959-3 contributor: fullname: AH Durfee – volume: 79 start-page: 457 issue: 4 year: 1981 ident: 22_CR16 publication-title: Commun. Math. Phys. doi: 10.1007/BF01209307 contributor: fullname: PK Mitter – volume-title: Differential Topology year: 1974 ident: 22_CR10 contributor: fullname: V Guillemin – volume: 8 start-page: 2209 issue: 4 year: 2008 ident: 22_CR19 publication-title: Algebr. Geom. Topol. doi: 10.2140/agt.2008.8.2209 contributor: fullname: DA Ramras – volume: 28 start-page: 17 year: 1954 ident: 22_CR22 publication-title: Comment. Math. Helv. doi: 10.1007/BF02566923 contributor: fullname: R Thom – volume-title: Algebraic Topology year: 2002 ident: 22_CR11 contributor: fullname: A Hatcher – volume: 3 start-page: 719 issue: 14 year: 1964 ident: 22_CR13 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/s3-14.4.719 contributor: fullname: JFP Hudson – volume: 299 start-page: 171 issue: 1 year: 1994 ident: 22_CR4 publication-title: Math. Ann. doi: 10.1007/BF01459778 contributor: fullname: B Fine – ident: 22_CR17 doi: 10.1007/978-1-4612-9906-6 – ident: 22_CR5 – ident: 22_CR9 – ident: 22_CR6 – volume: 7 start-page: 2239 year: 2007 ident: 22_CR18 publication-title: Algebr. Geom. Topol. doi: 10.2140/agt.2007.7.2239 contributor: fullname: DA Ramras – volume-title: Model Categories and Their Localizations, Volume 99 of Mathematical Surveys and Monographs year: 2003 ident: 22_CR12 contributor: fullname: PS Hirschhorn – volume: 171 start-page: 23 year: 1972 ident: 22_CR8 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1972-0309111-X contributor: fullname: DH Gottlieb |
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| Snippet | Let
X
⊂
R
n
be a real algebraic set and
M
a smooth, closed manifold. We show that all continuous maps
M
→
X
are homotopic (in
X
) to
C
∞
maps. We apply this... |
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| SubjectTerms | Algebraic Geometry Convex and Discrete Geometry Differential Geometry Hyperbolic Geometry Mathematics Mathematics and Statistics Original Paper Projective Geometry Topology |
| Title | Smoothing maps into algebraic sets and spaces of flat connections |
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