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Intersection homology of stratified fibrations and neighborhoods. (English) Zbl 1160.55003

This paper contains a detailed study of the intersection homology of stratified fibrations and neighborhoods of singular sets in stratified spaces. The setting is in the widely applicable theory of manifold homotopically stratified spaces introduced by F. Quinn [J. Am. Math. Soc. 1, No. 2, 441–499 (1988; Zbl 0655.57010)] as a means of studying purely topological (as opposed to smooth or PL) phenomena. The author derives spectral sequences for the intersection homology of stratified fibrations. He is then able to apply this to the intersection homology of neighborhoods in manifold homotopically stratified spaces because neighborhoods of singular sets have homotopy models that are mapping cylinders of stratified fibrations.

MSC:

55N33 Intersection homology and cohomology in algebraic topology
57N80 Stratifications in topological manifolds
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55R55 Fiberings with singularities in algebraic topology
55R65 Generalizations of fiber spaces and bundles in algebraic topology

Citations:

Zbl 0655.57010
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References:

[1] Borel, A., Intersection Cohomology, Progr. Math., vol. 50 (1984), Birkhäuser: Birkhäuser Boston, MA
[2] Bredon, Glen, Sheaf Theory (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0874.55001
[3] Cappell, Sylvain E.; Shaneson, Julius L., Singular spaces, characteristic classes, and intersection homology, Ann. of Math. (2), 134, 325-374 (1991) · Zbl 0759.55002
[4] Cappell, Sylvain E.; Shaneson, Julius L., The mapping cone and cylinder of a stratified map, (Prospects in Topology, Princeton, NJ. Prospects in Topology, Princeton, NJ, Ann. of Math. Stud., vol. 138 (1994), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ), 58-66 · Zbl 0932.57022
[5] Chapman, T. A., Concordances of Hilbert cube manifolds and tubular neighborhoods of finite-dimensional manifolds, (Cantrell, James C., Geometric Topology (1970), Academic Press: Academic Press New York), 581-596 · Zbl 0465.57003
[6] Fadell, Edward, Generalized normal bundles for locally-flat embedding, Trans. Amer. Math. Soc., 114, 488-513 (1965) · Zbl 0129.39503
[7] Greg Friedman, Intersection homology and Poincaré duality on homotopically stratified spaces, submitted for publication; see http://www.arxiv.org/abs/math.GT/0702087; Greg Friedman, Intersection homology and Poincaré duality on homotopically stratified spaces, submitted for publication; see http://www.arxiv.org/abs/math.GT/0702087 · Zbl 1180.55003
[8] Friedman, Greg, Alexander polynomials of non-locally-flat knots, Indiana Univ. Math. J., 52, 1479-1578 (2003) · Zbl 1065.57026
[9] Friedman, Greg, Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata, Topology Appl., 134, 69-109 (2003) · Zbl 1032.55004
[10] Friedman, Greg, Intersection Alexander polynomials, Topology, 43, 71-117 (2004) · Zbl 1045.57011
[11] Friedman, Greg, Intersection homology of regular and cylindrical neighborhoods, Topology Appl., 149, 97-148 (2005) · Zbl 1092.55004
[12] Friedman, Greg, Superperverse intersection cohomology: Stratification (in)dependence, Math. Z., 252, 49-70 (2006) · Zbl 1106.55003
[13] Friedman, Greg, Singular chain intersection homology for traditional and super-perversities, Trans. Amer. Math. Soc., 359, 1977-2019 (2007) · Zbl 1109.55004
[14] Goresky, Mark; MacPherson, Robert, Intersection homology theory, Topology, 19, 135-162 (1980) · Zbl 0448.55004
[15] Goresky, Mark; MacPherson, Robert, Intersection homology II, Invent. Math., 72, 77-129 (1983) · Zbl 0529.55007
[16] Hughes, Bruce, Stratifications of mapping cylinders, Topology Appl., 94, 127-145 (1999) · Zbl 0928.57027
[17] Hughes, Bruce, Stratifications of teardrops, Fund. Math., 161, 305-324 (1999) · Zbl 0942.57021
[18] Hughes, Bruce, Stratified path spaces and fibrations, Proc. Roy. Soc. Edinburgh Sect. A, 129, 351-384 (1999) · Zbl 0923.55007
[19] Hughes, Bruce, The approximate tubular neighborhood theorem, Ann. of Math. (2), 156, 867-889 (2002) · Zbl 1030.57034
[20] Hughes, Bruce; Taylor, Laurence R.; Weinberger, Shmuel; Williams, Bruce, Neighborhoods in stratified spaces with two strata, Topology, 39, 873-919 (2000) · Zbl 0954.57008
[21] Hurewicz, Witold; Wallman, Henry, Dimension Theory (1948), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0036.12501
[22] King, Henry C., Topological invariance of intersection homology without sheaves, Topology Appl., 20, 149-160 (1985) · Zbl 0568.55003
[23] Laurentiu, Maxim, Intersection homology and Alexander modules of hypersurface complements, Comment. Math. Helv., 81, 123-155 (2006) · Zbl 1111.32030
[24] Munkres, James R., Elements of Algebraic Topology (1984), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0673.55001
[25] Quinn, Frank, Intrinsic skeleta and intersection homology of weakly stratified sets, (Geometry and Topology. Geometry and Topology, Athens, GA, 1985. Geometry and Topology. Geometry and Topology, Athens, GA, 1985, Lecture Notes in Pure and Appl. Math., vol. 105 (1987), Dekker: Dekker New York), 225-241 · Zbl 0619.57007
[26] Quinn, Frank, Homotopically stratified sets, J. Amer. Math. Soc., 1, 441-499 (1988) · Zbl 0655.57010
[27] Spanier, E., Singular homology and cohomology with local coefficients and duality for manifolds, Pacific J. Math., 160, 165-200 (1993) · Zbl 0806.55005
[28] Swan, Richard G., The Theory of Sheaves (1964), Univ. Chicago Press: Univ. Chicago Press Chicago, IL · Zbl 0119.25801
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