|
Rational homotopy theory for non-simply connected spaces
Author(s):
Antonio
Gómez-Tato;
Stephen
Halperin;
Daniel
Tanré
Journal:
Trans. Amer. Math. Soc.
352
(2000),
1493-1525.
MSC (1991):
Primary 55P62, 55R25
Posted:
November 18, 1999
MathSciNet review:
1653355
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.
References:
- 1.
- A.K. Bousfield, V.K.A.M. Gugenheim, On PL DeRham theory and rational homotopy type, Memoirs of the Amer. Math. Soc. 179, (1976).MR 54:13906
- 2.
- A.K. Bousfield, D.M. Kan, Homotopy Limits, Completions and Localisations, Lecture Notes 304, (1972), Springer Verlag. MR 51:1825
- 3.
- E.H. Brown, R.H. Szczarba, Continuous cohomology and real homotopy type, Trans. Amer. Math. Soc. 311, (1989), 57-106. MR 89f:55005
- 4.
- E.H. Brown, R.H. Szczarba, Rational and real homotopy theory with arbitrary fundamental groups, Duke Math. J. 71, (1993), 299-316. MR 94i:55017
- 5.
- E.H. Brown, R.H. Szczarba, Continuous cohomology and real homotopy type II, Astérisque 191, (1990), 45-70. MR 92d:55004
- 6.
- P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Inventiones Math. 29, (1975), 245-274. MR 52:3584
- 7.
- A. Dold, Partitions of unity in the theory of fibrations, Ann. of Math. 78, (1963), 223-255. MR 27:5264
- 8.
- S. Eilenberg, S. MacLane, Relations between homology and homotopy groups of spaces, Ann. of Math. 46, (1945), 480-509. MR 7:137g
- 9.
- Y. Félix, S. Halperin, Rational LS-category and its applications, Trans. Amer. Math. Soc. 273, (1982), 1-37. MR 84h:55011
- 10.
- Y. Félix, S. Halperin, J.-C. Thomas, Rational Homotopy and its Applications, in preparation.
- 11.
- A. Gómez-Tato, Théorie de Sullivan pour la cohomologie à coefficients locaux, Trans. Amer. Math. Soc. 330, (1992), 235-305. MR 92f:55012
- 12.
- J. O. Goyo, The Sullivan model of the homotopy-fixed-point set, Thesis, University of Toronto, (1989).
- 13.
- S. Halperin, Lectures on minimal models, Mémoires S.M.F. Nouvelle série 9-10, (1983). MR 85i:55009
- 14.
- S. Halperin, J. Stasheff, Obstructions to homotopy equivalences, Advances in Math. 32, (1979), 233-279. MR 80j:55016
- 15.
- S. Halperin, D. Tanré, Fibrés
 et Homotopie filtrée, Illinois J. of Math., 34, (1990), 284-324.MR 91g:55018 - 16.
- J.P. May, Simplicial objects in algebraic topology, Midway reprint, University Chicago Press, Chicago 1982. MR 93m:55025
- 17.
- G.D. Mostow, Factor spaces of solvable groups, Ann. of Math. 60, (1954), 1-27. MR 15:853g; MR 19:752d
- 18.
- D. Quillen, Rational homotopy theory, Ann. of Math. 90, (1969), 205-295. MR 41:2678
- 19.
- S. Schlessinger, J. Stasheff, Deformation theory and rational homotopy type, Preprint.
- 20.
- D. Sullivan, Geometric Topology, part I; Localization, Periodicity, and Galois Symmetry, M.I.T. Press, Cambridge, (1970). MR 58:13006a
- 21.
- D. Sullivan, Infinitesimal Computations in Topology, Publications I.H.E.S. 47, (1977), 269-331.MR 58:31119
- 22.
- D. Tanré, Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Math. 1025, Springer-Verlag, (1983).MR 86b:55010
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (1991):
55P62, 55R25
Retrieve articles in all Journals with
MSC (1991):
55P62, 55R25
Additional Information:
Antonio
Gómez-Tato
Affiliation:
Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, Santiago de Compostela, 15706 España
Email:
agtato@zmat.usc.es
Stephen
Halperin
Affiliation:
College of Computer, Mathematical and Physical Science, University of Maryland, College Park, Maryland 20742-3281
Email:
shalper@deans.umd.edu
Daniel
Tanré
Affiliation:
U.R.A. CNRS 0751, U.F.R. de Mathématiques, Université des Sciences et Technologies de Lille, 59655 Villeneuve d'Ascq Cedex, France
Email:
Daniel.Tanre@agat.univ-lille1.fr
DOI:
10.1090/S0002-9947-99-02463-0
PII:
S 0002-9947(99)02463-0
Keywords:
Fundamental group,
simplicial set,
Sullivan model,
rational homotopy,
rational vector bundle,
linear connections
Received by editor(s):
November 20, 1997
Posted:
November 18, 1999
Additional Notes:
The first author's research was partially supported by an ``Action Intégrée'' and a Xunta of Galicia grant, the second author's research was partially supported by a NATO grant and a NSERC grant and the third author's research was partially supported by a NATO grant.
Copyright of article:
Copyright
2000,
American Mathematical Society
|
