Intersection Cohomology pp 221-233

Part of the Progress in Mathematics book series (volume 50)

Problems and Bibliography on Intersection Homology

  • M. Goresky
  • R. MacPherson

Abstract

Before considering various possible extensions of intersection homology (such as intersection K-theory), we wish to reflect on small resolutions[D], [F]. For a small resolution π: \( \tilde X \to X \) there is a canonical isomorphism \( H_* (\tilde X) \cong IH_* (X) \). We might expect other “intersection functors” to satisfy a similar identity. A severe limitation on the existence of such functors is therefore provided by the existence of spaces which have two different small resolutions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography for Intersection Homology

General References

  1. [A]
    A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux Pervers, Soc. Math. de France, Asteristique #100 (1983).Google Scholar
  2. [B]
    J. L. Brylinski, (Co)-homologie d’intersection et faisceaux pervers. Sem. Bourbaki #585, Soc. Math. de France Asterisque 92–93 (1982), pp. 129–158.MathSciNetGoogle Scholar
  3. [C]
    M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), pp. 135–162.MATHCrossRefMathSciNetGoogle Scholar
  4. [D]
    M. Goresky and R. MacPherson, Intersection Homology II, Inv. Math. 72 (1983), pp. 77–130.MATHCrossRefMathSciNetGoogle Scholar
  5. [E]
    R. MacPherson, Intersection Homology (Hermann Weyl Lectures) to appear in Annals of Mathematics Studies, Princeton University Press.Google Scholar
  6. [F]
    R. MacPherson, Global questions in the topology of singular spaces, lecture delivered to ICM (Warsaw, Poland), Aug. 1983.Google Scholar
  7. [G]
    T. A. Springer, Quelques applications de la cohomologie d’ intersection. Séminaire Bourbaki #589, Soc. Math. de France Astérisque # 92–93 (1982) pp. 249–274.MathSciNetGoogle Scholar
  8. [H]
    Seminar on Intersection Homology, Bern Switzerland (A. Borel, ed). This seminar.Google Scholar
  9. [I]
    Analyse et Topologie sur les Espaces Singuljers, Soc. Math. de France Astérisque #101, 102.Google Scholar

Topological Papers

  1. [1]
    J. Cheeger, On the Hodge theory of Riemannian pseudo-manifolds. Proc. Symposia Pure Math. 36, Providence: AMS (1980), pp. 91–146.Google Scholar
  2. [2]
    A. Chou, The Dirac operator on spaces with conical singularities and positive scalar curvature, to appear.Google Scholar
  3. [3]
    W. Fulton and R. MacPherson, Categorical Framework for the study of singular spaces. Mem. Amer. Math. Soc. 243 (1981) A.M.S. Providence RI (1981).Google Scholar
  4. [4]
    M. Goresky, Intersection homology operations, to appear in Comment. Math. Helv.Google Scholar
  5. [5]
    M. Goresky and R. MacPherson, La dualité de Poincaré pour les espaces singuliers, C.R. Acad. Sci. t. 284 (Serie A) (1977), 1549–1551.MATHMathSciNetGoogle Scholar
  6. [6]
    M. Goresky and R. MacPherson, The Lefschetz fixed point theorem and intersection homology, to appear. (See also [H]).Google Scholar
  7. [7]
    M. Goresky and P. Siegel, Linking pairings on singular spaces, Comment. Math. Helv. 58 (1983) pp. 96–110.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    N. Habegger, Obstructions to immersions in intersection homology theory, preprint. Univ. of Geneva, 1983.Google Scholar
  9. [9]
    H. King, Intersection homology and homology manifolds. Topology 21 (1982).Google Scholar
  10. [10]
    H. King, Topological invariance of intersection homology without sheaves. (to appear in Topology).Google Scholar
  11. [11]
    R. MacPherson and K. Vilonen, Construction elementaire des faisceaux pervers, preprint, 1983.Google Scholar
  12. [12]
    M, Nagase, L2 cohomology and intersection cohomology of stratified spaces. Duke Math J. 50 (1983), pp. 329–368.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    W. Pardon, Cobordism groups of integral Witt spaces, to appear.Google Scholar
  14. [14]
    P. Siegel, Witt spaces, a geometric cycle theory for KO homology at odd primes. Ph.D. thesis (M.I.T), 1979. (to appear in Amer. J. of Math.)Google Scholar
  15. [15]
    J. Cheeger, M. Goresky, and R. MacPherson, L2-cohomology and intersection homology for singular algebraic varieties, proceedings of year in differential geometry, I.A.S., S. Yau, ed. (1981) Annals of Math. Studies, Princeton Univ. Press.Google Scholar
  16. [16]
    P. Deligne, Pureté de la cohomologie de MacPherson-Goresky, d’après un expose de O. Gabber, rédigé par P. Deligne, IHES preprint, Fév. 1981.Google Scholar
  17. [17]
    P. Deligne, Applications de la Pureté, Proc. of D-modules et singularités, Astérisque, to appear.Google Scholar
  18. [18]
    W. Fulton and R. Lazarsfeld, The numerical positivity of ample vector bundles. Ann. Math. 118 (1983) pp. 35–60.CrossRefMathSciNetGoogle Scholar
  19. [19]
    M. Goresky and R. MacPherson, Morse theory and intersection homology, in Soc. Math. de France Astérisque #101, 102 [I] pp. 135–192Google Scholar
  20. [20]
    M. Goresky and R. MacPherson, On the topology of complex algebraic maps, Algebraic Geometry-Proceedings, La Rabida, Springer Lect. Notes in Math. No. 961 (1982), pp. 119–129.CrossRefMathSciNetGoogle Scholar
  21. [21]
    W. C. Hsiang and V. Pati, L2 cohomology of normal algebraic surfaces, preprint, Princeton Univ., 1983.Google Scholar
  22. [22]
    J. Steenbrink, Mixed Hodge structures associated with isolated singularities, in Singularities, Proc. of Symp. in pure Math. vol. 40 #2 pp. 513–536. Amer. Math. Soc. Providence RI (1983).MathSciNetGoogle Scholar
  23. [23]
    J.L. Verdier, Specialization de faisceaux et Monodromie Moderee, in Analyse et Topologie sur les Espaces Singuljers, Soc. Math. de France Astérisque #101, 102 [I], pp. 332–364.Google Scholar
  24. [24]
    J. Bernstein, Algebraic theory of D-modules, proc. of “D-modules et singularities”, Astérisque, to appear.Google Scholar
  25. [25]
    J.-L. Brylinski, Modules holonomes a singularitiés régulières et filtration de Hodge I, Algebraic Geometry (Proceedings La Rabida) Lect. Notes in Math. No. 961, Springer-Verlag (1982) pp. 1–21.CrossRefMathSciNetGoogle Scholar
  26. [26]
    J.L. Brylinski, Modules holonomes à singularités régulières et filtrations de Hodge II, in Soc. Math. de France Astérisque #101, 102 [I] pp. 75–117.Google Scholar
  27. [27]
    J.L. Brylinski, Transformations canoniques, dualité projective, transformation de Fourier, et sommes trigonmetriques, to appear in Astérisque.Google Scholar
  28. [28]
    J.L. Brylinski, A. Dubson, and M. Kashiwara, Formule de l’indice pour les modules holnomes et obstruction d’Euler locale, C.R.A.S., (1981)Google Scholar
  29. [29]
    J.L. Brylinski, B. Malgrange, and J.-L. Verdier, Transformations de Fourier géométrique I, C.R. Acad. Sci. Paris 297 (1983), 55–58.MATHMathSciNetGoogle Scholar
  30. [30]
    A. Galligo, M. Granger, and P. Maisonobe, D-modules et faisceaux pervers dont le support singulier est un croisement normal, preprint, Univ. of Nice (1983).Google Scholar
  31. [31]
    M. Kashiwara, The Riemann-Hilbert problem, preprint, R.I.M.S. (1983).Google Scholar
  32. [32]
    Le D.T. and Z. Mebkhout, Introduction to linear differential systems, in Singularities Proc. of Symp. in pure Math. #40 (Part 2) pp. 31–64, Amer. Math. Soc. Providence, RI (1983).Google Scholar
  33. [33]
    T. Oda, Introduction to algebraic analysis on complex manifolds, Adv. Stud. in Pure Math.1 (1983) pp. 29–48, Jaoan Math. Soc.MathSciNetGoogle Scholar
  34. [34]
    A. Beilinson and J. Bernstein, Localisation des G-modules, C.R.A.S., No. 292 (1981), pp. 15–18.MATHMathSciNetGoogle Scholar
  35. [35]
    A. Borel, L2 cohomology and intersection cohomology of certain arithmetic varieties, Proc. of E. Noether Symposium at Bryn Mawr, Springer Verlag.Google Scholar
  36. [36]
    W. Borho and R. MacPherson, Representations des groupes de Weyl et homologie d’intersection pour les varieties de nilpotents, C.R. Acad. Sc. Paris, 292 (1981), pp. 707–710.MATHMathSciNetGoogle Scholar
  37. [37]
    W. Borho and R. MacPherson, Partial resolution of nilpotent varieties, in Soc. Math. de France Astérisque #101, 102 [I] pp. 23–74.Google Scholar
  38. [38]
    J. L. Brylinski and M. Kashiwara, Démonstration de la Conjecture de Kazhdan et Lusztig sur les modules de Verma (C.R. Acad. Sci. Paris t.291 (1980) pp. 373–376).MATHMathSciNetGoogle Scholar
  39. [39]
    J. L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. math. 64 (1981), pp. 387–410.MATHCrossRefMathSciNetGoogle Scholar
  40. [40]
    S. Gelfand, R. MacPherson, Verma modules and Schubert cells; a dictionary, Séminaire d’Algèbre. Lecture Notes in Math., vol 924, Berlin-Heidelberg-New York: Springer 1982, pp. 1–50.Google Scholar
  41. [41]
    D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Inv. Math. 53 (1979), pp. 165–184.MATHCrossRefMathSciNetGoogle Scholar
  42. [42]
    D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality, Proc. Symp. Pure Math. vol. 36, pp. 185–203, Amer. Math. Soc. (1980).MathSciNetGoogle Scholar
  43. [43]
    G. Lusztig, Representations of Hecke algebras and Coxeter groups (preprint MIT 1982)-edited by D. King.Google Scholar
  44. [44]
    G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math. 42 (1981), pp. 169–178.MATHCrossRefMathSciNetGoogle Scholar
  45. [45]
    G. Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc. Symp. Pure Math. vol. 37, Amer. Math. Soc. (1980), pp. 313–317.MathSciNetGoogle Scholar
  46. [46]
    G. Lusztig, Equivariant sheaves on reductive groups I, preprint, M.I.T. 1982.Google Scholar
  47. [47]
    G. Lusztig and D. Vogan, Singularities of closures of K-orbits on flag manifolds, Inv. Math. 71, No. 2 (1983), pp. 365–380.MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    G. Lusztig, Intersection homology complexes on reductive groups, Inv. Math. 75 (1984), 205–272.MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    T.A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Inv. Math. 36 (1976), pp. 173–207.MATHCrossRefMathSciNetGoogle Scholar
  50. [50]
    A. Zelevinsky, A p-adic analogue of the Kazhdan-Lusztig conjecture (in Russian), Funct. Anal. and its Appl. 15 (1981), pp. 9–21.Google Scholar
  51. [51]
    S. Zucker, Hodge theory with degenerating coefficients I, Ann. of Math. 109 (1979), pp. 415–476.CrossRefMathSciNetGoogle Scholar
  52. [52]
    S. Zucker, L2-cohomology of warped products and arithmetic groups. Inventiones Math. 70 (1982), pp. 169–218.MATHCrossRefMathSciNetGoogle Scholar
  53. [53]
    S. Zucker, L2-cohomology and intersection homology of locally symmetric varieties, Proc. Symp. pure math. #40 A.M.S. (1983).Google Scholar
  54. [54]
    S. Zucker, L2-cohomology and intersection homology of locally summetric varieties II. preprint (1983).Google Scholar
  55. [55]
    S. Zucker, Hodge theory and arithmetic groups, in Soc. Math. de France Astérisque #101, 102 [I] pp. 365–381.Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1984

Authors and Affiliations

  • M. Goresky
    • 1
  • R. MacPherson
    • 2
  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Department of MathematicsBrown UniversityProvidenceUSA

Personalised recommendations