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O-minimal spectra, infinitesimal subgroups and cohomology

Published online by Cambridge University Press:  12 March 2014

Alessandro Berarducci
Alessandro Berarducci*
Affiliation:
Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy. E-mail: berardu@mail.dm.unipi.it URL: www.dm.unipi.it/~berardu
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Abstract

By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G00 such that the quotient G/G00, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G ↦ G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum of G. We prove that G/G00 is a topological quotient of . We thus obtain a natural homomorphism Ψ* from the cohomology of G/G00 to the (Čech-)cohomology of . We show that if G00 satisfies a suitable contractibility conjecture then is acyclic in Čech cohomology and Ψ is an isomorphism. Finally we prove the conjecture in some special cases.

Keywords

o-minimalitydefinable grouptype-definable groupLie group
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1177 - 1193
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Berarducci, A., Zero-groups and maximal tori, Logic Colloquium '04 (Andretta, A., Kearnes, K., and Zambella, D., editors), Lecture Notes in Logic, vol. 29, ASL, 2007, pp. 33–45.Google Scholar
[2]Berarducci, A. and Otero, M., Intersection theory for o-minimal manifolds, Annals of Pure and Applied Logic, vol. 107 (2001), no. 1-3, pp. 87–119.CrossRefGoogle Scholar
[3]Berarducci, A., Otero, M., Peterzil, Y., and Pillay, A., A descending chain condition for groups definable in o-minimal structures, Annals of Pure and Applied Logic, vol. 134 (2005), pp. 303–313.CrossRefGoogle Scholar
[4]Bredon, G. E., Sheaf theory, second ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997.CrossRefGoogle Scholar
[5]Carral, M. and Coste, M., Normal spectral spaces and their dimension, Journal of Pure and Applied Algebra, vol. 30 (1983), pp. 227–235.CrossRefGoogle Scholar
[6]Coste, M. and Roy, M.-F., La topologie du Spectre Reel, Ordered fields and real algebraic geometry (Dubois, D. W. and Recio, T., editors), Contemporary Mathematics, vol. 8, American Mathematical Society, 1982, pp. 27–59.CrossRefGoogle Scholar
[7]Delfs, H., Kohomologie affiner semialgebraischer Räume, Dissertation, Regensburg, 1980.Google Scholar
[8]Delfs, H., The homotopy axiom in semialgebraic cohomology, Journal für die reine und angewandte Mathematik, vol. 355 (1985), pp. 108–128.Google Scholar
[9]Delfs, H., Homology of locally semialgebraic spaces, Lecture Notes in Mathematics, vol. 1484, Springer-Verlag, Berlin, 1991.CrossRefGoogle Scholar
[10]Delfs, H. and Knebusch, M., On the homology of algebraic varieties over real closed fields, Journal für die reine und angewandte Mathematik, vol. 335 (1982), pp. 122–163.Google Scholar
[11]Edmundo, M., A remark on divisibility of definable groups, Mathematical Logic Quarterly, vol. 51 (2005), no. 6, pp. 639–641.CrossRefGoogle Scholar
[12]Edmundo, M., Jones, G. O., and Peatfield, N. J., Sheaf cohomology in o-minimal structures, Journal of Mathematical Logic, vol. 6 (2006), no. 2, pp. 163–179.CrossRefGoogle Scholar
[13]Edmundo, M. and Otero, M., Definably compact abelian groups, Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 163–180.CrossRefGoogle Scholar
[14]Godement, R., Topologie algébrique et théorie des faisceaux, Actualités scientifiques et industrielles, vol. XIII, Publications de l'Institut de Mathématique de l'Université de Strasbourg, no. 1252, Hermann, Paris, troisième ed., 1973.Google Scholar
[15]Halmos, P. R., Measure theory, Springer-Verlag, 1974.Google Scholar
[16]Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures and the NIP, Journal of the American Mathematical Society, (2007), pp. 1–34, Article electronically published.Google Scholar
[17]Knebusch, M., Semialgebraic topology in the last ten years, Real algebraic geometry (Rennes, 1991), Lecture Notes in Mathematics, vol. 1524, Springer, Berlin, 1992, pp. 1–36.CrossRefGoogle Scholar
[18]Lane, S. Mac, Homology, reprint of the first ed., Die Grundlehren der mathematischen Wissenschaften, vol. 114, Springer-Verlag, 1967.Google Scholar
[19]Lascar, D. and Pillay, A., Hyperimaginaries and automorphism groups, this JOurnal, vol. 66 (2001), pp. 127–143.Google Scholar
[20]Munkres, J. R., Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984.Google Scholar
[21]Otero, M., A survey on groups definable in o-minimal structures, preprint, 2006.Google Scholar
[22]Prterzil, Y. and Pillay, A., Generic sets in definably compact groups, Fundamenta Mathematicae, vol. 193 (2007), pp. 153–170.CrossRefGoogle Scholar
[23]Peterzil, Y., Pillay, A., and Starchenko, S., Linear groups definable in o-minimal structures, Journal of Algebra, vol. 247 (2002), no. 1, pp. 1–23.CrossRefGoogle Scholar
[24]Peterzil, Y. and Starchenko, S., Uniform definability of the Weiestrass P -functions and generalized tori of dimension one, Selecta Mathematica, New Series, vol. 10 (2004), pp. 525–550.CrossRefGoogle Scholar
[25]Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society, vol. 59 (1999), pp. 769–786.CrossRefGoogle Scholar
[26]Pillay, A., On groups and rings definable in o-minimal structures, Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239–255.CrossRefGoogle Scholar
[27]Pillay, A., Sheaves of continuous definable functions, this Journal, vol. 53 (1988), no. 4, pp. 1165–1169.Google Scholar
[28]Pillay, A., Type-definability, compact Lie groups, and o-minimality, Journal of Mathematical Logic, vol. 4 (2004), pp. 147–162.CrossRefGoogle Scholar
[29]Razeni, V., One-dimensional groups over an o-minimal structure, Annals of Pure and Applied Logic, vol. 53 (1991), no. 3, pp. 269–277.CrossRefGoogle Scholar
[30]Shelah, S., Minimal bounded index subgroup for dependent theories, preprint, 2005, ArXiv: math. L0/0603652vl.Google Scholar
[31]Spanier, E. H., Algebraic topology, Springer-Verlag, New York-Berlin, 1981, corrected reprint.CrossRefGoogle Scholar
[32]van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Notes, vol. 248, Cambridge University Press, 1998.CrossRefGoogle Scholar
[33]Woerheide, A., O-minimal homology, Ph.D. thesis, University of Illinois at Urbana–Champaign, 1996.Google Scholar