Skip to main content
SpringerLink
Log in

Blown-up intersection cochains and Deligne’s sheaves

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In a series of papers the authors introduced the so-called blown-up intersection cochains. These cochains are suitable to study products and cohomology operations of intersection cohomology of stratified spaces. The aim of this paper is to prove that the sheaf versions of the functors of blown-up intersection cochains are realizations of Deligne’s sheaves. This proves that Deligne’s sheaves can be incarnated at the level of complexes of sheaves by soft sheaves of perverse differential graded algebras. We also study Poincaré and Verdier dualities of blown-up intersections sheaves with the use of Borel–Moore chains of intersection.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borel, A., et al.: Intersection Cohomology, Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2008). (Notes on the seminar held at the University of Bern, Bern, 1983, Reprint of the 1984 edition)

    Google Scholar 

  2. Borel, A., Moore, J.C.: Homology theory for locally compact spaces. Mich. Math. J. 7, 137–159 (1960)

    Article  MathSciNet  Google Scholar 

  3. Bredon, G.E.: Sheaf Theory, second ed., Graduate Texts in Mathematics, vol. 170. Springer, New York (1997)

    Google Scholar 

  4. Brown, R.: Topology and Groupoids. BookSurge, LLC, Charleston (2006). (Third edition of ıt Elements of modern topology [McGraw-Hill, New York, 1968; MR0227979], With 1 CD-ROM (Windows, Macintosh and UNIX))

    MATH  Google Scholar 

  5. Chataur, D., Saralegi-Aranguren, M., Tanré, D.: Intersection Homology. General Perversities and Topological Invariance. arXiv:1602.03009. (To appear in Illinois J. of Math.) (2016)

  6. Chataur, D., Saralegi-Aranguren, M., Tanré, D.: Steenrod squares on intersection cohomology and a conjecture of M Goresky and W Pardon. Algebraic Geom. Topol. 16(4), 1851–1904 (2016)

    Article  MathSciNet  Google Scholar 

  7. Chataur, D., Saralegi-Aranguren, M., Tanré, D.: Singular decompositions of a cap product. Proc. Am. Math. Soc. 145(8), 3645–3656 (2017)

    Article  MathSciNet  Google Scholar 

  8. Chataur, D., Saralegi-Aranguren, M., Tanré, D.: Blown-Up Intersection Cohomology, An Alpine Bouquet of Algebraic Topology, Contemporary Mathematics, vol. 708, pp. 45–102. American Mathematical Society, Providence (2018)

    Book  Google Scholar 

  9. Chataur, D., Saralegi-Aranguren, M., Tanré, D.: Intersection cohomology, simplicial blow-up and rational homotopy. Mem. Am. Math. Soc. 254(1214), viii+108 (2018)

    MathSciNet  MATH  Google Scholar 

  10. Chataur, D., Saralegi-Aranguren, M., Tanré, D.: Poincaré duality with cap products in intersection homology. Adv. Math. 326, 314–351 (2018)

    Article  MathSciNet  Google Scholar 

  11. Deligne, P.: Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. (40), 5–57 (1971)

  12. Friedman, G.: Singular intersection homology. (2019). http://faculty.tcu.edu/gfriedman/IHbook.pdf

  13. Friedman, Greg: Singular chain intersection homology for traditional and super-perversities. Trans. Am. Math. Soc. 359(5), 1977–2019 (2007). (electronic)

    Article  MathSciNet  Google Scholar 

  14. Friedman, G.: Intersection homology and Poincaré duality on homotopically stratified spaces. Geom. Topol. 13(4), 2163–2204 (2009)

    Article  MathSciNet  Google Scholar 

  15. Friedman, G.: Intersection homology with general perversities. Geom. Dedicata 148, 103–135 (2010)

    Article  MathSciNet  Google Scholar 

  16. Friedman, G., McClure, J.E.: Cup and cap products in intersection (co)homology. Adv. Math. 240, 383–426 (2013)

    Article  MathSciNet  Google Scholar 

  17. Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19(2), 135–162 (1980)

    Article  MathSciNet  Google Scholar 

  18. Goresky, M., MacPherson, R.: Intersection homology. II. Invent. Math. 72(1), 77–129 (1983)

    Article  MathSciNet  Google Scholar 

  19. Hovey, M.: Intersection Homological Algebra, New Topological Contexts for Galois Theory and Algebraic Geometry (BIRS 2008), Geometry & Topology Monographs, vol. 16, pp. 133–150. Geometry & Topology Publisher, Coventry (2009)

    Google Scholar 

  20. Iversen, B.: Cohomology of Sheaves. Universitext. Springer, Berlin (1986)

    Book  Google Scholar 

  21. King, H.C.: Topological invariance of intersection homology without sheaves. Topol. Appl. 20(2), 149–160 (1985)

    Article  MathSciNet  Google Scholar 

  22. Laitinen, E.: End homology and duality. Forum Math. 8(1), 121–133 (1996)

    MathSciNet  MATH  Google Scholar 

  23. Pollini, G.: Intersection differential forms. Rend. Sem. Mat. Univ. Padova 113, 71–97 (2005)

    MathSciNet  MATH  Google Scholar 

  24. Quinn, F.: Homotopically stratified sets. J. Am. Math. Soc. 1(2), 441–499 (1988)

    Article  MathSciNet  Google Scholar 

  25. Saralegi-Aranguren, M.: de Rham intersection cohomology for general perversities. Ill. J. Math. 49(3), 737–758 (2005)

    Article  MathSciNet  Google Scholar 

  26. Spanier, E.: Singular homology and cohomology with local coefficients and duality for manifolds. Pac. J. Math. 160(1), 165–200 (1993)

    Article  MathSciNet  Google Scholar 

  27. Whitehead, G.W.: Elements of Homotopy Theory, Graduate Texts in Mathematics, vol. 61. Springer, New York (1978)

    Book  Google Scholar 

  28. Willard, S.: General Topology. Addison-Wesley Publishing Co., Reading (1970)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lamfa, Université de Picardie Jules Verne, 33, rue Saint-Leu, 80039, Amiens Cedex 1, France

    David Chataur

  2. Laboratoire de Mathématiques de Lens, EA 2462, Université d’Artois, SP18, rue Jean Souvraz, 62307, Lens Cedex, France

    Martintxo Saralegi-Aranguren

  3. Département de Mathématiques, UMR 8524, Université de Lille, 59655, Villeneuve d’Ascq Cedex, France

    Daniel Tanré

Authors
  1. David Chataur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martintxo Saralegi-Aranguren
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Tanré
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Tanré.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported through the program “Research in Pairs” at the Mathematisches Forschunginstitut Oberwolfach in 2016. The authors thank the MFO for its generosity and hospitality. The third author was also supported by the MINECO and FEDER research Project MTM2016-78647-P.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chataur, D., Saralegi-Aranguren, M. & Tanré, D. Blown-up intersection cochains and Deligne’s sheaves. Geom Dedicata 204, 315–337 (2020). https://doi.org/10.1007/s10711-019-00458-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-019-00458-w

Keywords

Mathematics Subject Classification (2010)

Search

Navigation