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On the toral rank conjecture and some consequences. (English) Zbl 1387.57049

Summary: The aim of this work is to improve the lower bound of the Puppe inequality, cf. [V. Puppe, Georgian Math. J. 16, No. 2, 369–379 (2009; Zbl 1190.57023)]. His Theorem 1.1 states that the sum of all Betti numbers of a well-behaved space \(X\) is at least equal to \(2n\), where \(n\) is the rank of an \(n\)-torus \(T^n\) acting almost freely on \(X\).

MSC:

57S10 Compact groups of homeomorphisms
57R20 Characteristic classes and numbers in differential topology
57R91 Equivariant algebraic topology of manifolds
55P91 Equivariant homotopy theory in algebraic topology

Citations:

Zbl 1190.57023
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References:

[1] C. Allday and S. Halperin, Lie group actions on spaces of finite rank, Quart. J. Math. Oxford (2) 29, pp. 63-76, (1978). · Zbl 0395.57024
[2] C. Allday and V. Puppe, Cohomological methods in transformation groups, volume 32 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, (1993). · Zbl 0799.55001
[3] C. Allday and V. Puppe, On the localization theorem at the cochain level and free torus actions, Algebraic topology Göttingen 84, Procedings, Springer lect. Notes in Math 1172, pp. 1-16, (1985). · Zbl 0596.57023
[4] M. Amann, Cohomological consequences of almost free torus actions arXiv:1204.6276, Vol. 1 27, April (2012).
[5] A. Borel, Seminar on transformation groups Ann. of math Studies n° 46. Princeton New Jersey.
[6] E. H. Brown, Twisted tensor product I, Ann. of Math vol. 69, pp. 223-246, (1959). · Zbl 0199.58201
[7] Y. Félix, S. Halperin, and J.-C. Thomas, Rational homotopy theory, volume 205 of Graduate Texts in Mathematics. Springer-Verlag, New York, (2001). · Zbl 0961.55002
[8] Y. Félix, J. Oprea, and D. Tanré, Algebraic models in geometry, volume 17 of Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, (2008). · Zbl 1149.53002
[9] S. Halperin, Finiteness in the minimal models of Sullivan, Trans. A. M. S. 230, pp. 173-199, (1977). · Zbl 0364.55014
[10] S. Halperin, Rational homotopy and torus actions, London Math. Soc. Lecture Note Series 93, Cambridge Univ. Press, pp. 293-306, (1985). · Zbl 0562.57015
[11] M. R. Hilali, Sur la conjecture de Halperin relative au rang torique. Bull. Belg. Math. Soc. Simon Stevin 7, No. 2, pp. 221-227, (2000). · Zbl 0973.55008
[12] M. R. Hilali, Actions du tore 𝑇 𝑛 sur les espaces simplement connexes. Thèse à l’Université catholique de Louvain, (1990).
[13] W. Y. Hsiang, Cohomology theory of topological transformation groups, Berlin-Heidelberg-New York, Springer, (1975). · Zbl 0429.57011
[14] I. M. James, reduced product spaces, Ann. of math 82, pp. 170-197, (1995).
[15] V. Puppe, Multiplicative aspects of the Halperin-Carlsson conjecture, Georgian Mathematical Journal, (2009), 16:2, pp. 369-379, arXiv 0811.3517. · Zbl 1190.57023
[16] V. Puppe, On the torus rank of topological spaces, Proceding Baker, (1987).
[17] I. R. Shafarevich, Basic Algebraic Geometry, 2 Vols., Springer, (1994). · Zbl 0797.14002
[18] Y. Ustinovskii, On almost free torus actions and Horroks conjecture, (2012), arXiv 1203.3685v2.
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