Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Some remarks on configuration spaces


Author: George Raptis
Journal: Proc. Amer. Math. Soc. 139 (2011), 1879-1887
MSC (2010): Primary 55R80; Secondary 57N99
DOI: https://doi.org/10.1090/S0002-9939-2010-10580-4
Published electronically: October 6, 2010
MathSciNet review: 2763775
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies the homotopy type of the configuration spaces $ F_n(X)$ by introducing the idea of configuration spaces of maps. For every map $ f: X \to Y$, the configuration space $ F_n(f)$ is the space of configurations in $ X$ that have distinct images in $ Y$. We show that the natural maps $ F_n(X) \leftarrow F_n(f) \rightarrow F_n(Y)$ are homotopy equivalences when $ f$ is a proper cell-like map between $ d$-manifolds. We also show that the best approximation to $ X \mapsto F_n(X)$ by a homotopy invariant functor is given by the $ n$-fold product map.


References [Enhancements On Off] (What's this?)

  • 1. M. Aouina, J. R. Klein, On the homotopy invariance of configuration spaces, Algebr. Geom. Topol. 4 (2004), 813-827 (electronic). MR 2100681 (2005k:55021)
  • 2. S. Armentrout, Cellular decompositions of $ 3$-manifolds that yield $ 3$-manifolds, Mem. Amer. Math. Soc., No. 107, American Mathematical Society, 1971. MR 0413104 (54:1225)
  • 3. C.-F. Bödigheimer, Stable splittings of mapping spaces, Algebraic topology (Seattle, Wash., 1985), Lecture Notes in Math., No. 1286, pp. 174-187, Springer, Berlin, 1987. MR 922926 (89c:55011)
  • 4. M. Brown, A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. MR 0117695 (22:8470b)
  • 5. T. A. Chapman, Cell-like mappings, Algebraic and geometrical methods in topology, Lecture Notes in Math., Vol. 428, pp. 230-240, Springer, Berlin, 1974. MR 0383423 (52:4304)
  • 6. T. A. Chapman, Homotopy conditions which detect simple homotopy equivalences, Pacific J. Math. 80 (1979), no. 1, 13-46. MR 534693 (81f:57011)
  • 7. M. M. Cohen, A course in simple-homotopy theory. Graduate Texts in Mathematics, Vol. 10, Springer-Verlag, New York-Berlin, 1973. MR 0362320 (50:14762)
  • 8. E. Fadell, L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118. MR 0141126 (25:4537)
  • 9. P. S. Hirschhorn, Model categories and their localizations. Mathematical Surveys and Monographs, Vol. 99, American Mathematical Society, 2003. MR 1944041 (2003j:18018)
  • 10. M. Hovey, Model Categories. Mathematical Surveys and Monographs, Vol. 63, American Mathematical Society, 1999. MR 1650134 (99h:55031)
  • 11. R. C. Lacher, Cell-like mappings of ANRs, Bull. Amer. Math. Soc. 74 (1968), 933-935. MR 0244963 (39:6276)
  • 12. R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717-731. MR 0251714 (40:4941)
  • 13. N. Levitt, Spaces of arcs and configuration spaces of manifolds, Topology 34 (1995), no. 1, 217-230. MR 1308497 (95k:55015)
  • 14. R. Longoni, P. Salvatore, Configuration spaces are not homotopy invariant, Topology 44 (2005), no. 2, 375-380. MR 2114713 (2005k:55024)
  • 15. W. J. R. Mitchell, D. Repovš, The topology of cell-like mappings, Conference on Differential Geometry and Topology, Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), 265-300. MR 1122860 (92f:54012)
  • 16. J. H. Roberts, N. E. Steenrod, Monotone transformations of two-dimensional manifolds, Ann. of Math. (2) 39 (1938), no. 4, 851-862. MR 1503441
  • 17. F. Quinn, Ends of maps. III: Dimensions $ 4$ and $ 5$, J. Differential Geom. 17 (1982), no. 3, 503-521. MR 679069 (84j:57012)
  • 18. G. Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213-221. MR 0331377 (48:9710)
  • 19. L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271-294. MR 0295365 (45:4431)
  • 20. J. W. T. Youngs, Homeomorphic approximations to monotone mappings, Duke Math. J. 15 (1948), 87-94. MR 0024623 (9:524a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 55R80, 57N99

Retrieve articles in all journals with MSC (2010): 55R80, 57N99


Additional Information

George Raptis
Affiliation: Institut für Mathematik, Universität Osnabrück, Albrechtstrasse 28a, 49069 Osnabrück, Germany
Email: graptis@mathematik.uni-osnabrueck.de

DOI: https://doi.org/10.1090/S0002-9939-2010-10580-4
Keywords: Configuration spaces, cell-like maps
Received by editor(s): April 30, 2010
Received by editor(s) in revised form: May 11, 2010
Published electronically: October 6, 2010
Communicated by: Brooke Shipley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society