Poincaré - Verdier duality in o-minimal structures
Annales de l'Institut Fourier, Volume 60 (2010) no. 4, p. 1259-1288

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.

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.

DOI : https://doi.org/10.5802/aif.2554
Classification:  03C64,  55N30
Keywords: O-minimal structures, sheaf cohomology 
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     pages = {1259-1288},
     doi = {10.5802/aif.2554},
     mrnumber = {2722241},
     zbl = {pre05793932},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_4_1259_0}
}

Edmundo, Mário J.; Prelli, Luca. Poincaré - Verdier duality in o-minimal structures. Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1259-1288. doi : 10.5802/aif.2554. http://www.numdam.org/item/AIF_2010__60_4_1259_0/

[1] Berarducci, A.; Edmundo, M.; Otero, M. Corrigendum to “Transfer methods for o-minimal topology”, J. Symb. Logic, Tome 72 (2007) no. 3, p. 1079-1080 | Article | MR 2354917 | Zbl 1119.03334

[2] Berarducci, A.; Otero, M. Transfer methods for o-minimal topology, J. Symb. Logic, Tome 68 (2003) no. 3, pp. 785-794 | Article | MR 2000077 | Zbl 1060.03059

[3] Bochnak, J.; Coste, M.; Roy, M.-F. Real Algebraic Geometry, Springer-Verlag (1998) | MR 1659509 | Zbl 0912.14023

[4] Bredon, G. Sheaf theory, Springer-Verlag, Second Edition (1997) | MR 1481706 | Zbl 0874.55001

[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 2432120

[6] Coste, M.; Roy, M.-F. La topologie du spectre réel, in Ordered fields and real algebraic geometry, Contemporary Mathematics, Tome 8 (1982), pp. 27-59 | MR 653174 | Zbl 0485.14007

[7] Delfs, H. The homotopy axiom in semi-algebraic sheaf cohomology, J. reine angew. Maths., Tome 355 (1985), pp. 108-128 | Article | MR 772485 | Zbl 0543.55004

[8] Delfs, H. Homology of locally semialgebraic spaces, Springer-Verlag, LNM 1484 (1991) | MR 1176311 | Zbl 0751.14033

[9] Delfs, H.; Knebusch, M. On the homology of algebraic varieties over real closed fields, J. reine u.angew. Math., Tome 335 (1982), pp. 122-163 | Article | MR 667464 | Zbl 0484.14006

[10] Dold, A. Lectures on algebraic topology, Springer-Verlag (1995) | MR 1335915 | Zbl 0872.55001

[11] Edmundo, M. Covering definable manifolds by open definable subsets, Lecture Notes in Logic, In: Logic Colloquium ’05, Tome 28 (2008) ((ed., C. Dimitracopoulos et al.) Cambridge University Press) | MR 2395801 | Zbl 1135.03012

[12] Edmundo, M.; Jones, G.; Peatfield, N. Sheaf cohomology in o-minimal structures, J. Math. Logic, Tome 6 (2006) no. 2, pp. 163-179 | Article | MR 2317425 | Zbl 1120.03024

[13] Edmundo, M.; Otero, M. Definably compact abelian groups, J. Math. Log., Tome 4 (2004) no. 2, pp. 163-180 | Article | MR 2114966 | Zbl 1070.03025

[14] Edmundo, M.; Woerheide, A. Comparation theorems for o-minimal singular (co)homology, Trans. Amer. Math. Soc., Tome 360 (2008) no. 9, pp. 4889-4912 | Article | MR 2403708 | Zbl 1153.03010

[15] Edmundo, M.; Woerheide, A. The Lefschetz coincidence theorem in o-minimal expansions of fields, Topology Appl., Tome 156 (2009) no. 15, pp. 2470-2484 | Article | MR 2546949 | Zbl 1179.55002

[16] Godement, R. Théorie des faisceaux, Hermann (1958) | MR 102797

[17] Iversen, B. Cohomology of sheaves, Springer Verlag (1986) | MR 842190

[18] Kashiwara, M.; Schapira, P. Sheaves on manifolds, Springer Verlag (1990) | MR 1074006 | Zbl 0709.18001

[19] Kashiwara, M.; Schapira, P. Ind-sheaves, Astérisque Tome 271 (2001) (136 pp) | MR 1827714 | Zbl 0993.32009

[20] Kashiwara, M.; Schapira, P. Categories and sheaves, Springer Verlag (2005) | MR 2182076 | Zbl 1118.18001

[21] Pillay, A. Sheaves of continuous definable functions, J. Symb. Logic, Tome 53 (1988) no. 4, pp. 1165-1169 | Article | MR 973106 | Zbl 0683.03018

[22] Prelli, L. Sheaves on subanalytic sites, Rend. Sem. Mat. Univ. Padova, Tome 120 (2008), pp. 167-216 | Numdam | MR 2492657 | Zbl 1171.32002

[23] Van Den Dries, L. Tame topology and o-minimal structures, Cambridge University Press (1998) | MR 1633348 | Zbl 0953.03045

[24] Wilkie, A. Covering definable open subsets by open cells, Cuvillier Verlag, 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.) (2005)

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