Abstract
Within its traditional range of perversity parameters, intersection cohomology is a topological invariant of pseudomanifolds. This is no longer true once one allows superperversities, perversities with . In this case, intersection cohomology may depend on the choice of the stratification by which it is defined. Topological invariance also does not hold if one allows stratifications with codimension one strata. Nonetheless, both errant situations arise in important situations, the former in the Cappell-Shaneson superduality theorem and the latter in any discussion of pseudomanifold bordism. We show that while full invariance of intersection cohomology under restratification does not hold in this generality, it does hold up to restratifications that fix the the top stratum.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Borel, A. et al.: Intersection cohomology. Progress in Mathematics, vol. 50, Birkhauser, Boston, 1984
Bredon, G.: Topology and geometry. Springer-Verlag, New York, 1993
Cappell, S.E., Shaneson, J.L.: Singular spaces, characteristic classes, and intersection homology. Annals of Mathematics 134, 325–374 (1991)
Friedman, G.: Singular chain intersection homology for traditional and superperversities. To appear in Trans. Amer. Math. Soc.; see also http://www.arxiv.org/abs/math.GT/0407301
Friedman, G.: Intersection alexander polynomials. Topology 43, 71–117 (2004) see also http://www.arxiv.org/abs/math.GT/0307153
Goresky, M., MacPherson, R.: Intersection homology theory. Topology 19, 135–162 (1980)
Goresky, M., MacPherson, R.: Intersection homology II. Invent. Math. 72, 77–129 (1983)
Habegger, N., Saper, L.: Intersection cohomology of cs-spaces and Zeeman's filtration. Invent. Math. 105, 247–272 (1991)
Hughes, B., Weinberger, S.: Surgery and stratified spaces. Surveys on Surgery Theory Vol. 2 (Princeton, N.J.) (Sylvain Cappell, Andrew Ranicki, and Jonathan Rosenberg, eds.), Annals of Mathematical Studies, vol. 149, Princeton University Press, 2001, pp. 319–352
King, H.C.: Topological invariance of intersection homology without sheaves. Topology Appl. 20, 149–160 (1985)
Kirwan, F.: An introduction to intersection homology theory. Pitman Research Notes in Mathematics Series, vol. 187, Longman Scientific and Technical, Harlow, 1988
Kleiman, S.: The development of intersection homology theory. A Century of Mathematics in America Part II (Providence, R.I.), Hist. Math., vol. 2, Amer. Math. Soc., 1989, pp. 543–585
Lusztig, G.: Intersection cohomology methods in representation theory. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (Tokyo), Math. Soc. Japan, 1991, pp. 155–174
Maxim, L.: Intersection homology and Alexander modules of hypersurface complements. Thesis. See http://www.arxiv.org/abs/math.AT/0409412
Siegel, P.H.: Witt spaces: a geometric cycle theory for ko-homology at odd primes. American J. Math. 105, 1067–1105 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Friedman, G. Superperverse intersection cohomology: stratification (in)dependence. Math. Z. 252, 49–70 (2006). https://doi.org/10.1007/s00209-005-0844-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-005-0844-3