Invariance of $o$-minimal cohomology with definably compact supports

Edmundo, Mário J.; Prelli, Luca

Invariance of

o

-minimal cohomology with definably compact supports

Edmundo, Mário J. ; Prelli, Luca

Confluentes Mathematici, Tome 7 (2015), p. 35-53 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé

In this paper we find general criteria for invariance and finiteness results for $o$ -minimal cohomology in an arbitrary $o$ -minimal structure. We apply our criteria and obtain new invariance and finiteness results for $o$ -minimal cohomology in $o$ -minimal expansions of ordered groups and for the $o$ -minimal cohomology of definably compact definable groups in arbitrary $o$ -minimal structures.

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/cml.17
Classification: 03C64, 55N30

@article{CML_2015__7_1_35_0,
     author = {Edmundo, M\'ario J. and Prelli, Luca},
     title = {Invariance of $o$-minimal cohomology with definably compact supports},
     journal = {Confluentes Mathematici},
     volume = {7},
     year = {2015},
     pages = {35-53},
     doi = {10.5802/cml.17},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CML_2015__7_1_35_0}
}

Edmundo, Mário J.; Prelli, Luca. Invariance of $o$-minimal cohomology with definably compact supports. Confluentes Mathematici, Tome 7 (2015) pp. 35-53. doi : 10.5802/cml.17. http://gdmltest.u-ga.fr/item/CML_2015__7_1_35_0/

Bibliographie

[1] A. Berarducci and A. Fornasiero O-minimal cohomology: finiteness and invariance results J. Math. Logic 9 (2) (2009) 167–182. | MR 2679438 | Zbl 1236.03030

[2] A. Berarducci and M. Otero O-minimal fundamental group, homology and manifolds J. London Math. Soc. 65 (2) (2002) 257–270. | MR 1883182 | Zbl 1013.03044

[3] J. Bochnak, M. Coste and M-F. Roy Real algebraic geometry Springer-Verlag 1998. | MR 1659509 | Zbl 0633.14016

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

[5] M. Coste An introduction to $o$ -minimal geometry Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa (2000).

[6] M. Carral and M. Coste Normal spectral spaces and their dimensions J. Pure and Appl. Algebra 30 (3) (1983) 227–235. | MR 724034 | Zbl 0525.14015

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

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

[9] L. van den Dries Tame topology and $o$ -minimal structures Cambridge University Press 1998. | MR 1633348 | Zbl 0953.03045

[10] M. Edmundo, G. Jones and N. Peatfield Sheaf cohomology in $o$ -minimal structures J. Math. Logic 6 (2) (2006) 163–179. | MR 2317425 | Zbl 1120.03024

[11] M. Edmundo, M. Mamino and L. Prelli On definably proper maps Fund. Math. (to appear).

[12] M. Edmundo and M. Otero Definably compact abelian groups J. Math. Logic 4 (2) (2004) 163–180. | MR 2114966 | Zbl 1070.03025

[13] M. Edmundo and L. Prelli Poincaré - Verdier duality in $o$ -minimal structures Ann. Inst. Fourier Grenoble 60 (4) (2010) 1259–1288. | Numdam | MR 2722241 | Zbl 1235.03070

[14] M. Edmundo and L. Prelli Sheaves on T-topologies J. Math. Soc. Japan (to appear).

[15] M. Edmundo and G. Terzo A note on generic subsets of definable groups Fund. Math. 215 (1) (2011) 53–65. | MR 2851701 | Zbl 1244.03115

[16] M. Edmundo and A. Woerheide Comparision theorems for $o$ -minimal singular (co)homology Trans. Amer. Math. Soc. 360 (9) (2008) 4889–4912. | MR 2403708 | Zbl 1153.03010

[17] P. Eleftheriou A semi-linear group which is not affine Ann. Pure Appl. Logic 156 (2008) 287 – 289. | MR 2484486 | Zbl 1155.03020

[18] P. Eleftheriou, Y. Peterzil and J. Ramakrishnan Interpretable groups are definable J. Math. Log. 14 1450002 (2014) [47 pages]. | MR 3225588

[19] A. Fornasiero O-minimal spectrum Unpublished, 33pp, 2006. http://www.dm.unipi.it/~fornasiero/articles/spectrum.pdf

[20] R. Godement Théorie des faisceaux Hermann 1958. | MR 102797

[21] B. Iversen Cohomology of sheaves Springer Verlag 1986. | MR 842190 | Zbl 1272.55001

[22] M. Kashiwara and P. Schapira Sheaves on manifolds Springer Verlag 1990. | MR 1074006 | Zbl 0709.18001

[23] M. Otero A survey on groups definable in $o$ -minimal structures in Model Theory with Applications to Algebra and Analysis, vol. 2, Editors: Z. Chatzidakis, D. Macpherson, A. Pillay and A. Wilkie, LMS LNS 350 Cambridge Univ. Press (2008) 177–206 | MR 2436142 | Zbl 1166.03016

[24] Y. Peterzil and C. Steinhorn Definable compactness and definable subgroups of $o$ -minimal groups J. London Math. Soc. 59 (2) (1999) 769–786. | MR 1709079 | Zbl 0935.03047

[25] A. Pillay On groups and fields definable in $o$ -minimal structures J. Pure Appl. Algebra 53 (1988) 239 – 255. | MR 961362 | Zbl 0662.03025

[26] A. Pillay Sheaves of continuous definable functions J. Symb. Logic 53 (4) (1988) 1165–1169. | MR 973106 | Zbl 0683.03018