Spaces of algebraic and continuous maps between real algebraic varieties
Adamaszek, Michał, Kozlowski, Andrzej and Yamaguchi, Kohhei. (2011) Spaces of algebraic and continuous maps between real algebraic varieties. Quarterly Journal of Mathematics, Vol.62 (No.4). pp. 771-790. ISSN 0033-5606Full text not available from this repository.
Official URL: http://dx.doi.org/10.1093/qmath/haq029
We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a 'minimum degree' filtration by finite-dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation of these spaces. This result can be seen as a generalization of the results of Mostovoy, Vassiliev and others on the topology of the space of real rational maps and the space of real polynomials without n-fold roots. It can also be viewed as a real analogue of Mostovoy's work on the topology of the space of holomorphic maps between complex projective spaces, which generalizes Segal's work on the space of complex rational maps.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Mathematics|
|Library of Congress Subject Headings (LCSH):||Holomorphic mappings, Topology, Algebraic varieties|
|Journal or Publication Title:||Quarterly Journal of Mathematics|
|Publisher:||Oxford University Press|
|Page Range:||pp. 771-790|
|Access rights to Published version:||Restricted or Subscription Access|
|Funder:||Engineering and Physical Sciences Research Council (EPSRC), Japan. Monbu Kagakushō [Japan. Ministry of Education, Culture, Sports, Science and Technology] (MK)|
|Grant number:||EP/D063191/1 (EPSRC), 19540068 (MK)|
|References:||1. M. Adamaszek, Spaces of rational functions, Master Thesis,Warsaw University, 2007 (in Polish). 2. J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 36, Springer, Berlin, 1998. 3. C. P. Boyer, J. C. Hurtubise and R. J. Milgram, Stability theorems for spaces of rational curves, Int. J. Math. 12 (2001), 223–262. 4. C. P. Boyer, J. C. Hurtubise, B. M. Mann and R. J. Milgram, The topology of the space of rational maps into generalised flag manifolds, Acta Math. 173 (1994), 61–101. 5. R. L. Cohen, J. D. S. Jones and G. B. Segal, Stability for holomorphic spheres and Morse theory, Contemp. Math. 258 (2000), 87–106. 6. F. R. Cohen, J. C. Moore and J. A. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. Math. 110 (1979) 549–565. 7. M. Goresky and R. MacPherson, Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Springer, Berlin, 1980. 8. J. Gravesen, On the topology of spaces of holomorphic maps, Acta Math. 162 (1989), 247–286. 9. M. Gromov, Oka’s principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989) 851–897. 10. M. A. Guest, Topology of the space of absolute minima of the energy functional, Amer. J. Math. 106 (1984), 21–42. 11. M. A. Guest, Topology of the space of rational curves on a toric variety, Acta Math. 174 (1995), 119–145. 12. M. A. Guest, A. Kozlowski and K.Yamaguchi, The topology of spaces of coprime polynomials, Math. Z. 217 (1994), 435–446. 13. M. A. Guest, A. Kozlowski and K. Yamaguchi, Spaces of polynomials with roots of bounded multiplicity, Fund. Math. 116 (1999), 93–117. 14. A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2001. 15. J. Havlicek, On spaces of holomorphic maps from two copies of the Riemann sphere to complex Grassmannians, Contemp. Math. 182 (1995), 83–116. 16. D. Husemoller, Fiber Bundles, Graduate Texts in Mathematics 20, Springer, London, 1993. 17. F. Kirwan, On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles, Ark. Math. 24 (1986), 221–275. 18. A. Kozlowski and K. Yamaguchi, Topology of complements of discriminants and resultants, J. Math. Soc. Japan 52 (2000), 949–959. 19. A.Kozlowski and K.Yamaguchi, Spaces of holomorphic maps between complex projective spaces of degree one, Topology Appl. 132 (2003), 139–145. 20. J. Mostovoy, Spaces of rational loops on a real projective space, Trans. Amer. Math. Soc. 353 (2001), 1959–1970. 21. J. Mostovoy, Spaces of rational maps and the Stone–Weierstrass theorem, Topology 45 (2006), 281–293. 22. J. Mostovoy, Truncated simplicial resolutions and spaces of rational maps, Q. J. Math., doi: 10.1093/qmath/HAQ031, to appear. 23. G. B. Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), 39–72. 24. E. H. Spanier, Algebraic Topology, Springer, Berlin, 1981. 25. V. A. Vassiliev, Complements of Discriminants of Smooth Maps, Topology and Applications, Translations of Mathematical Monographs 98, American Mathematical Society, Providence, RI, 1992 (revised edition 1994). 26. K.Yamaguchi, Complements of resultants and homotopy types, J. Math. Kyoto Univ. 39 (1999), 675–684.|
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