Singular chain intersection homology for traditional and super-perversities

Friedman, Greg

doi:10.1090/S0002-9947-06-03962-6

Singular chain intersection homology for traditional and super-perversities
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Trans. Amer. Math. Soc. 359 (2007), 1977-2019 Request permission

Abstract:

We introduce a singular chain intersection homology theory which generalizes that of King and which agrees with the Deligne sheaf intersection homology of Goresky and MacPherson on any topological stratified pseudomanifold, compact or not, with constant or local coefficients, and with traditional perversities or superperversities (those satisfying $\bar p(2)>0$). For the case $\bar p(2)=1$, these latter perversities were introduced by Cappell and Shaneson and play a key role in their superduality theorem for embeddings. We further describe the sheafification of this singular chain complex and its adaptability to broader classes of stratified spaces.

References

Markus Banagl, Extending intersection homology type invariants to non-Witt spaces, Mem. Amer. Math. Soc. 160 (2002), no. 760, x+83. MR 1937924, DOI 10.1090/memo/0760
Intersection cohomology, Progress in Mathematics, vol. 50, Birkhäuser Boston, Inc., Boston, MA, 1984. Notes on the seminar held at the University of Bern, Bern, 1983; Swiss Seminars. MR 788171
Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
Sylvain E. Cappell and Julius L. Shaneson, Singular spaces, characteristic classes, and intersection homology, Ann. of Math. (2) 134 (1991), no. 2, 325–374. MR 1127478, DOI 10.2307/2944349
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Greg Friedman, Superperverse intersection cohomology: stratification (in)dependence, Math. Z. 252 (2006), no. 1, 49–70. MR 2209151, DOI 10.1007/s00209-005-0844-3
Greg Friedman, Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata, Topology Appl. 134 (2003), no. 2, 69–109. MR 2009092, DOI 10.1016/S0166-8641(03)00088-9
Greg Friedman, Intersection Alexander polynomials, Topology 43 (2004), no. 1, 71–117. MR 2030588, DOI 10.1016/S0040-9383(03)00030-2
Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. MR 572580, DOI 10.1016/0040-9383(80)90003-8
Mark Goresky and Robert MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77–129. MR 696691, DOI 10.1007/BF01389130
Nathan Habegger and Leslie Saper, Intersection cohomology of cs-spaces and Zeeman’s filtration, Invent. Math. 105 (1991), no. 2, 247–272. MR 1115543, DOI 10.1007/BF01232267
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
Henry C. King, Topological invariance of intersection homology without sheaves, Topology Appl. 20 (1985), no. 2, 149–160. MR 800845, DOI 10.1016/0166-8641(85)90075-6
Steven L. Kleiman, The development of intersection homology theory, A century of mathematics in America, Part II, Hist. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1989, pp. 543–585. MR 1003155, DOI 10.24033/bsmf.2027
Robert MacPherson, Intersection homology and perverse sheaves, unpublished AMS monograph, 1990.
Robert MacPherson and Kari Vilonen, Elementary construction of perverse sheaves, Invent. Math. 84 (1986), no. 2, 403–435. MR 833195, DOI 10.1007/BF01388812
John W. Milnor, Infinite cyclic coverings, Conference on the Topology of Manifolds (Michigan State Univ., E. Lansing, Mich., 1967) Prindle, Weber & Schmidt, Boston, Mass., 1968, pp. 115–133. MR 0242163
James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
Frank Quinn, Intrinsic skeleta and intersection homology of weakly stratified sets, Geometry and topology (Athens, Ga., 1985) Lecture Notes in Pure and Appl. Math., vol. 105, Dekker, New York, 1987, pp. 233–249. MR 873296
Frank Quinn, Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988), no. 2, 441–499. MR 928266, DOI 10.1090/S0894-0347-1988-0928266-2
Colin Patrick Rourke and Brian Joseph Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982. Reprint. MR 665919
Richard G. Swan, The theory of sheaves, University of Chicago Press, Chicago and London, 1964.