Poincaré - Verdier duality in o-minimal structures
[Dualité de Poincaré - Verdier pour les structures o-minimales]
Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1259-1288.

On démontre une dualité de Poincaré - Verdier dans le cadre de la cohomologie o-minimale des faisceaux avec support compact et définissable sur des espaces définissablement normaux, définissablement localement compacts dans une structure o-minimale arbitraire.

Here we prove a Poincaré - Verdier duality theorem for the o-minimal sheaf cohomology with definably compact supports of definably normal, definably locally compact spaces in an arbitrary o-minimal structure.

DOI : 10.5802/aif.2554
Classification : 03C64, 55N30
Keywords: O-minimal structures, sheaf cohomology
Mot clés : Structure o-minimale, cohomologie des faisceaux
Edmundo, Mário J. 1 ; Prelli, Luca 2

1 Universidade Aberta & CMAF Universidade de Lisboa Av. Prof. Gama Pinto 2 1649-003 Lisboa (Portugal)
2 Università di Padova Dipartimento di Matematica Pura ed Applicata Via Trieste 63 35121 Padova (Italy)
@article{AIF_2010__60_4_1259_0,
     author = {Edmundo, M\'ario J. and Prelli, Luca},
     title = {Poincar\'e - {Verdier} duality in o-minimal structures},
     journal = {Annales de l'Institut Fourier},
     pages = {1259--1288},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     doi = {10.5802/aif.2554},
     mrnumber = {2722241},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2554/}
}
TY  - JOUR
AU  - Edmundo, Mário J.
AU  - Prelli, Luca
TI  - Poincaré - Verdier duality in o-minimal structures
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 1259
EP  - 1288
VL  - 60
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2554/
DO  - 10.5802/aif.2554
LA  - en
ID  - AIF_2010__60_4_1259_0
ER  - 
%0 Journal Article
%A Edmundo, Mário J.
%A Prelli, Luca
%T Poincaré - Verdier duality in o-minimal structures
%J Annales de l'Institut Fourier
%D 2010
%P 1259-1288
%V 60
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2554/
%R 10.5802/aif.2554
%G en
%F AIF_2010__60_4_1259_0
Edmundo, Mário J.; Prelli, Luca. Poincaré - Verdier duality in o-minimal structures. Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1259-1288. doi : 10.5802/aif.2554. http://www.numdam.org/articles/10.5802/aif.2554/

[1] Berarducci, A.; Edmundo, M.; Otero, M. Corrigendum to “Transfer methods for o-minimal topology”, J. Symb. Logic, Volume 72 (2007) no. 3, pp. 1079-1080 | DOI | MR | Zbl

[2] Berarducci, A.; Otero, M. Transfer methods for o-minimal topology, J. Symb. Logic, Volume 68 (2003) no. 3, pp. 785-794 | DOI | MR | Zbl

[3] Bochnak, J.; Coste, M.; Roy, M.-F. Real Algebraic Geometry, Springer-Verlag, 1998 | MR | Zbl

[4] Bredon, G. Sheaf theory, Second Edition, Springer-Verlag, 1997 | MR | Zbl

[5] Coste, M. An introduction to o-minimal geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000). Available in RAAG preprint server (http://ihp-raag.org/) | MR

[6] Coste, M.; Roy, M.-F. La topologie du spectre réel, in Ordered fields and real algebraic geometry, Contemporary Mathematics, Volume 8 (1982), pp. 27-59 | MR | Zbl

[7] Delfs, H. The homotopy axiom in semi-algebraic sheaf cohomology, J. reine angew. Maths., Volume 355 (1985), pp. 108-128 | DOI | MR | Zbl

[8] Delfs, H. Homology of locally semialgebraic spaces, LNM 1484, Springer-Verlag, 1991 | MR | Zbl

[9] Delfs, H.; Knebusch, M. On the homology of algebraic varieties over real closed fields, J. reine u.angew. Math., Volume 335 (1982), pp. 122-163 | DOI | MR | Zbl

[10] Dold, A. Lectures on algebraic topology, Springer-Verlag, 1995 | MR | Zbl

[11] Edmundo, M. Covering definable manifolds by open definable subsets, In: Logic Colloquium ’05, 28, Lecture Notes in Logic, 2008 (ed., C. Dimitracopoulos et al.) Cambridge University Press | MR | Zbl

[12] Edmundo, M.; Jones, G.; Peatfield, N. Sheaf cohomology in o-minimal structures, J. Math. Logic, Volume 6 (2006) no. 2, pp. 163-179 | DOI | MR | Zbl

[13] Edmundo, M.; Otero, M. Definably compact abelian groups, J. Math. Log., Volume 4 (2004) no. 2, pp. 163-180 | DOI | MR | Zbl

[14] Edmundo, M.; Woerheide, A. Comparation theorems for o-minimal singular (co)homology, Trans. Amer. Math. Soc., Volume 360 (2008) no. 9, pp. 4889-4912 | DOI | MR | Zbl

[15] Edmundo, M.; Woerheide, A. The Lefschetz coincidence theorem in o-minimal expansions of fields, Topology Appl., Volume 156 (2009) no. 15, pp. 2470-2484 | DOI | MR | Zbl

[16] Godement, R. Théorie des faisceaux, Hermann, 1958 | MR

[17] Iversen, B. Cohomology of sheaves, Springer Verlag, 1986 | MR

[18] Kashiwara, M.; Schapira, P. Sheaves on manifolds, Springer Verlag, 1990 | MR | Zbl

[19] Kashiwara, M.; Schapira, P. Ind-sheaves, 271, Astérisque, 2001 (136 pp) | Numdam | MR | Zbl

[20] Kashiwara, M.; Schapira, P. Categories and sheaves, Springer Verlag, 2005 | MR | Zbl

[21] Pillay, A. Sheaves of continuous definable functions, J. Symb. Logic, Volume 53 (1988) no. 4, pp. 1165-1169 | DOI | MR | Zbl

[22] Prelli, L. Sheaves on subanalytic sites, Rend. Sem. Mat. Univ. Padova, Volume 120 (2008), pp. 167-216 | Numdam | MR | Zbl

[23] van den Dries, L. Tame topology and o-minimal structures, Cambridge University Press, 1998 | MR | Zbl

[24] Wilkie, A. Covering definable open subsets by open cells, In: O-minimal Structures, Proceedings of the RAAG Summer School Lisbon 2003, Lecture Notes in Real Algebraic and Analytic Geometry (M. Edmundo, D. Richardson and A. Wilkie eds.), Cuvillier Verlag, 2005

[25] Woerheide, A. O-minimal homology, PhD. Thesis (1996), University of Illinois at Urbana-Champaign

Cité par Sources :