Cambridge University Press 978-1-107-15074-4 — Singular Intersection Homology Greg Friedman Index More Information

Index

KEY: Numbers in bold indicate where terms are deﬁned

∂-pseudomanifold, see pseudomanifold, effects of simplicial subdivision on, 96–98 ∂-pseudomanifold in small degrees, 130 ∂-stratified pseudomanifold, see stratified on regular strata, 96, 130 pseudomanifold, ∂-stratified pseudomanifold PL chain, 121 abstract simplicial complex, see simplicial complex, simplex, 92 abstract singular chain, 129 acknowledgments, xiv singular simplex, 129 acyclic model, 199 strata vs. skeleta, 105–107 Acyclic Model Theorem, 368 Aluffi, Paolo, 707 admissible triangulation, see triangulation, admissible assembly map, 697, 702 agreeable triple of perversities, 372, 376 AT map, 751 ( , ) AT space, 744, 751 Qp¯ ,t¯Y Qt¯X ,q¯ ; Q , 434 (0¯, t¯; 0¯), 373 augmentation cocycle (1), xxi (n¯, n¯; 0¯), 374 augmentation map (a), xx see (p¯, Dp¯; 0¯), 373 augmented intersection chain complex, (p¯, t¯;¯p), 374 intersection chain complex, augmented (t¯, 0;¯ 0¯), 373 balls, 189 and diagonal maps, 372 Banagl, Markus, xiv, 695, 697, 699, 701, 704, 707, 708 and projection maps, 398 basic sets, 607 in terms of dual perversities, 373 Bernstein, Joseph, 710 Akin, Ethan, 692, 706 Beshears, Aaron, 59 Albin, Pierre, 706, 707, 709 Be˘ılinson, Alexander, 710 Alexander–Whitney map, 199, 365–367 bilinear pairing, see pairing intersection version, see intersection Bockstein map, 233 Alexander–Whitney map (IAW) bordism, 689–702 algebra background, 713–738 bordism group, 690 algebraic diagonal map (d¯), 366, 368, 375 bordism homology theory, 690 co-unitality, 398 ofs ¯-duality spaces, 698 coassociativity, 408 of mod 2 Euler spaces, 692 cocommutativity, 394 of L-spaces, 700 locality, 453, 472 of all pseudomanifolds, 692 naturality, 384 of IP spaces, 697 algebraic mapping cone, 421, 723 and k-theory, 697 and boundary maps, 422, 424, 724 computations, 697 compatibility with products, 425, 427 of locally square-free spaces, 698 allowable of manifolds, 690–691 chain, 5, 92 of pseudomanifolds, 691–702

787

788 Index

of Witt spaces, 693–698 Chriestenson, Bryce, 708 and k-theory, 693, 694 closed cone, see cone, closed computations, 694, 696 closed star, 741 vs. cobordism, 689 coarsening, 249 Borel, Armand, 353, 704 cobordism, see bordism boundary (of pseudomanifold), 52 cochain cross product, 365 filtration of, 153 codimension, xviii, 21 boundary map, 91, 213 Cohen, Daniel, 302 boundary, partial, see partial boundary cohomology cross product, 367, 374 Bourbaki, Nicolas, xiv and evaluation, 468 Brasselet, Jean-Paul, 11, 704, 707, 709 as a cup product, 434, 470 Bredon, Glen, xiv, 705 associativity, 405, 466 Brylinski, Jean-Luc, 709 commutativity, 393, 467 bundle morphism, extension of, 503 evaluation, 403 bundle of groups, 299 for ∂-pseudomanifolds, 476 properties, 477–479 cap product, 376 interchange with cap product, 442, 448, 451, 472 and evaluation, 402, 468 interchange with cup product, 436, 441, 448, 450, associativity, 410, 411, 413, 470 471 boundary formula, 377 naturality, 382, 383, 466 compatibility of ordinary and intersection cap projection pullback, 402, 468 products, 388 stability, 429, 469 ∂ for -pseudomanifolds, 476 unitality, 402, 468 properties, 477–479 cohomology with compact supports, 480, see also stability, 480 intersection cohomology with compact supports interchange with cross product, 442, 448, 451, 472 cohomotopy, 651, 651, 666–669 naturality, 387, 466 cohomotopy groups, 651, 667 ordinary cap product filters through intersection commutative cochain problem, 597 homology, 530–535 complementary perversity, see perversity, dual ordinary homology, 368 completely interior simplex, 175 philosophy, 363–368 allowability, 175 stability, 414, 418, 469 completely regular, 34 topological invariance, 390 composition map, 718 unitality, 401, 468 cone, xviii well-defined, 378 closed, xviii, 23 with compact supports, 487 of a PL space, 755 and Mayer–Vietoris sequence, 488 open, xviii, 22, 23,26 with inverse limits, 487 cone filtration, 22, 26 Cappell, Sylvain, xiv, 59, 695, 701, 707, 711, 712 cone formula, see intersection homology, cone formula chain group or intersection cohomology, cone formula simplicial, 91 conventions, xvii–xxiii singular, 128 convex cell, 740 chain homotopy, 140, 716, 716–717, 720–722 coordinate map, 745, 745 and Hom, 720, 721 compatibility, 745 composition, 720 criss-crosses, 432 prism construction, 140 cross product, xxi, 284–286, 368, 370 tensor product, 721, 722 and evaluation, 403, 468 chain map, 716, 716–717 associativity, 210, 286, 466 and Hom, 719 cohomology, see cohomology cross product chain complex of, 715–717 commutativity, 212, 286, 467 tensor product, 717, 719 compatibility of simplicial and PL, 205–208, 290 Chataur, David, xiv, 534, 709 compatibility of simplicial and singular, 202, 208, Cheeger, Jeff, xiii, 302, 699, 708 290

Index 789

interchange with cap product, 442, 448, 451, 472 de Cataldo, Mark Andrea, 710 intersection homology, 204, 208 de Rham invariant, 697 naturality, 209, 286, 465 Dedekind domain, 217, 224, 225, 727 of intersection chains torsion-free implies flat, 727 is a chain homotopy equivalence, 321 degree, xx PL, 207, 286 Deligne, Pierre, 10, 710 singular, 202, 285 depth, 27 of partial boundary pairs, 474 diagonal map (d), xviii, 365, 372 of singular chains, 199–204 diagonal map, algebraic, see algebraic diagonal map PL, 205, 204–209 diffeomorphic implies PL homeomorphic, 189 PL relative, 207, 286 Dimca, Alexandru, 706 relative, 203, 286 dimension simplicial, 201, 765, 765–768 formal, see formal dimension singular, 201 simplicial, 740 stability, 214, 215, 286, 469 dimension theory, 53, 294, 502 unitality, 213, 286, 467 dimensional Homogeneity, 299 with coefficients, 224–226 dimensional homogeneity, see also homogenization, CS model, 476 291–780 CS set, 29 dimensionally homogeneous, 293 local finiteness of stratification, 31 direct limit open subset is a CS set, 33 commutativity of direct limits, 484 orientable, see orientation over increasing sequence of subsets, 193, 194 recursive, 32 direct system of groups, 480 satisfies Frontier Condition, 31 directed set, 480 topological properties, 34 disjoint union, xviii cup product, 376 distinguished neighborhood, 28 as pullback of the cross product, 376, 432, 470 intersection homology invariance of, 228, 287 associativity, 409, 411, 413, 467 Dold, Albrecht, xiv commutativity, 394, 467 Double Suspension Theorem, 30 compatibility of ordinary and intersection cup dual perversity, see perversity, dual products, 388 duality, see Poincaré duality or Lefschetz duality for ∂-pseudomanifolds, 476 duality map (D), xxi, 537, 536–540, 559 properties, 477–479 signs, 537 front face/back face construction, 364, 366 eigenvalues of symmetric real matrices are real, 632 interchange with cross product, 436, 441, 448, 450, Eilenberg–Zilber map, see cross product 471 Eilenberg–Zilber product triangulation, 762–764 naturality, 386, 466 Eilenberg–Zilber Theorem, 365 ordinary cohomology, 366 engulfing, 678 philosophy, 363–368 Eppelmann, Thorsten, 697 stability, 430, 470 Euclidean polyhedron, see polyhedron topological invariance, 390 evaluation map, 718 unitality, 401, 468 face, see simplex, face cup product pairing, 5, 571, see also cup product filtered collar, 51 dual to intersection pairing, 603, 609, 610 intersection homology invariance of, 245 see image pairing, image pairing filtered homeomorphism, 29 is nonsingular, 571 filtered pair, 148 symmetry for IP spaces, 631 filtered space, xviii, 2, 20 symmetry for Witt spaces, 630 manifold with submanifold, 22 topological invariance, 587 PL, 42 cup-i product, 597 filtration, xviii, 20 Dai, Xianzhe, 708 cone, see cone filtration Davis, James, xiv intrinsic, see intrinsic filtration

790 Index

intrinsic PL, see intrinsic PL filtration Hector, Gilbert, 709 join, see join filtration higher signature, 701 product, see product filtration Hilton, Peter, xiv subspace, see subspace filtration Hirzebruch Signature Theorem, 649 suspension, see suspension filtration Hirzebruch, Friedrich, 649 trivial, see trivial filtration Hom∗ chain complex, 715–717 filtration-preserving, 29 homeomorphism, filtered, see filtered Five Lemma, 255 homeomorphism flat module, 217, 224, 727 homogenization, 293 formal dimension, xviii, 17, 20 of a product, 298 vs. geometric dimension, 21 homology forward tame, 58 singular, 128 Frontier Condition, 24, 25, 31 with local coefficients, 300 full subcomplex, 122 homotopically stratified space, 58 full triangulation, 122 manifold homotopically stratified space (MHSS), 59 Fulton, William, 61 homotopy link (holink), 58 fundamental class, xxi, 508, 508, 509 Hudson, John, xiv and change of perversity, 523–526 Hughes, C. Bruce, 59 and change of stratification, 526–530 Hunsicker, Eugénie, 647, 709, 712 in L-homology, 697, 702 identity map (id), xxiii in ko-homology, 694, 697, 700 image pairing, 591, 590–596 in a distinguished neighborhood, 512–515, 520 is a cup product pairing, 635 in signature homology, 700, 702 is nondegenerate, 591, 593, 594 local, 512 may not be nonsingular, 592 of ∂-pseudomanifold, 549 nonsingularity and torsion, 637 topological invariance, 555–558 inclusion map, xviii of a boundary, 551 inherited perversity, see subspace perversity of a manifold, 501 inner product, 693 of a product, 535 inner product space, see symmetric inner product requirement thatp ¯ ≥ 0,¯ 505, 507, 515, 521 space topological invariance, 530 interchange map, see transposition Gelfand, Sergei, 706 intersection Alexander module, 301 general position, 4, 104, 597, 600 intersection Alexander–Whitney map (IAW), 368, failure for pseudomanifolds, 604 371, 374 in PL manifolds, 600 for partial boundary pairs, 474 stratified, see stratified general position intersection chain, 5, see also intersection chain generalized homology theory, 690 complex generic point, 650, 660 allowability by strata vs. skeleta, 105–107 inverse image of, 650, 660 splitting, 337–352 see GM intersection homology, intersection homology intersection chain complex see GM perversity, perversity, GM augmented, 154 see GM stratified homotopy, stratified homotopy, GM GM Godement, Roger, 705 PL, 120 Goresky, Mark, xiii, 216, 226, 302, 530, 596, 696, simplicial, 92 698, 707, 710 singular, 129 Goresky–MacPherson duality, 601, 606 GM and non-GM agree below t¯, 273 see Goresky–MacPherson perversity, perversity, GM map to point, 397 grading, homological vs. cohomological, 714–715 non-GM, 267, 269, 271 Gromov, Mikhail, 613 all definitions agree, 269, 271 Habegger, Nathan, 12 first alternative definition, 269 Hartshorne, Robin, 706 PL, 269, 271 Hatcher, Allen, xiv second alternative definition, 271

Index 791

simplicial, 267, 269, 271 relative, 157, 275 singular, 267, 269, 271 efficient perversities, see perversity, efficient PL excision, 160, 176, 281 GM, 120 finite generation, 258, 257–261, 291 non-GM, 269, 271 GM, xxii PL chain modules are flat, 277 PL, 120 PL chain modules are not necessarily free, 134 simplicial, 92, 90–107 projection maps, 396, 397 simplicial, behavior under subdivision, 96–98 relative, see relative intersection chain complex singular, 129 simplicial GM and non-GM agree below t¯, 263, 273 GM, 92 GM vs. non-GM, 11–13, 263–265 non-GM, 267, 269, 271 cone formula, 264–265 singular duality needs non-GM, 263–265 GM, 129 invariance for distinguished neighborhoods, 228 non-GM, 267, 269, 271 invariance for filtered collars, 245 singular chain modules are projective, 277 invariance for links, 228 singular chain modules over Z are free, 133 inverse limit over compact subsets, 487 intersection cochain, see also intersection cochain is ordinary cohomology for normal pseudomanifold complex and zero perversity, 363 other meanings, 353 is ordinary homology for high perversities, 195 intersection cochain complex, 355 is ordinary homology for normal pseudomanifold intersection cohomology, xxii, 355 and top perversity, 195 cap product, see cap product Lefschetz duality, see Lefschetz duality cone formula, 356 long exact sequence of a pair, 150, 275 cross product, see cohomology cross product map from ordinary cohomology to intersection cup product, see cup product homology (αp¯ ), 530, 532, 610 excision, 358 map to point, 397 is ordinary cohomology for normal pseudomanifold Mayer–Vietoris sequence, 162, 180, 282 and top perversity, 362 motivation, 104 Lefschetz duality, see Lefschetz duality non-GM, xxii, 267, 269, 271, 262–276 long exact sequence of a pair, 358 and sheaf theory, 265 Mayer–Vietoris sequence, 359 can’t be reduced, 274 PL first alternative definition, 269 difficulties, 10, 354, 356, 358, 361 is direct sum over regular strata, 292, 295 PL and singular are isomorphic over a field, 361 is not relative intersection homology, 265 Poincaré duality, see Poincaré duality motivation, 262–265 products on ∂-pseudomanifolds, 473–480 PL, 269, 271 relative Mayer–Vietoris sequence, 359 second alternative definition, 271 topological invariance, 362 simplicial, 267, 269, 271 vs. sheaf cohomology, 353 singular, 267, 269, 271 with compact supports, 480 of X+, 183–185, 282 behavior under direct limits, 482 of a cone, see intersection homology, cone formula excision, 482 of a point, 129 functoriality, 481 of a suspension, 183, 181–183, 282 Mayer–Vietoris sequence, 483 of cone on a manifold, 131 of a cone, 481 of link depends only on stratum, 229, 287 intersection homology, xxii, 5 PL, 120–121 behavior under direct limits, 194, 283 excision, 160, 158–163, 281 behavior under normalization, 197, 283 GM, 120 cone formula, 143–146, 223 Mayer–Vietoris sequence, 162, 282 GM, 143 non-GM, 269, 271 GM vs. non-GM, 264–265 PL and singular are isomorphic, 237, 239, 240, 289, non-GM, 274 289, 290

792 Index

Poincaré duality, see Poincaré duality Ji, Lizhen, 302 projection maps, 396, 397 join, 75–81, 303 reduced, 154, 154–155 of PL spaces, 755 relative, see relative intersection homology join filtration, 76 relative Mayer–Vietoris sequence, 186 of pseudomanifold is a pseudomanifold, 79 simplicial Künneth Theorem, 314, 302–325 behavior under subdivision, 122, 289 algebraic, 217, 326–332 GM, 92 representation of classes in torsion summand, 330 non-GM, 267, 269, 271 chain homotopy equivalence, 321 simplicial and singular are isomorphic, 241, 290 cohomology, 370, 461, 473 simplicial vs. PL, 121–127 field coefficients, 317 singular, 128–134 for ∂-pseudomanifolds, 314 excision, 176, 281 local splitting behavior, 308, 325, 332–337 GM, 129 needs non-GM intersection homology, 318 Mayer–Vietoris sequence, 180, 282 product of cones, 303 non-GM, 267, 269, 271 relative, 319 subdivision, 175–179 single perversity, 302 topological invariance, 244, 241–257 with a manifold factor, 216, 219, 286 with coefficients, 221, 220–234 with coefficients, 225 with local coefficients, 299–302 Kashiwara, Masaki, 706 intersection number, 598 King, Henry, 11, 63, 128, 188, 191, 193, 216, 234 intersection pairing, 4, 5, 8, 263, 264, 364, 596–612, Kirk, Paul, xiv see also intersection product Kirwan, Frances, xiii, 704 dual to cup product pairing, 603, 609, 610 Kleiman, Steven, 10 on topological pseudomanifolds, 609 Klimczak, Mathieu, 708 intersection Poincaré space, see IP space knot theory, 301 intersection product, xxii, 6, 364, 601, see also Koszul sign convention, xxii, 364, 713–719 intersection pairing Kreck, Matthias, 700 boundary formula, 602 L-classes, 652, 669, 648–689 commutativity, 602 L0 formula, 656, 681 of intersection chains, 606–608 axiomatic characterization, 658–659, 681 of non-GM intersection chains, 608 evaluation formula, 653, 680 intrinsic filtration, 66, 63–74, 249 in small degrees, 654, 673 is coarsest filtration, 66 multiplication by spheres, 679 see of ∂-pseudomanifold, natural filtration of smooth manifolds, 653, 672 of open subset is intrinsic, 68 outline of construction, 650–659 of product with manifold, 68 philosophy, 648–650 intrinsic PL filtration, 69 L-groups, 697 of a PL pseudomanifold is a PL pseudomanifold, 72 L-space, 699 of open subset is intrinsic, 70 L-spectrum, 697 of PL ∂-pseudomanifold, see natural filtration Lagrangian structure, 699 of product of pseudomanifolds, 82 Lagrangian subspace, see pairing, Lagrangian of product with manifold, 70 subspace inverse limit, 487 Lam, Tsit Yuen, xiv invertible cobordism, 63 Lang, Serge, xiv IP space, 622, 621–623 Laures, Gerd, 697, 707 is a Witt space, 622, 623 Lefschetz duality, 560, 562 product of IP spaces is IP, 626 general form, 562 stratification independence, 629 manifold Lefschetz duality as intersection symmetry of cup product pairing, 631 homology duality, 8, 98–99 symmetry of torsion pairing, 631 necessity of boundary collars, 565 Iversen, Birger, 706 topological invariance, 565

Index 793

Lefschetz, Solomon, 600 mapping cone, algebraic, see algebraic mapping cone Leichtnam, Éric, 706, 707 Maslov index, 645 Libgober, Anatoly, 712 Massey, David, 710, 711 link, xix, 28 Mather, John, 56 intersection homology of, 229, 287 Maxim, Laurent,iu, xiv, 707, 708, 711, 712 invariance of intersection homology, 228, 287 Mayer–Vietoris argument, 188–198 link of a link is a link, 38 for CS sets, 191 non-uniqueness, 30 for manifolds, 188, 190 linking pairing, see torsion pairing Mayer–Vietoris sequence maps, xxi local coefficients, 299 Mazzeo, Rafe, 706, 707 localization, 696 McClure, James E., xiv, 188, 368, 598, 603, 697, 707 locally cone-like, 28 McCrory, Clint, 6, 605 locally finite, 740 Migliorini, Luca, 710 locally finitely generated, 258, 291 Miller, David A., 59 for product space, 323 Milnor, John, 649 locally torsion-free, 226, 287, 622 Minatta, Augusto, 700, 702, 707 always holds for top perversity, 227 Morgan, John, 697, 699 forp ¯ implies for Dp¯, 546 Morton, Hugh, 84 for product space, 322 Munkres, James, xiv GM vs. non-GM, 288 natural filtration, 73 independence of stratification, 248 nearly stratum-preserving, 58 Looijenga, Eduard, 711 negative definite, 731 lower middle perversity, see perversity, lower middle Noetherian ring, 261, 291 MacPherson, Robert, xiii, 61, 216, 302, 530, 596, 710, Nollet, Scott, xiv 711 non-GM intersection homology, see intersection manifold, xvii homology PL, see PL manifold nondegenerate pairing, see pairing, nondegenerate manifold stratified space, 26 nonsingular pairing, see pairing, nonsingular ∂-manifold, xvii normal stratified pseudomanifold, see stratified Manin, Yuri, 706 pseudomanifold, normal map normalization of pseudomanifolds, 49–51, 197–198 Alexander–Whitney, see Alexander–Whitney map normally nonsingular inclusion, 61 algebraic diagonal, see algebraic diagonal map normally nonsingular map, 61 boundary, see boundary map normally nonsingular projection, 61 chain, see chain map normally nonsingular subspace, 61, 151 see diagonal, see diagonal map PL trivial, PL trivial normally nonsingular duality, see duality map subspace Eilenberg–Zilber, see cross product notation, xvii–xxiii interchange, see transposition Novikov additivity, 643, 643–647 intersection Alexander–Whitney, see intersection Novikov Conjecture, 701–702 Alexander–Whitney map stratified, 702 normally nonsingular, see normally nonsingular now what?, 703 map open cone, see cone, open piecewise linear, see piecewise linear map orientable, see orientation PL, see PL map orientation PL chain, see PL chain map of a product, 535–536 placid, see placid map and change of perversity, 523–526 shift, see shift map and change of stratification, 526–530 simplicial, see simplicial map and fundamental class, 508, 509 simplicial chain, see simplicial chain map in a distinguished neighborhood, 510, 512–515, 520 stratified, see stratified map local, 512 umkehr, see umkehr map of ∂-pseudomanifold, 549

794 Index

of a boundary, 549 efficient, 284 of a CS set, 501, 501–505 GM, 88, 243 and change of stratification, 501, 502, 505 lower middle (m ¯ ), xxi, 89, 614 and codimension-one strata, 502, 505 on regular strata, 87, 130 of a manifold, 500, 499–501 product orientation bundle, 500 necessity for Künneth Theorem, 304–306 of a product, 510 sufficiency for Künneth Theorem, 306–311 of a pseudomanifold, see orientation, of a CS set product (Q), 304, 314 of a simplex, 91 top (t¯), xxi, 6, 88 of a stratified pseudomanifold, see orientation, of a upper middle (n ¯), xxi, 89, 614 CS set and product interchange properties, 452 orientation sheaf, 506, 505–508 is (n¯, n¯)-compatible, 317 and homology classes, 508 Peters, Chris, 711 extension of sections, 508 Piazza, Paolo, 706, 707 of a product, 510, 535 piecewise linear, see PL orientation section, 508 piecewise linear map of polyhedra, 743 requirement thatp ¯ ≥ 0,¯ 515 composition, 743 is simplicial, 743 p¯-allowable, see allowable products of, 744 pairing, 568, 568–571 PL Approximation Theorem, 750 adjoint formulation, 569, 733 PL category antisymmetric, 630, 633, 736–738 AT, 42, 751 Lagrangian subspace, 738 AT and PL are equivalent, 753 see cup product, cup product pairing PL, 42, 747 see image pairing, image pairing PL chain Lagrangian subspace, 734 characterization using supports and homology, negative definite, 731 115–120 nondegenerate, 569, 733 PL chain complex, 110 nonsingular, 4, 5, 7–9, 263, 264, 364, 569, 733 PL chain map, 113 and determinant, 633, 734 PL homology, 9–10, 112, 108–115 positive definite, 731 isomorphic to simplicial homology, 112 see skew symmetric, pairing, antisymmetric PL homology map, 113 symmetric, 630, 730 PL intersection homology, see intersection homology, is diagonalizable, 633, 731 PL Pardon, William, 621, 696–698, 707 PL manifold, 745 parentheses, xxiii PL map, 40 Park, Efton, xiv consistency of coordinate and simplicial definitions, partial boundary, 473 747 partial boundary pair, 473 coordinate definition, 747 ∂-pseudomanifolds, 473 induces PL chain map, 113 products of ∂-pseudomanifolds, 473 represented by simplicial map, 40 partially ordered set (poset), 759 simplicial definition, 747 product of, 760, 762 triangulation of, 748 Parusiński, Adam, 707 PL pseudomanifold, see pseudomanifold, PL perverse sheaves, xiii PL space, 39, 38–42, 745 perverse signature, 614, 647–648 cone of, 755 Wall non-additivity, 648 is a CS set, 44 perversity, xxi, 3, 87 is triangulable, 746 (p¯, q¯)-compatible, 314, 370 join of, 755 ˆ a Qp¯ ,q¯ , 444 product of, 755, 764 maximal (Qp¯ ,q¯ ), 371, 375 with a dense manifold is a pseudomanifold, 45 0,¯ xxi, 88 PL stratified pseudomanifold, see stratified dual, xxii, 3, 89 pseudomanifold, PL

Index 795

PL structure, 745 classical, 35 and triangulation, 749 codimension-one strata, 35 base of, 745 has regular strata, 35 PL subspace, 41, 754 irreducible complex varieties, 45, 57 closed, triangulation of, 41, 754 orbit spaces, 57 open, 42, 754 orientable, see orientation subcomplex, 41, 754 PL, 43 PL topology, 739–768 classical, 43 PL trivial nns, see PL trivial normally nonsingular triangulation, 46, 72 subspace triangulation of PL ∂-pseudomanifold, 73 PL trivial normally nonsingular subspace, 656, 681 simplicial, 43 as inverse of regular point, 682 classical, 46–48 of a Witt space, 682 suspension, 36 placid map, 137 Quinn, Frank, 58 PM-convex set, 516 Poincaré duality, 540 rational homotopy theory, 597 see for ordinary (co)homology when 0¯ and t¯ recursive CS set, CS set, recursive see intersection homology agree, 534 reduced intersection homology, intersection Lefschetz duality, see Lefschetz duality homology, reduced necessity of torsion-free condition, 539 references, standard, xiv signs, 537 regular point, 682 topological invariance, 547 inverse is a PL trivial normally nonsingular Poincaré duality map, see duality map subspace, 682 see polyhedral link, xix regular strata, strata, regular polyhedron, 742, 742–744 relative homology, 146 is the space of a simplicial complex, 743 relative intersection chain complex products of, 743 GM, 147, 149 positive definite, 731 non-GM, 275–276 (p¯, q¯)-stratified map, see stratified map relative intersection homology (p¯, q¯)GM-stratified map, see stratified map, GM GM, 149, 146–150 Prüfer domain, 225 Mayer–Vietoris sequence, 186 prerequisites, xiv non-GM, 275 prism construction, 126 Richardson, Ken, xiv product filtration, 22, 26, 74, 303 Rotman, Joseph, xiv of ∂-pseudomanifolds, 80 Rourke, Colin, xiv of intrinsic PL filtrations, 82 Saito, Morihiko, 711 of pseudomanifolds is a pseudomanifold, 79 Sanderson, Brian, xiv product of PL spaces, 755 Saper, Leslie, 12, 709 product perversity, see perversity, product Saralegi-Aranguren, Martintxo, xiv, 12, 188, 265, 534, projective module, 224, 277, 724 709, 712 is summand of free module, 725 Schapira, Pierre, 706 quasi-isomorphisms induce chain homotopy Schürmann, Jörg, 707, 708, 710 equivalences, 726 Schütz, Dirk, 712 split short exact sequences, 725 Seade, José, 707 projective resolution, 727 Serre’s Theorem, 651, 669 Prokhorenkov, Igor, xiv Serre, Jean-Pierre, 651, 669 properties of intersection (co)homology products, Shaneson, Julius, 59, 695, 701, 707, 711, 712 465–473 sheaf, 506 pseudomanifold, 35 orientation sheaf, see orientation ∂-pseudomanifold, 52 section, 506 classical, 52 sheaf theory, avoidance of, 10–11 co(homology) products on, 473–480 shift map, 421, 723, 723 necessity of boundary collars for duality, 565 shuffle, 199, 757, 762, 763, 765

796 Index

shuffle product, see cross product singular intersection homology, see intersection Siebenmann, Laurence, 16, 31, 33 homology, singular Siegel, Paul, 216, 226, 615, 693, 697, 707 singular locus (Σ), xviii, 2, 23 sign conventions, xxii singular set, see singular locus signature, 590, 613 singular simplex, 128 and isometries, 633 singular strata, see strata, singular and Lagrangian subspace, 633, 734 singular subdivision, see subdivision, singular in terms of positive/negative definite subspaces, 731 singularity, 2, 18 Novikov additivity, 643–647 skeleton, xviii, 20 of a manifold, 634 simplicial, 21, 23, 740 of a matrix, 632, 729 space, filtered, see filtered space block sum, 633, 735 Spanier, Edwin, xiv of a symmetric pairing, 632, 730 Spectral Theorem, 731 block sum, 633, 735 Spice, Loren, xiv of a Witt space, see Witt signature split inner product, see symmetric inner product space, of image pairing, 634 split Wall non-additivity, 645–648 Stallings, John, 63 signature homology, 700, 702 Stammbach, Urs, xiv signature theorem, see Hirzebruch Signature Theorem standard mistake, 157, 337 signs, 364, 537, 602, 713–719 Stanley, Richard, 712 simplex, xx, 91 Stasheff, James, 649 face, 740 Steenbrink, Joseph, 711 proper, 740 Steenrod square, 597 geometric, 740 strata, xviii, 2, 21 open, xx partial ordering of, 25 oriented, 91 regular, xviii, 2, 23 vertex, 740 filtered spaces without, 23–24, 27, 150 simplicial chain map, 113 singular, xviii, 2, 23 simplicial complex, 740, 739–742 stratification, see filtration abstract, 758 stratified general position, 604 associated to a poset, 759, 760 stratified homeomorphism, 60 realization of, 758 induces intersection cohomology isomorphism, 357 canonical PL structure, 746 induces intersection homology isomorphism, 139, dimension, 740 279 finite, 741 vs. filtered homeomorphism, 60 is a polyhedron, 743 stratified homotopy, 62 is a PL space, 745 GM, 139 locally finite, 39 induces homotopic intersection chain maps, 139, subcomplex, 741 279 subdivision, see subdivision induces homotopic intersection cochain maps, 357 simplicial filtered space, 91 non-GM, 279 simplicial filtration, 21, 23 vs. stratum-preserving homotopy, 62 simplicial intersection homology, see intersection stratified homotopy equivalence, 62 homology, simplicial induces intersection cohomology isomorphism, 358 simplicial isomorphism, 742, 749 induces intersection homology isomorphism, 142, simplicial map, 742 280 is piecewise linear, 743 stratified map, 60, 137 is PL, 40, 748 functoriality, 139 simplicial pseudomanifold, see pseudomanifold, GM, 137 simplicial induces intersection chain map, 138, 278, 279 simplicial skeleton, see skeleton, simplicial induces intersection cochain map, 357 singular chain complex, 128 maps of relative intersection homology, 153, 279 singular cone (of a singular simplex), 131, 145 non-GM, 278

Index 797

stratified pseudomanifold, 35, 37 Taylor, Laurence, 59 ∂-stratified pseudomanifold, 51 Tennison, B.R., 705 co(homology) products on, 473–480 Thom class, 681 necessity of boundary collars for duality, 565 Thom Isomorphism Theorem, 657 open subset is a ∂-stratified pseudomanifold, 54 Thom, René, 56, 648 vs. ∂-manifold, 52 Thom–Mather space, 56 classical, 35, 37 top perversity, see perversity, top dependence on filtration, 36 topological group, 597 links are stratified pseudomanifolds, 36, 37 torsion pairing, 574, 574–587 normal, 48, 195, 362 alternative definition, 586–587 and ordinary Poincaré duality, 534 image pairing, see image pairing open subset is a stratified pseudomanifold, 36 in terms of cap product, 581 orientable, see orientation in terms of cup product, 581 PL, 43 is nonsingular, 574, 583 classical, 43, 44 symmetry, 574, 583, 631 links are unique, 44 topological invariance, 587 vs. manifold with boundary, 36 torsion submodule, 569 stratified space, 25 and Ext, 570 stratifold, 700 and Hom, 547, 569, 570 stratum-preserving, 58 torsion-free quotient, 569 subcomplex, see simplicial complex, subcomplex and Hom, 547, 569, 570 subdivision, 96–98, 159, 751 transposition, 213, 717 and intersection homology, 280 transversality, 4, 104, 598 barycentric, 159, 235–237, 741 triangulation, 39, 746, 751 chain map, 109–110 admissible, 39, 744, 751 directed sets of, 108 and PL structure, 749 of intersection chains, 280–281 compatible, 751 of simplicial complex, 741 of a product, 762–764 of simplicial intersection chain, 120 PL, 746 singular, 163–175, 280 subdivision, 39 and intersection homology, 170, 174 trivial filtration, 26, 27 of intersection chains, 169 umkehr map, 658, 657–658, 681 of singular chain, 168 and products with spheres, 683 of singular simplex, 166 composition of, 683 submanifold as a skeleton, 22 unfiltered, 27 subspace filtration, 22 Universal Coefficient Theorem inclusion map is stratified, 137, 148 intersection cohomology, 355 subspace perversity, 148 intersection homology, 230, 226–234, 287 suggestions for further reading, 703–712 upper middle perversity, see perversity, upper middle Sullivan, Dennis, 692, 694, 697 Useful Lemma, 115 support, xv, xx, 91, 115–120 Valette, Guillaume, 565 suspension, xviii, 26 vertex, see simplex, vertex filtration of, 23, 26, 30 weak boundary, 575 intersection homology of, 99–104 Weibel, Charles, xiv Witt signature is zero, 643 Weinberger, Shmuel, 59, 712 Suwa, Tatsuo, 707, 708 Well-Ordering Theorem, 124, 126, 164 Swan, Richard, xiv, 705 whew, 497 Sylvester’s Law of Inertia, 730 Whitney conditions, 55 693 symmetric inner product space, Whitney stratified space, 55 split, 693 Whitney umbrella, 55 symmetric signature, 697 Whitney, Hassler, 55 Tanré, Daniel, xiv, 534, 709 Williams, Bruce, 59

798 Index

Williamson, Geordie, 711 Witt condition, 616 Witt group, 615, 693 W (Q), 694 Witt signature, 614, 633, 634, 633–647 and orientation, 638 cobordism invariance, 638 Novikov additivity, 643 of a boundary is 0, 638 of a product, 638 of a suspension is zero, 643 of disjoint union, 638 of image pairing, 634 of Witt space with boundary, 634–637 alternative definition, 634–637 is a Witt signature without boundary, 637 topological invariance, 638 Witt space, 616, 614–621 m¯ n¯ I H∗ (X) I H∗ (X), 617 and field characteristic, 621 boundary is Witt, 616 can’t have codimension-one strata, 616 dependence on coefficients, 618–621 examples, 617–618 is not necessarily an IP space, 623 product of Witt spaces is Witt, 624 stratification independence, 628 symmetry of cup product pairing, 630 Wolak, Robert, 712 Woolf, Jonathan, xiii, xiv, 59, 695, 704, 707, 712 Yan, Min, 59 Yokura, Shoji, 708 0¯ perversity, see perversity, 0¯ Zorn’s Lemma, 190, 192 Zucker, Steven, 709