Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-30T17:13:16.097Z Has data issue: false hasContentIssue false
  • English
  • Français

Orbit spaces of gradient vector fields

Published online by Cambridge University Press:  30 October 2012

JACK S. CALCUT  and
ROBERT E. GOMPF
JACK S. CALCUT
Affiliation:
Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA (email: [email protected])
ROBERT E. GOMPF
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA (email: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting property.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013 , pp. 1732 - 1747
Copyright
Copyright © 2012 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Arm82]Armstrong, M. A.. Calculating the fundamental group of an orbit space. Proc. Amer. Math. Soc. 84 (1982), 267–271.Google Scholar
[Arn93]Arnol’d, V. I.. Problems on singularities and dynamical systems. Developments in Mathematics: the Moscow School. Chapman and Hall, London, 1993, pp. 251–274.Google Scholar
[Bar11]Barmak, J.. Algebraic Topology of Finite Topological Spaces and Applications. Springer, Berlin, 2011.Google Scholar
[Bre72]Bredon, G. E.. Introduction to Compact Transformation Groups. Academic Press, New York, 1972.Google Scholar
[CG11]Calcut, J. S. and Gompf, R. E.. Orbit spaces of gradient vector fields. Preprint, http://arxiv.org/abs/1105.5286v1 [math.DS], 2011, 18 pp.Google Scholar
[CGM12]Calcut, J. S., Gompf, R. E. and McCarthy, J. D.. On fundamental groups of quotient spaces. Topology Appl. 159 (2012), 322–330.CrossRefGoogle Scholar
[Con60]Conner, P. E.. Retraction properties of the orbit space of a compact topological transformation group. Duke Math. J. 27 (1960), 341–357.CrossRefGoogle Scholar
[DT58]Dold, A. and Thom, R.. Quasifaserungen und unendliche symmetrische Produkte. Ann. of Math. (2) 67 (1958), 239–281.Google Scholar
[GN85]Goel, S. K. and Neumann, D. A.. Completely unstable dynamical systems. Trans. Amer. Math. Soc. 291 (1985), 639–668.Google Scholar
[Hat02]Hatcher, A.. Algebraic Topology. Cambridge University Press, Cambridge, 2002.Google Scholar
[Hir76]Hirsch, M. W.. Differential Topology. Springer, New York, 1994, corrected reprint of the 1976 original.Google Scholar
[McCa79]McCann, R. C.. Dynamical systems whose orbit spaces are nearly Hausdorff. Proc. Amer. Math. Soc. 76 (1979), 258–262.Google Scholar
[McCo66]McCord, M. C.. Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J. 33 (1966), 465–474.CrossRefGoogle Scholar
[Mic56]Michael, E.. Continuous selections. I. Ann. of Math. (2) 63 (1956), 361–382.Google Scholar
[Mil63]Milnor, J.. Morse Theory, based on lecture notes by M. Spivak and R. Wells. Princeton University Press, Princeton, NJ, 1963.Google Scholar
[Mil65]Milnor, J. Lectures on the $h$-cobordism Theorem, notes by L.C. Siebenmann and J. Sondow. Princeton University Press, Princeton, NJ, 1965.Google Scholar
[MY57]Montgomery, D. and Yang, C. T.. The existence of a slice. Ann. of Math. (2) 65 (1957), 108–116.CrossRefGoogle Scholar
[Nak96]Nakayama, H.. On strongly Hausdorff flows. Fund. Math. 149 (1996), 167–170.CrossRefGoogle Scholar
[NO61]Nomizu, K. and Ozeki, H.. The existence of complete Riemannian metrics. Proc. Amer. Math. Soc. 12 (1961), 889–891.Google Scholar
[Oli76]Oliver, R.. A proof of the Conner conjecture. Ann. of Math. (2) 103 (1976), 637–644.Google Scholar
[Pal61]Palais, R. S.. On the existence of slices for actions of non-compact Lie groups. Ann. of Math. (2) 73 (1961), 295–323.Google Scholar
[Paj03]Pajitnov, A. V.. $C^0$-topology in Morse theory. Preprint, arXiv:math/0303195v2 [math.DG], 2003, 22 pp.Google Scholar
[Paj06]Pajitnov, A. V.. Circle-valued Morse Theory. de Gruyter, Berlin, 2006.CrossRefGoogle Scholar
[Rai09]Rainer, A.. Orbit projections as fibrations. Czechoslovak Math. J. 59(134) (2009), 529–538.Google Scholar