4580:
4231:
4262:
4929:
3903:
424:—these ideas break down. For example, it is no longer possible to make sense of the notion of "general position" for cycles. Goresky and MacPherson introduced a class of "allowable" cycles for which general position does make sense. They introduced an equivalence relation for allowable cycles (where only "allowable boundaries" are equivalent to zero), and called the group
4674:
4575:{\displaystyle {\begin{matrix}{\mathcal {H}}^{2}\left(\mathbf {R} i_{*}\mathbb {Q} _{U}|_{p=0}\right)&=&\mathbb {Q} _{p=0}\\{\mathcal {H}}^{1}\left(\mathbf {R} i_{*}\mathbb {Q} _{U}|_{p=0}\right)&=&\mathbb {Q} _{p=0}^{\oplus 2}\\{\mathcal {H}}^{0}\left(\mathbf {R} i_{*}\mathbb {Q} _{U}|_{p=0}\right)&=&\mathbb {Q} _{p=0}\end{matrix}}}
1716:. If the perversity is not specified, then one usually means the lower middle perversity. If a space can be stratified with all strata of even dimension (for example, any complex variety) then the intersection homology groups are independent of the values of the perversity on odd integers, so the upper and lower middle perversities are equivalent.
4226:{\displaystyle {\begin{aligned}H^{0}(U;\mathbb {Q} )&\cong H^{0}(X;\mathbb {Q} )=\mathbb {Q} \\H^{1}(U;\mathbb {Q} )&\cong H^{1}(X;\mathbb {Q} )=\mathbb {Q} ^{\oplus 2}\\H^{2}(U;\mathbb {Q} )&\cong H^{1}(X;\mathbb {Q} )=\mathbb {Q} ^{\oplus 2}\\H^{3}(U;\mathbb {Q} )&\cong H^{2}(X;\mathbb {Q} )=\mathbb {Q} \\\end{aligned}}}
2681:
4924:{\displaystyle {\begin{matrix}{\mathcal {H}}^{0}(IC_{\mathbb {V} (f)})&=&\mathbb {Q} _{\mathbb {V} (f)}\\{\mathcal {H}}^{1}(IC_{\mathbb {V} (f)})&=&\mathbb {Q} _{p=0}^{\oplus 2}\\{\mathcal {H}}^{i}(IC_{\mathbb {V} (f)})&=&0&{\text{for }}i\neq 0,1\end{matrix}}}
5310:. Moreover, the complex is uniquely characterized by these conditions, up to isomorphism in the derived category. The conditions do not depend on the choice of stratification, so this shows that intersection cohomology does not depend on the choice of stratification either.
3633:
237:
2460:
629:
3518:
1207:
1863:
2281:
2042:
has a triangulation compatible with the stratification, then simplicial intersection homology groups can be defined in a similar way, and are naturally isomorphic to the singular intersection homology groups.
3404:
1347:
2141:
There is a variety with two different small resolutions that have different ring structures on their cohomology, showing that there is in general no natural ring structure on intersection (co)homology.
3125:
4628:
1979:
that consists of all singular chains such that both the chain and its boundary are linear combinations of allowable singular simplexes. The singular intersection homology groups (with perversity
3908:
1464:
521:, though the groups often turn out to be independent of the choice of stratification. There are many different definitions of stratified spaces. A convenient one for intersection homology is an
2873:
814:
3003:
1105:
5276:
5182:
3274:
5103:
5019:
2676:{\displaystyle IC_{p}(X)=\tau _{\leq p(n)-n}\mathbf {R} i_{n*}\tau _{\leq p(n-1)-n}\mathbf {R} i_{n-1*}\cdots \tau _{\leq p(2)-n}\mathbf {R} i_{2*}\mathbb {C} _{X\setminus X_{n-2}}}
1006:
694:
121:
3821:
4669:
3441:
3347:
3306:
2825:
3205:
2954:
2912:
2780:
2452:
2413:
936:
3076:
1273:
3870:
3757:
422:
895:
2714:
2030:
1401:
1379:
1156:
1127:
1934:
1771:
720:
2374:
2323:
464:
2101:
1973:
1907:
1553:
287:
3467:
1668:
766:
5422:
3160:
1512:
1233:
2741:
1624:
507:
378:
330:
5451:
3682:
1050:
is any complex quasi-projective variety (possibly with singularities) then its underlying space is a topological pseudomanifold, with all strata of even dimension.
5504:
4257:
3708:
3513:
837:
3890:
3841:
3777:
3728:
3653:
3487:
3225:
3023:
1714:
547:
3352:
2130:. Roughly speaking, this means that most fibers are small. In this case the morphism induces an isomorphism from the (intersection) homology of
1164:
5476:
473:-dimensional allowable cycles modulo this equivalence relation "intersection homology". They furthermore showed that the intersection of an
1787:
2156:
63:
5602:
5533:
52:
1280:
1129:, which measures how far cycles are allowed to deviate from transversality. (The origin of the name "perversity" was explained by
5463:
3628:{\displaystyle \mathbf {R} ^{k}i_{*}\mathbb {Q} _{U}|_{p=0}=\mathop {\underset {V\subset U}{\text{colim}}} H^{k}(V;\mathbb {Q} )}
2057:
is a topological manifold, then the intersection homology groups (for any perversity) are the same as the usual homology groups.
59:
5673:
5355:
3084:
518:
344:-dimensional cycle. One may prove that the homology class of this cycle depends only on the homology classes of the original
3469:
the derived pushforward is the identity map on a smooth point, hence the only possible cohomology is concentrated in degree
4585:
5630:
2956:
with a local system, one can use
Deligne's formula to define intersection cohomology with coefficients in a local system.
1409:
5683:
5663:
2830:
5668:
5625:
5389:
Warning: there is more than one convention for the way that the perversity enters
Deligne's construction: the numbers
2066:
1627:
775:
5508:
5678:
5350:
2970:
1353:
The second condition is used to show invariance of intersection homology groups under change of stratification.
5581:
232:{\displaystyle H_{i}(X,\mathbb {Q} )\times H_{n-i}(X,\mathbb {Q} )\to H_{0}(X,\mathbb {Q} )\cong \mathbb {Q} .}
1062:
5217:
5123:
3230:
5047:
4963:
965:
653:
5620:
3782:
4633:
3409:
3311:
3279:
2785:
3165:
5370:
5345:
2920:
2878:
2746:
2418:
2379:
1470:
Intersection homology groups of complementary dimension and complementary perversity are dually paired.
902:
3028:
1241:
3846:
3733:
398:
848:
2689:
5360:
2326:
1989:
247:
109:
1384:
1362:
1139:
1110:
509:-dimensional allowable cycle gives an (ordinary) zero-cycle whose homology class is well-defined.
5563:
5498:
1912:
1749:
1024:
44:
2454:
and then truncating it in the derived category; more precisely it is given by
Deligne's formula
699:
2340:
2289:
430:
5598:
5529:
5482:
5472:
5471:. E. Cattani, Fouad El Zein, Phillip Griffiths, DĆ©ng TrĂĄng LĂȘ., eds. Princeton. 21 July 2014.
3893:
2074:
1942:
1871:
1517:
530:
256:
243:
40:
3446:
1633:
733:
5567:
5392:
3130:
2333:(considered as an element of the derived category, so the cohomology on the right means the
1482:
1215:
333:
5588:
2719:
1568:
480:
351:
303:
5585:
5427:
5313:
3658:
2334:
534:
381:
94:
4236:
3687:
3492:
624:{\displaystyle \emptyset =X_{-1}\subset X_{0}\subset X_{1}\subset \cdots \subset X_{n}=X}
819:
5365:
3875:
3826:
3762:
3713:
3638:
3472:
3210:
3008:
2965:
82:
17:
1673:
5657:
5571:
3897:
336:, then their intersection is a finite collection of points. Using the orientation of
113:
90:
86:
67:
2046:
The intersection homology groups are independent of the choice of stratification of
340:
one may assign to each of these points a sign; in other words intersection yields a
5648:
5546:
5521:
48:
3079:
32:
3406:
This can be computed explicitly by looking at the stalks of the cohomology. At
5647:(includes discussion on the etymology of the term "intersection homology") â
5486:
5490:
102:
28:
5608:
1202:{\displaystyle \mathbf {p} \colon \mathbb {Z} _{\geq 2}\to \mathbb {Z} }
1741:
517:
Intersection homology was originally defined on suitable spaces with a
380:-dimensional cycles; one may furthermore prove that this pairing is
5644:
5550:
1858:{\displaystyle \sigma ^{-1}\left(X_{n-k}\setminus X_{n-k-1}\right)}
55:
in the fall of 1974 and developed by them over the next few years.
3730:
can be refined by considering the intersection of an open disk in
2276:{\displaystyle I^{p}H_{n-i}(X)=I^{p}H^{i}(X)=H_{c}^{i}(IC_{p}(X))}
5591:
This gives a sheaf-theoretic approach to intersection cohomology.
395:—that is, when the space has places that do not look like
3399:{\displaystyle \tau _{\leq 1}\mathbf {R} i_{*}\mathbb {Q} _{U}}
4956:
On the complement of some closed set of codimension 2, we have
1670:. Its complement is the upper middle perversity, with values
2376:
is given by starting with the constant sheaf on the open set
4844:
4763:
4685:
4609:
4592:
4477:
4371:
4273:
1342:{\displaystyle \mathbf {p} (k+1)-\mathbf {p} (k)\in \{0,1\}}
5542:
C.R. Acad. Sci. t. 284 (1977), pp. 1549â1551 Serie A .
3227:, and the derivatives are homogeneous of degree 2. Setting
2150:
Deligne's formula for intersection cohomology states that
5616:
Hist. Math. 2, Amer. Math. Soc., 1989, pp. 543â585.
4582:
Truncating this gives the nontrivial cohomology sheaves
3120:{\displaystyle \mathbb {V} (f)\subset \mathbb {C} ^{3}}
4679:
4267:
5430:
5395:
5220:
5126:
5050:
4966:
4677:
4636:
4623:{\displaystyle {\mathcal {H}}^{0},{\mathcal {H}}^{1}}
4588:
4265:
4239:
3906:
3878:
3849:
3829:
3785:
3765:
3736:
3716:
3690:
3661:
3641:
3521:
3495:
3475:
3449:
3412:
3355:
3314:
3282:
3233:
3213:
3168:
3133:
3087:
3031:
3011:
2973:
2923:
2881:
2833:
2788:
2749:
2722:
2692:
2463:
2421:
2382:
2343:
2292:
2159:
2077:
1992:
1975:
is a subcomplex of the complex of singular chains on
1945:
1915:
1874:
1790:
1752:
1676:
1636:
1571:
1520:
1485:
1412:
1387:
1365:
1283:
1244:
1218:
1167:
1142:
1113:
1065:
968:
905:
851:
822:
778:
736:
702:
656:
550:
483:
433:
401:
354:
306:
259:
124:
3207:
vanish. This is because it is homogeneous of degree
5645:
What is the etymology of the term "perverse sheaf"?
5584:72 (1983), no. 1, 77–129. 10.1007/BF01389130
5552:
What is the etymology of the term "perverse sheaf"?
5540:
La dualité de
Poincaré pour les espaces singuliers.
1459:{\displaystyle \mathbf {p} (k)+\mathbf {q} (k)=k-2}
5445:
5416:
5270:
5176:
5097:
5013:
4923:
4663:
4622:
4574:
4251:
4225:
3884:
3864:
3835:
3815:
3771:
3751:
3722:
3702:
3676:
3647:
3627:
3507:
3481:
3461:
3435:
3398:
3341:
3300:
3268:
3219:
3199:
3154:
3119:
3070:
3017:
2997:
2948:
2906:
2867:
2819:
2774:
2735:
2708:
2675:
2446:
2407:
2368:
2325:is the intersection complex, a certain complex of
2317:
2275:
2095:
2024:
1967:
1928:
1901:
1857:
1765:
1708:
1662:
1618:
1547:
1506:
1458:
1395:
1373:
1341:
1267:
1227:
1201:
1150:
1121:
1099:
1000:
930:
889:
831:
808:
760:
714:
688:
623:
501:
458:
416:
372:
324:
281:
231:
2868:{\displaystyle \mathbb {C} _{X\setminus X_{n-2}}}
2716:is a truncation functor in the derived category,
5610:The development of intersection homology theory.
5278:is zero except on a set of codimension at least
5184:is zero except on a set of codimension at least
3127:has an isolated singularity at the origin since
2415:and repeatedly extending it to larger open sets
1514:. Its complement is the maximal perversity with
809:{\displaystyle U\cong \mathbb {R} ^{i}\times CL}
5596:An Introduction to Intersection Homology Theory
246:—this duality was understood in terms of
242:Classically—going back, for instance, to
58:Intersection cohomology was used to prove the
5614:A Century of Mathematics in America, Part II,
5528:. Progress in Mathematics, Birkhauser Boston
8:
3308:the inclusion map, the intersection complex
3263:
3257:
1336:
1324:
772:, and a filtration-preserving homeomorphism
1733:with some stratification, and a perversity
1027:such that every simplex is contained in an
5503:: CS1 maint: location missing publisher (
5037:the groups form the constant local system
4233:hence the cohomology sheaves at the stalk
3005:defined by a cubic homogeneous polynomial
2998:{\displaystyle X\subset \mathbb {CP} ^{2}}
5429:
5394:
5259:
5246:
5241:
5225:
5219:
5165:
5152:
5147:
5131:
5125:
5086:
5073:
5068:
5055:
5049:
5002:
4989:
4984:
4971:
4965:
4897:
4867:
4866:
4865:
4849:
4843:
4842:
4828:
4817:
4813:
4812:
4786:
4785:
4784:
4768:
4762:
4761:
4741:
4740:
4739:
4735:
4734:
4708:
4707:
4706:
4690:
4684:
4683:
4678:
4676:
4646:
4645:
4644:
4635:
4614:
4608:
4607:
4597:
4591:
4590:
4587:
4556:
4552:
4551:
4526:
4521:
4514:
4510:
4509:
4502:
4493:
4482:
4476:
4475:
4461:
4450:
4446:
4445:
4420:
4415:
4408:
4404:
4403:
4396:
4387:
4376:
4370:
4369:
4352:
4348:
4347:
4322:
4317:
4310:
4306:
4305:
4298:
4289:
4278:
4272:
4271:
4266:
4264:
4238:
4215:
4214:
4204:
4203:
4188:
4170:
4169:
4154:
4137:
4133:
4132:
4121:
4120:
4105:
4087:
4086:
4071:
4054:
4050:
4049:
4038:
4037:
4022:
4004:
4003:
3988:
3976:
3975:
3965:
3964:
3949:
3931:
3930:
3915:
3907:
3905:
3877:
3856:
3852:
3851:
3848:
3828:
3806:
3805:
3790:
3784:
3764:
3743:
3739:
3738:
3735:
3715:
3689:
3660:
3640:
3618:
3617:
3602:
3577:
3562:
3557:
3550:
3546:
3545:
3538:
3528:
3523:
3520:
3515:the cohomology is more interesting since
3494:
3474:
3448:
3420:
3419:
3411:
3390:
3386:
3385:
3378:
3369:
3360:
3354:
3324:
3323:
3322:
3313:
3281:
3241:
3240:
3232:
3212:
3173:
3167:
3132:
3111:
3107:
3106:
3089:
3088:
3086:
3062:
3049:
3036:
3030:
3010:
2989:
2985:
2982:
2981:
2972:
2934:
2922:
2892:
2880:
2851:
2840:
2836:
2835:
2832:
2799:
2787:
2760:
2748:
2727:
2721:
2697:
2691:
2659:
2648:
2644:
2643:
2633:
2624:
2600:
2578:
2569:
2539:
2526:
2517:
2493:
2471:
2462:
2432:
2420:
2393:
2381:
2351:
2342:
2300:
2291:
2255:
2239:
2234:
2212:
2202:
2174:
2164:
2158:
2076:
2035:are the homology groups of this complex.
2007:
1997:
1991:
1950:
1944:
1920:
1914:
1873:
1832:
1813:
1795:
1789:
1757:
1751:
1695:
1675:
1652:
1635:
1605:
1570:
1519:
1484:
1430:
1413:
1411:
1388:
1386:
1366:
1364:
1307:
1284:
1282:
1245:
1243:
1217:
1195:
1194:
1182:
1178:
1177:
1168:
1166:
1143:
1141:
1114:
1112:
1082:
1071:
1070:
1064:
1039:-simplexes, then the underlying space of
986:
973:
967:
916:
904:
875:
856:
850:
821:
791:
787:
786:
777:
735:
701:
674:
661:
655:
609:
590:
577:
561:
549:
482:
441:
432:
408:
404:
403:
400:
353:
305:
264:
258:
222:
221:
211:
210:
195:
181:
180:
159:
145:
144:
129:
123:
1100:{\displaystyle I^{\mathbf {p} }H_{i}(X)}
43:especially well-suited for the study of
5382:
5271:{\displaystyle H^{-i}(j_{x}^{!}IC_{p})}
5177:{\displaystyle H^{-i}(j_{x}^{*}IC_{p})}
3269:{\displaystyle U=\mathbb {V} (f)-\{0\}}
2927:
2885:
2844:
2792:
2753:
2652:
2425:
2386:
1825:
1130:
1035:â1 simplex is contained in exactly two
979:
909:
667:
5496:
5098:{\displaystyle H^{i}(j_{x}^{*}IC_{p})}
5014:{\displaystyle H^{i}(j_{x}^{*}IC_{p})}
1001:{\displaystyle X_{i}\setminus X_{i-1}}
689:{\displaystyle X_{i}\setminus X_{i-1}}
3816:{\displaystyle H^{k}(U;\mathbb {Q} )}
3779:, we can just compute the cohomology
950:is a topological pseudomanifold, the
7:
5538:Mark Goresky and Robert MacPherson,
4664:{\displaystyle IC_{\mathbb {V} (f)}}
3436:{\displaystyle p\in \mathbb {V} (f)}
3342:{\displaystyle IC_{\mathbb {V} (f)}}
3301:{\displaystyle i:U\hookrightarrow X}
2820:{\displaystyle X\setminus X_{n-k-1}}
5576:Goresky, Mark; MacPherson, Robert,
5558:Goresky, Mark; MacPherson, Robert,
5306:is the complementary perversity to
3200:{\displaystyle \partial _{i}f(0)=0}
2917:By replacing the constant sheaf on
108:have a fundamental property called
3170:
2949:{\displaystyle X\setminus X_{n-2}}
2907:{\displaystyle X\setminus X_{n-2}}
2775:{\displaystyle X\setminus X_{n-k}}
2447:{\displaystyle X\setminus X_{n-k}}
2408:{\displaystyle X\setminus X_{n-2}}
1917:
1754:
931:{\displaystyle X\setminus X_{n-1}}
551:
25:
5566:19 (1980), no. 2, 135–162.
4630:, hence the intersection complex
3071:{\displaystyle x^{3}+y^{3}+z^{3}}
1725:Fix a topological pseudomanifold
1268:{\displaystyle \mathbf {p} (2)=0}
1107:depend on a choice of perversity
5594:Frances Kirwan, Jonathan Woolf,
4494:
4388:
4290:
3865:{\displaystyle \mathbb {C} ^{*}}
3823:. This can be done by observing
3752:{\displaystyle \mathbb {C} ^{3}}
3524:
3370:
2625:
2570:
2518:
2134:to the intersection homology of
2126:is of codimension greater than 2
1431:
1414:
1389:
1367:
1308:
1285:
1246:
1169:
1144:
1115:
1072:
1043:is a topological pseudomanifold.
417:{\displaystyle \mathbb {R} ^{n}}
4952:) has the following properties
3872:bundle over the elliptic curve
2118:> 0, the space of points of
1777:(a singular simplex) is called
890:{\displaystyle X_{n-1}=X_{n-2}}
638:by closed subspaces such that:
5440:
5434:
5405:
5399:
5356:Topologically stratified space
5265:
5234:
5171:
5140:
5092:
5061:
5008:
4977:
4882:
4877:
4871:
4855:
4801:
4796:
4790:
4774:
4751:
4745:
4723:
4718:
4712:
4696:
4656:
4650:
4522:
4416:
4318:
4208:
4194:
4174:
4160:
4125:
4111:
4091:
4077:
4042:
4028:
4008:
3994:
3969:
3955:
3935:
3921:
3810:
3796:
3671:
3665:
3622:
3608:
3558:
3430:
3424:
3334:
3328:
3292:
3251:
3245:
3188:
3182:
3143:
3137:
3099:
3093:
2709:{\displaystyle \tau _{\leq p}}
2613:
2607:
2558:
2546:
2506:
2500:
2483:
2477:
2363:
2357:
2312:
2306:
2270:
2267:
2261:
2245:
2224:
2218:
2192:
2186:
2138:(with the middle perversity).
2122:where the fiber has dimension
2087:
2019:
2013:
1962:
1956:
1896:
1890:
1721:Singular intersection homology
1703:
1692:
1680:
1677:
1649:
1637:
1613:
1602:
1590:
1587:
1581:
1575:
1530:
1524:
1495:
1489:
1441:
1435:
1424:
1418:
1318:
1312:
1301:
1289:
1256:
1250:
1191:
1094:
1088:
768:-dimensional stratified space
755:
737:
696:, there exists a neighborhood
496:
484:
453:
447:
367:
355:
319:
307:
276:
270:
215:
201:
188:
185:
171:
149:
135:
64:RiemannâHilbert correspondence
1:
4934:Properties of the complex IC(
2337:of the complex). The complex
2025:{\displaystyle I^{p}H_{i}(X)}
1059:Intersection homology groups
5572:10.1016/0040-9383(80)90003-8
5560:Intersection homology theory
3162:and all partial derivatives
1396:{\displaystyle \mathbf {p} }
1374:{\displaystyle \mathbf {q} }
1151:{\displaystyle \mathbf {p} }
1122:{\displaystyle \mathbf {p} }
5626:Encyclopedia of Mathematics
5336:) in the derived category.
3900:gives the cohomology groups
2067:resolution of singularities
1929:{\displaystyle \Delta ^{i}}
1766:{\displaystyle \Delta ^{i}}
1479:The minimal perversity has
77:GoreskyâMacPherson approach
66:. It is closely related to
60:KazhdanâLusztig conjectures
5700:
1740:A map Ï from the standard
1235:to the integers such that
715:{\displaystyle U\subset X}
527:topological pseudomanifold
332:-dimensional cycle are in
296:-dimensional cycle. If an
5578:Intersection homology. II
5424:are sometimes written as
2875:is the constant sheaf on
2369:{\displaystyle IC_{p}(X)}
2318:{\displaystyle IC_{p}(X)}
459:{\displaystyle IH_{i}(X)}
5582:Inventiones Mathematicae
2096:{\displaystyle f:X\to Y}
1968:{\displaystyle I^{p}(X)}
1902:{\displaystyle i-k+p(k)}
1548:{\displaystyle q(k)=k-2}
1474:Examples of perversities
1358:complementary perversity
282:{\displaystyle H_{j}(X)}
5621:"Intersection homology"
5526:Intersection Cohomology
5120: > 0 then
4671:has cohomology sheaves
3462:{\displaystyle p\neq 0}
1663:{\displaystyle (k-2)/2}
761:{\displaystyle (n-i-1)}
18:Intersection cohomology
5447:
5418:
5417:{\displaystyle p(k)-n}
5272:
5178:
5099:
5015:
4925:
4665:
4624:
4576:
4253:
4227:
3886:
3866:
3837:
3817:
3773:
3753:
3724:
3704:
3678:
3649:
3629:
3509:
3483:
3463:
3437:
3400:
3343:
3302:
3270:
3221:
3201:
3156:
3155:{\displaystyle f(0)=0}
3121:
3072:
3019:
2999:
2950:
2908:
2869:
2821:
2776:
2737:
2710:
2677:
2448:
2409:
2370:
2319:
2277:
2097:
2026:
1969:
1930:
1903:
1859:
1767:
1710:
1664:
1620:
1549:
1508:
1507:{\displaystyle p(k)=0}
1460:
1397:
1375:
1343:
1269:
1229:
1228:{\displaystyle \geq 2}
1203:
1152:
1123:
1101:
1002:
932:
891:
833:
810:
762:
716:
690:
625:
541:that has a filtration
503:
460:
418:
374:
326:
283:
233:
5674:Generalized manifolds
5507:) CS1 maint: others (
5448:
5419:
5371:Mixed Hodge structure
5346:Decomposition theorem
5273:
5179:
5100:
5016:
4926:
4666:
4625:
4577:
4254:
4228:
3887:
3867:
3838:
3818:
3774:
3754:
3725:
3705:
3679:
3655:where the closure of
3650:
3630:
3510:
3484:
3464:
3438:
3401:
3344:
3303:
3271:
3222:
3202:
3157:
3122:
3073:
3020:
3000:
2951:
2909:
2870:
2822:
2777:
2738:
2736:{\displaystyle i_{k}}
2711:
2678:
2449:
2410:
2371:
2327:constructible sheaves
2320:
2278:
2106:of a complex variety
2098:
2027:
1970:
1931:
1904:
1860:
1768:
1711:
1665:
1621:
1619:{\displaystyle m(k)=}
1550:
1509:
1461:
1398:
1376:
1344:
1270:
1230:
1204:
1153:
1124:
1102:
1003:
933:
892:
834:
811:
763:
717:
691:
626:
504:
502:{\displaystyle (n-i)}
461:
419:
375:
373:{\displaystyle (n-i)}
327:
325:{\displaystyle (n-i)}
284:
234:
37:intersection homology
5446:{\displaystyle p(k)}
5428:
5393:
5351:BorelâMoore homology
5218:
5124:
5048:
4964:
4675:
4634:
4586:
4263:
4237:
3904:
3876:
3847:
3827:
3783:
3763:
3734:
3714:
3688:
3684:contains the origin
3677:{\displaystyle i(V)}
3659:
3639:
3519:
3493:
3473:
3447:
3410:
3353:
3312:
3280:
3231:
3211:
3166:
3131:
3085:
3029:
3009:
2971:
2921:
2879:
2831:
2786:
2747:
2743:is the inclusion of
2720:
2690:
2461:
2419:
2380:
2341:
2290:
2157:
2075:
1990:
1943:
1913:
1872:
1868:is contained in the
1788:
1750:
1674:
1634:
1569:
1518:
1483:
1410:
1385:
1363:
1281:
1242:
1216:
1165:
1140:
1111:
1063:
966:
903:
849:
839:is the open cone on
820:
776:
734:
700:
654:
548:
481:
431:
399:
352:
304:
300:-dimensional and an
292:is represented by a
257:
122:
5684:Cohomology theories
5664:Intersection theory
5361:Intersection theory
5251:
5157:
5078:
5029:≠ 0, and for
4994:
4836:
4469:
4252:{\displaystyle p=0}
3703:{\displaystyle p=0}
3508:{\displaystyle p=0}
2244:
646:and for each point
248:intersection theory
5669:Algebraic topology
5443:
5414:
5268:
5237:
5174:
5143:
5095:
5064:
5011:
4980:
4921:
4919:
4811:
4661:
4620:
4572:
4570:
4444:
4249:
4223:
4221:
3882:
3862:
3833:
3813:
3769:
3749:
3720:
3700:
3674:
3645:
3625:
3593:
3505:
3479:
3459:
3433:
3396:
3339:
3298:
3266:
3217:
3197:
3152:
3117:
3068:
3015:
2995:
2946:
2904:
2865:
2817:
2772:
2733:
2706:
2673:
2444:
2405:
2366:
2315:
2273:
2230:
2093:
2022:
1965:
1926:
1899:
1855:
1763:
1706:
1660:
1616:
1545:
1504:
1456:
1393:
1371:
1339:
1265:
1225:
1199:
1148:
1119:
1097:
1025:simplicial complex
998:
928:
887:
832:{\displaystyle CL}
829:
806:
758:
712:
686:
621:
499:
456:
414:
370:
322:
279:
229:
39:is an analogue of
5607:Kleiman, Steven.
5478:978-0-691-16134-1
5332: = dim(
5282:for the smallest
5188:for the smallest
5113: < 0
4900:
3894:hyperplane bundle
3885:{\displaystyle X}
3836:{\displaystyle U}
3772:{\displaystyle U}
3723:{\displaystyle V}
3710:. Since any such
3648:{\displaystyle V}
3581:
3578:
3482:{\displaystyle 0}
3220:{\displaystyle 3}
3018:{\displaystyle f}
2061:Small resolutions
1560:middle perversity
53:Robert MacPherson
41:singular homology
16:(Redirected from
5691:
5679:Duality theories
5634:
5555:
5514:
5512:
5502:
5494:
5489:. Archived from
5470:
5460:
5454:
5452:
5450:
5449:
5444:
5423:
5421:
5420:
5415:
5387:
5277:
5275:
5274:
5269:
5264:
5263:
5250:
5245:
5233:
5232:
5183:
5181:
5180:
5175:
5170:
5169:
5156:
5151:
5139:
5138:
5104:
5102:
5101:
5096:
5091:
5090:
5077:
5072:
5060:
5059:
5020:
5018:
5017:
5012:
5007:
5006:
4993:
4988:
4976:
4975:
4930:
4928:
4927:
4922:
4920:
4901:
4898:
4881:
4880:
4870:
4854:
4853:
4848:
4847:
4835:
4827:
4816:
4800:
4799:
4789:
4773:
4772:
4767:
4766:
4755:
4754:
4744:
4738:
4722:
4721:
4711:
4695:
4694:
4689:
4688:
4670:
4668:
4667:
4662:
4660:
4659:
4649:
4629:
4627:
4626:
4621:
4619:
4618:
4613:
4612:
4602:
4601:
4596:
4595:
4581:
4579:
4578:
4573:
4571:
4567:
4566:
4555:
4542:
4538:
4537:
4536:
4525:
4519:
4518:
4513:
4507:
4506:
4497:
4487:
4486:
4481:
4480:
4468:
4460:
4449:
4436:
4432:
4431:
4430:
4419:
4413:
4412:
4407:
4401:
4400:
4391:
4381:
4380:
4375:
4374:
4363:
4362:
4351:
4338:
4334:
4333:
4332:
4321:
4315:
4314:
4309:
4303:
4302:
4293:
4283:
4282:
4277:
4276:
4258:
4256:
4255:
4250:
4232:
4230:
4229:
4224:
4222:
4218:
4207:
4193:
4192:
4173:
4159:
4158:
4145:
4144:
4136:
4124:
4110:
4109:
4090:
4076:
4075:
4062:
4061:
4053:
4041:
4027:
4026:
4007:
3993:
3992:
3979:
3968:
3954:
3953:
3934:
3920:
3919:
3891:
3889:
3888:
3883:
3871:
3869:
3868:
3863:
3861:
3860:
3855:
3842:
3840:
3839:
3834:
3822:
3820:
3819:
3814:
3809:
3795:
3794:
3778:
3776:
3775:
3770:
3758:
3756:
3755:
3750:
3748:
3747:
3742:
3729:
3727:
3726:
3721:
3709:
3707:
3706:
3701:
3683:
3681:
3680:
3675:
3654:
3652:
3651:
3646:
3634:
3632:
3631:
3626:
3621:
3607:
3606:
3594:
3592:
3579:
3573:
3572:
3561:
3555:
3554:
3549:
3543:
3542:
3533:
3532:
3527:
3514:
3512:
3511:
3506:
3488:
3486:
3485:
3480:
3468:
3466:
3465:
3460:
3442:
3440:
3439:
3434:
3423:
3405:
3403:
3402:
3397:
3395:
3394:
3389:
3383:
3382:
3373:
3368:
3367:
3348:
3346:
3345:
3340:
3338:
3337:
3327:
3307:
3305:
3304:
3299:
3275:
3273:
3272:
3267:
3244:
3226:
3224:
3223:
3218:
3206:
3204:
3203:
3198:
3178:
3177:
3161:
3159:
3158:
3153:
3126:
3124:
3123:
3118:
3116:
3115:
3110:
3092:
3077:
3075:
3074:
3069:
3067:
3066:
3054:
3053:
3041:
3040:
3024:
3022:
3021:
3016:
3004:
3002:
3001:
2996:
2994:
2993:
2988:
2955:
2953:
2952:
2947:
2945:
2944:
2913:
2911:
2910:
2905:
2903:
2902:
2874:
2872:
2871:
2866:
2864:
2863:
2862:
2861:
2839:
2826:
2824:
2823:
2818:
2816:
2815:
2781:
2779:
2778:
2773:
2771:
2770:
2742:
2740:
2739:
2734:
2732:
2731:
2715:
2713:
2712:
2707:
2705:
2704:
2682:
2680:
2679:
2674:
2672:
2671:
2670:
2669:
2647:
2641:
2640:
2628:
2623:
2622:
2592:
2591:
2573:
2568:
2567:
2534:
2533:
2521:
2516:
2515:
2476:
2475:
2453:
2451:
2450:
2445:
2443:
2442:
2414:
2412:
2411:
2406:
2404:
2403:
2375:
2373:
2372:
2367:
2356:
2355:
2324:
2322:
2321:
2316:
2305:
2304:
2282:
2280:
2279:
2274:
2260:
2259:
2243:
2238:
2217:
2216:
2207:
2206:
2185:
2184:
2169:
2168:
2112:small resolution
2102:
2100:
2099:
2094:
2031:
2029:
2028:
2023:
2012:
2011:
2002:
2001:
1974:
1972:
1971:
1966:
1955:
1954:
1935:
1933:
1932:
1927:
1925:
1924:
1908:
1906:
1905:
1900:
1864:
1862:
1861:
1856:
1854:
1850:
1849:
1848:
1824:
1823:
1803:
1802:
1772:
1770:
1769:
1764:
1762:
1761:
1715:
1713:
1712:
1709:{\displaystyle }
1707:
1699:
1669:
1667:
1666:
1661:
1656:
1625:
1623:
1622:
1617:
1609:
1554:
1552:
1551:
1546:
1513:
1511:
1510:
1505:
1465:
1463:
1462:
1457:
1434:
1417:
1403:is the one with
1402:
1400:
1399:
1394:
1392:
1380:
1378:
1377:
1372:
1370:
1348:
1346:
1345:
1340:
1311:
1288:
1274:
1272:
1271:
1266:
1249:
1234:
1232:
1231:
1226:
1208:
1206:
1205:
1200:
1198:
1190:
1189:
1181:
1172:
1157:
1155:
1154:
1149:
1147:
1128:
1126:
1125:
1120:
1118:
1106:
1104:
1103:
1098:
1087:
1086:
1077:
1076:
1075:
1007:
1005:
1004:
999:
997:
996:
978:
977:
937:
935:
934:
929:
927:
926:
896:
894:
893:
888:
886:
885:
867:
866:
838:
836:
835:
830:
815:
813:
812:
807:
796:
795:
790:
767:
765:
764:
759:
721:
719:
718:
713:
695:
693:
692:
687:
685:
684:
666:
665:
630:
628:
627:
622:
614:
613:
595:
594:
582:
581:
569:
568:
508:
506:
505:
500:
465:
463:
462:
457:
446:
445:
423:
421:
420:
415:
413:
412:
407:
379:
377:
376:
371:
334:general position
331:
329:
328:
323:
288:
286:
285:
280:
269:
268:
250:. An element of
238:
236:
235:
230:
225:
214:
200:
199:
184:
170:
169:
148:
134:
133:
110:Poincaré duality
47:, discovered by
21:
5699:
5698:
5694:
5693:
5692:
5690:
5689:
5688:
5654:
5653:
5641:
5619:
5545:
5518:
5517:
5495:
5493:on 15 Aug 2020.
5479:
5468:
5462:
5461:
5457:
5426:
5425:
5391:
5390:
5388:
5384:
5379:
5342:
5327:
5321:
5314:Verdier duality
5294:) â„ (
5255:
5221:
5216:
5215:
5161:
5127:
5122:
5121:
5082:
5051:
5046:
5045:
4998:
4967:
4962:
4961:
4947:
4940:
4918:
4917:
4895:
4890:
4885:
4861:
4841:
4838:
4837:
4809:
4804:
4780:
4760:
4757:
4756:
4733:
4731:
4726:
4702:
4682:
4673:
4672:
4640:
4632:
4631:
4606:
4589:
4584:
4583:
4569:
4568:
4550:
4548:
4543:
4520:
4508:
4498:
4492:
4488:
4474:
4471:
4470:
4442:
4437:
4414:
4402:
4392:
4386:
4382:
4368:
4365:
4364:
4346:
4344:
4339:
4316:
4304:
4294:
4288:
4284:
4270:
4261:
4260:
4235:
4234:
4220:
4219:
4184:
4177:
4150:
4147:
4146:
4131:
4101:
4094:
4067:
4064:
4063:
4048:
4018:
4011:
3984:
3981:
3980:
3945:
3938:
3911:
3902:
3901:
3874:
3873:
3850:
3845:
3844:
3825:
3824:
3786:
3781:
3780:
3761:
3760:
3737:
3732:
3731:
3712:
3711:
3686:
3685:
3657:
3656:
3637:
3636:
3598:
3582:
3556:
3544:
3534:
3522:
3517:
3516:
3491:
3490:
3471:
3470:
3445:
3444:
3408:
3407:
3384:
3374:
3356:
3351:
3350:
3318:
3310:
3309:
3278:
3277:
3229:
3228:
3209:
3208:
3169:
3164:
3163:
3129:
3128:
3105:
3083:
3082:
3058:
3045:
3032:
3027:
3026:
3007:
3006:
2980:
2969:
2968:
2964:Given a smooth
2962:
2930:
2919:
2918:
2888:
2877:
2876:
2847:
2834:
2829:
2828:
2795:
2784:
2783:
2756:
2745:
2744:
2723:
2718:
2717:
2693:
2688:
2687:
2655:
2642:
2629:
2596:
2574:
2535:
2522:
2489:
2467:
2459:
2458:
2428:
2417:
2416:
2389:
2378:
2377:
2347:
2339:
2338:
2335:hypercohomology
2296:
2288:
2287:
2251:
2208:
2198:
2170:
2160:
2155:
2154:
2148:
2073:
2072:
2063:
2003:
1993:
1988:
1987:
1946:
1941:
1940:
1916:
1911:
1910:
1870:
1869:
1828:
1809:
1808:
1804:
1791:
1786:
1785:
1753:
1748:
1747:
1723:
1672:
1671:
1632:
1631:
1567:
1566:
1516:
1515:
1481:
1480:
1476:
1408:
1407:
1383:
1382:
1361:
1360:
1279:
1278:
1240:
1239:
1214:
1213:
1176:
1163:
1162:
1138:
1137:
1109:
1108:
1078:
1066:
1061:
1060:
1057:
982:
969:
964:
963:
912:
901:
900:
871:
852:
847:
846:
818:
817:
785:
774:
773:
732:
731:
698:
697:
670:
657:
652:
651:
605:
586:
573:
557:
546:
545:
515:
513:Stratifications
479:
478:
437:
429:
428:
402:
397:
396:
350:
349:
302:
301:
260:
255:
254:
191:
155:
125:
120:
119:
114:perfect pairing
83:homology groups
79:
45:singular spaces
23:
22:
15:
12:
11:
5:
5697:
5695:
5687:
5686:
5681:
5676:
5671:
5666:
5656:
5655:
5652:
5651:
5640:
5639:External links
5637:
5636:
5635:
5617:
5605:
5592:
5574:
5556:
5543:
5536:
5516:
5515:
5477:
5455:
5442:
5439:
5436:
5433:
5413:
5410:
5407:
5404:
5401:
5398:
5381:
5380:
5378:
5375:
5374:
5373:
5368:
5366:Perverse sheaf
5363:
5358:
5353:
5348:
5341:
5338:
5323:
5317:
5300:
5299:
5267:
5262:
5258:
5254:
5249:
5244:
5240:
5236:
5231:
5228:
5224:
5208:
5200:) â„
5173:
5168:
5164:
5160:
5155:
5150:
5146:
5142:
5137:
5134:
5130:
5114:
5094:
5089:
5085:
5081:
5076:
5071:
5067:
5063:
5058:
5054:
5042:
5041:
5010:
5005:
5001:
4997:
4992:
4987:
4983:
4979:
4974:
4970:
4958:
4957:
4943:
4942:The complex IC
4939:
4932:
4916:
4913:
4910:
4907:
4904:
4896:
4894:
4891:
4889:
4886:
4884:
4879:
4876:
4873:
4869:
4864:
4860:
4857:
4852:
4846:
4840:
4839:
4834:
4831:
4826:
4823:
4820:
4815:
4810:
4808:
4805:
4803:
4798:
4795:
4792:
4788:
4783:
4779:
4776:
4771:
4765:
4759:
4758:
4753:
4750:
4747:
4743:
4737:
4732:
4730:
4727:
4725:
4720:
4717:
4714:
4710:
4705:
4701:
4698:
4693:
4687:
4681:
4680:
4658:
4655:
4652:
4648:
4643:
4639:
4617:
4611:
4605:
4600:
4594:
4565:
4562:
4559:
4554:
4549:
4547:
4544:
4541:
4535:
4532:
4529:
4524:
4517:
4512:
4505:
4501:
4496:
4491:
4485:
4479:
4473:
4472:
4467:
4464:
4459:
4456:
4453:
4448:
4443:
4441:
4438:
4435:
4429:
4426:
4423:
4418:
4411:
4406:
4399:
4395:
4390:
4385:
4379:
4373:
4367:
4366:
4361:
4358:
4355:
4350:
4345:
4343:
4340:
4337:
4331:
4328:
4325:
4320:
4313:
4308:
4301:
4297:
4292:
4287:
4281:
4275:
4269:
4268:
4248:
4245:
4242:
4217:
4213:
4210:
4206:
4202:
4199:
4196:
4191:
4187:
4183:
4180:
4178:
4176:
4172:
4168:
4165:
4162:
4157:
4153:
4149:
4148:
4143:
4140:
4135:
4130:
4127:
4123:
4119:
4116:
4113:
4108:
4104:
4100:
4097:
4095:
4093:
4089:
4085:
4082:
4079:
4074:
4070:
4066:
4065:
4060:
4057:
4052:
4047:
4044:
4040:
4036:
4033:
4030:
4025:
4021:
4017:
4014:
4012:
4010:
4006:
4002:
3999:
3996:
3991:
3987:
3983:
3982:
3978:
3974:
3971:
3967:
3963:
3960:
3957:
3952:
3948:
3944:
3941:
3939:
3937:
3933:
3929:
3926:
3923:
3918:
3914:
3910:
3909:
3881:
3859:
3854:
3832:
3812:
3808:
3804:
3801:
3798:
3793:
3789:
3768:
3746:
3741:
3719:
3699:
3696:
3693:
3673:
3670:
3667:
3664:
3644:
3624:
3620:
3616:
3613:
3610:
3605:
3601:
3597:
3591:
3588:
3585:
3576:
3571:
3568:
3565:
3560:
3553:
3548:
3541:
3537:
3531:
3526:
3504:
3501:
3498:
3478:
3458:
3455:
3452:
3432:
3429:
3426:
3422:
3418:
3415:
3393:
3388:
3381:
3377:
3372:
3366:
3363:
3359:
3336:
3333:
3330:
3326:
3321:
3317:
3297:
3294:
3291:
3288:
3285:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3243:
3239:
3236:
3216:
3196:
3193:
3190:
3187:
3184:
3181:
3176:
3172:
3151:
3148:
3145:
3142:
3139:
3136:
3114:
3109:
3104:
3101:
3098:
3095:
3091:
3065:
3061:
3057:
3052:
3048:
3044:
3039:
3035:
3014:
2992:
2987:
2984:
2979:
2976:
2966:elliptic curve
2961:
2958:
2943:
2940:
2937:
2933:
2929:
2926:
2901:
2898:
2895:
2891:
2887:
2884:
2860:
2857:
2854:
2850:
2846:
2843:
2838:
2814:
2811:
2808:
2805:
2802:
2798:
2794:
2791:
2769:
2766:
2763:
2759:
2755:
2752:
2730:
2726:
2703:
2700:
2696:
2684:
2683:
2668:
2665:
2662:
2658:
2654:
2651:
2646:
2639:
2636:
2632:
2627:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2599:
2595:
2590:
2587:
2584:
2581:
2577:
2572:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2538:
2532:
2529:
2525:
2520:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2492:
2488:
2485:
2482:
2479:
2474:
2470:
2466:
2441:
2438:
2435:
2431:
2427:
2424:
2402:
2399:
2396:
2392:
2388:
2385:
2365:
2362:
2359:
2354:
2350:
2346:
2314:
2311:
2308:
2303:
2299:
2295:
2284:
2283:
2272:
2269:
2266:
2263:
2258:
2254:
2250:
2247:
2242:
2237:
2233:
2229:
2226:
2223:
2220:
2215:
2211:
2205:
2201:
2197:
2194:
2191:
2188:
2183:
2180:
2177:
2173:
2167:
2163:
2147:
2144:
2104:
2103:
2092:
2089:
2086:
2083:
2080:
2062:
2059:
2033:
2032:
2021:
2018:
2015:
2010:
2006:
2000:
1996:
1964:
1961:
1958:
1953:
1949:
1923:
1919:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1866:
1865:
1853:
1847:
1844:
1841:
1838:
1835:
1831:
1827:
1822:
1819:
1816:
1812:
1807:
1801:
1798:
1794:
1760:
1756:
1722:
1719:
1718:
1717:
1705:
1702:
1698:
1694:
1691:
1688:
1685:
1682:
1679:
1659:
1655:
1651:
1648:
1645:
1642:
1639:
1615:
1612:
1608:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1565:is defined by
1556:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1503:
1500:
1497:
1494:
1491:
1488:
1475:
1472:
1468:
1467:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1433:
1429:
1426:
1423:
1420:
1416:
1391:
1369:
1351:
1350:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1310:
1306:
1303:
1300:
1297:
1294:
1291:
1287:
1276:
1264:
1261:
1258:
1255:
1252:
1248:
1224:
1221:
1212:from integers
1210:
1209:
1197:
1193:
1188:
1185:
1180:
1175:
1171:
1158:is a function
1146:
1131:Goresky (2010)
1117:
1096:
1093:
1090:
1085:
1081:
1074:
1069:
1056:
1053:
1052:
1051:
1044:
995:
992:
989:
985:
981:
976:
972:
944:
943:
925:
922:
919:
915:
911:
908:
898:
884:
881:
878:
874:
870:
865:
862:
859:
855:
844:
828:
825:
805:
802:
799:
794:
789:
784:
781:
757:
754:
751:
748:
745:
742:
739:
711:
708:
705:
683:
680:
677:
673:
669:
664:
660:
632:
631:
620:
617:
612:
608:
604:
601:
598:
593:
589:
585:
580:
576:
572:
567:
564:
560:
556:
553:
519:stratification
514:
511:
498:
495:
492:
489:
486:
467:
466:
455:
452:
449:
444:
440:
436:
411:
406:
369:
366:
363:
360:
357:
321:
318:
315:
312:
309:
290:
289:
278:
275:
272:
267:
263:
244:Henri Poincaré
240:
239:
228:
224:
220:
217:
213:
209:
206:
203:
198:
194:
190:
187:
183:
179:
176:
173:
168:
165:
162:
158:
154:
151:
147:
143:
140:
137:
132:
128:
78:
75:
31:, a branch of
24:
14:
13:
10:
9:
6:
4:
3:
2:
5696:
5685:
5682:
5680:
5677:
5675:
5672:
5670:
5667:
5665:
5662:
5661:
5659:
5650:
5646:
5643:
5642:
5638:
5632:
5628:
5627:
5622:
5618:
5615:
5612:
5611:
5606:
5604:
5603:1-58488-184-4
5600:
5597:
5593:
5590:
5587:
5583:
5579:
5575:
5573:
5569:
5565:
5561:
5557:
5554:
5553:
5548:
5547:Goresky, Mark
5544:
5541:
5537:
5535:
5534:0-8176-3274-3
5531:
5527:
5523:
5520:
5519:
5513:, pp. 281-282
5510:
5506:
5500:
5492:
5488:
5484:
5480:
5474:
5467:
5466:
5459:
5456:
5437:
5431:
5411:
5408:
5402:
5396:
5386:
5383:
5376:
5372:
5369:
5367:
5364:
5362:
5359:
5357:
5354:
5352:
5349:
5347:
5344:
5343:
5339:
5337:
5335:
5331:
5326:
5320:
5315:
5311:
5309:
5305:
5297:
5293:
5289:
5285:
5281:
5260:
5256:
5252:
5247:
5242:
5238:
5229:
5226:
5222:
5213:
5209:
5207:
5204: â
5203:
5199:
5195:
5191:
5187:
5166:
5162:
5158:
5153:
5148:
5144:
5135:
5132:
5128:
5119:
5115:
5112:
5109: +
5108:
5087:
5083:
5079:
5074:
5069:
5065:
5056:
5052:
5044:
5043:
5040:
5036:
5032:
5028:
5024:
5003:
4999:
4995:
4990:
4985:
4981:
4972:
4968:
4960:
4959:
4955:
4954:
4953:
4951:
4946:
4937:
4933:
4931:
4914:
4911:
4908:
4905:
4902:
4892:
4887:
4874:
4862:
4858:
4850:
4832:
4829:
4824:
4821:
4818:
4806:
4793:
4781:
4777:
4769:
4748:
4728:
4715:
4703:
4699:
4691:
4653:
4641:
4637:
4615:
4603:
4598:
4563:
4560:
4557:
4545:
4539:
4533:
4530:
4527:
4515:
4503:
4499:
4489:
4483:
4465:
4462:
4457:
4454:
4451:
4439:
4433:
4427:
4424:
4421:
4409:
4397:
4393:
4383:
4377:
4359:
4356:
4353:
4341:
4335:
4329:
4326:
4323:
4311:
4299:
4295:
4285:
4279:
4246:
4243:
4240:
4211:
4200:
4197:
4189:
4185:
4181:
4179:
4166:
4163:
4155:
4151:
4141:
4138:
4128:
4117:
4114:
4106:
4102:
4098:
4096:
4083:
4080:
4072:
4068:
4058:
4055:
4045:
4034:
4031:
4023:
4019:
4015:
4013:
4000:
3997:
3989:
3985:
3972:
3961:
3958:
3950:
3946:
3942:
3940:
3927:
3924:
3916:
3912:
3899:
3898:Wang sequence
3895:
3879:
3857:
3830:
3802:
3799:
3791:
3787:
3766:
3744:
3717:
3697:
3694:
3691:
3668:
3662:
3642:
3614:
3611:
3603:
3599:
3595:
3589:
3586:
3583:
3574:
3569:
3566:
3563:
3551:
3539:
3535:
3529:
3502:
3499:
3496:
3476:
3456:
3453:
3450:
3427:
3416:
3413:
3391:
3379:
3375:
3364:
3361:
3357:
3331:
3319:
3315:
3295:
3289:
3286:
3283:
3260:
3254:
3248:
3237:
3234:
3214:
3194:
3191:
3185:
3179:
3174:
3149:
3146:
3140:
3134:
3112:
3102:
3096:
3081:
3063:
3059:
3055:
3050:
3046:
3042:
3037:
3033:
3012:
2990:
2977:
2974:
2967:
2959:
2957:
2941:
2938:
2935:
2931:
2924:
2915:
2899:
2896:
2893:
2889:
2882:
2858:
2855:
2852:
2848:
2841:
2812:
2809:
2806:
2803:
2800:
2796:
2789:
2767:
2764:
2761:
2757:
2750:
2728:
2724:
2701:
2698:
2694:
2666:
2663:
2660:
2656:
2649:
2637:
2634:
2630:
2619:
2616:
2610:
2604:
2601:
2597:
2593:
2588:
2585:
2582:
2579:
2575:
2564:
2561:
2555:
2552:
2549:
2543:
2540:
2536:
2530:
2527:
2523:
2512:
2509:
2503:
2497:
2494:
2490:
2486:
2480:
2472:
2468:
2464:
2457:
2456:
2455:
2439:
2436:
2433:
2429:
2422:
2400:
2397:
2394:
2390:
2383:
2360:
2352:
2348:
2344:
2336:
2332:
2328:
2309:
2301:
2297:
2293:
2264:
2256:
2252:
2248:
2240:
2235:
2231:
2227:
2221:
2213:
2209:
2203:
2199:
2195:
2189:
2181:
2178:
2175:
2171:
2165:
2161:
2153:
2152:
2151:
2145:
2143:
2139:
2137:
2133:
2129:
2125:
2121:
2117:
2114:if for every
2113:
2109:
2090:
2084:
2081:
2078:
2071:
2070:
2069:
2068:
2060:
2058:
2056:
2051:
2049:
2044:
2041:
2036:
2016:
2008:
2004:
1998:
1994:
1986:
1985:
1984:
1982:
1978:
1959:
1951:
1947:
1937:
1921:
1893:
1887:
1884:
1881:
1878:
1875:
1851:
1845:
1842:
1839:
1836:
1833:
1829:
1820:
1817:
1814:
1810:
1805:
1799:
1796:
1792:
1784:
1783:
1782:
1780:
1776:
1758:
1746:
1744:
1738:
1736:
1732:
1729:of dimension
1728:
1720:
1700:
1696:
1689:
1686:
1683:
1657:
1653:
1646:
1643:
1640:
1629:
1610:
1606:
1599:
1596:
1593:
1584:
1578:
1572:
1564:
1561:
1557:
1542:
1539:
1536:
1533:
1527:
1521:
1501:
1498:
1492:
1486:
1478:
1477:
1473:
1471:
1453:
1450:
1447:
1444:
1438:
1427:
1421:
1406:
1405:
1404:
1359:
1354:
1333:
1330:
1327:
1321:
1315:
1304:
1298:
1295:
1292:
1277:
1262:
1259:
1253:
1238:
1237:
1236:
1222:
1219:
1186:
1183:
1173:
1161:
1160:
1159:
1136:
1132:
1091:
1083:
1079:
1067:
1054:
1049:
1045:
1042:
1038:
1034:
1031:-simplex and
1030:
1026:
1023:-dimensional
1022:
1018:
1014:
1013:
1012:
1009:
993:
990:
987:
983:
974:
970:
962:is the space
961:
957:
954:-dimensional
953:
949:
941:
923:
920:
917:
913:
906:
899:
882:
879:
876:
872:
868:
863:
860:
857:
853:
845:
842:
826:
823:
803:
800:
797:
792:
782:
779:
771:
752:
749:
746:
743:
740:
729:
725:
709:
706:
703:
681:
678:
675:
671:
662:
658:
649:
645:
641:
640:
639:
637:
618:
615:
610:
606:
602:
599:
596:
591:
587:
583:
578:
574:
570:
565:
562:
558:
554:
544:
543:
542:
540:
536:
532:
529:. This is a (
528:
525:-dimensional
524:
520:
512:
510:
493:
490:
487:
476:
472:
450:
442:
438:
434:
427:
426:
425:
409:
394:
393:singularities
390:
385:
383:
364:
361:
358:
347:
343:
339:
335:
316:
313:
310:
299:
295:
273:
265:
261:
253:
252:
251:
249:
245:
226:
218:
207:
204:
196:
192:
177:
174:
166:
163:
160:
156:
152:
141:
138:
130:
126:
118:
117:
116:
115:
112:: there is a
111:
107:
104:
101:-dimensional
100:
96:
92:
88:
84:
76:
74:
72:
70:
65:
61:
56:
54:
50:
46:
42:
38:
34:
30:
19:
5649:MathOverflow
5624:
5613:
5609:
5595:
5577:
5559:
5551:
5539:
5525:
5522:Armand Borel
5491:the original
5465:Hodge Theory
5464:
5458:
5385:
5333:
5329:
5324:
5318:
5312:
5307:
5303:
5301:
5295:
5291:
5287:
5283:
5279:
5214:> 0 then
5211:
5205:
5201:
5197:
5193:
5189:
5185:
5117:
5110:
5106:
5038:
5034:
5030:
5026:
5022:
4949:
4944:
4941:
4935:
2963:
2916:
2685:
2330:
2285:
2149:
2146:Sheaf theory
2140:
2135:
2131:
2127:
2123:
2119:
2115:
2111:
2110:is called a
2107:
2105:
2064:
2054:
2052:
2047:
2045:
2039:
2037:
2034:
1980:
1976:
1939:The complex
1938:
1909:skeleton of
1867:
1778:
1774:
1742:
1739:
1734:
1730:
1726:
1724:
1628:integer part
1562:
1559:
1558:The (lower)
1469:
1357:
1355:
1352:
1211:
1134:
1058:
1055:Perversities
1047:
1040:
1036:
1032:
1028:
1020:
1016:
1010:
959:
955:
951:
947:
945:
939:
938:is dense in
840:
769:
730:, a compact
727:
723:
647:
643:
635:
633:
538:
526:
522:
516:
474:
470:
468:
392:
388:
386:
345:
341:
337:
297:
293:
291:
241:
105:
98:
80:
68:
57:
49:Mark Goresky
36:
26:
5328:shifted by
3349:is given as
3080:affine cone
531:paracompact
33:mathematics
5658:Categories
5377:References
5302:As usual,
3896:, and the
3025:, such as
1135:perversity
1011:Examples:
71:cohomology
5631:EMS Press
5499:cite book
5487:861677360
5409:−
5227:−
5154:∗
5133:−
5105:is 0 for
5075:∗
5021:is 0 for
4991:∗
4906:≠
4899:for
4830:⊕
4504:∗
4463:⊕
4398:∗
4300:∗
4182:≅
4139:⊕
4099:≅
4056:⊕
4016:≅
3943:≅
3858:∗
3596:
3587:⊂
3540:∗
3454:≠
3417:∈
3380:∗
3362:≤
3358:τ
3293:↪
3255:−
3171:∂
3103:⊂
2978:⊂
2939:−
2928:∖
2897:−
2886:∖
2856:−
2845:∖
2810:−
2804:−
2793:∖
2765:−
2754:∖
2699:≤
2695:τ
2664:−
2653:∖
2638:∗
2617:−
2602:≤
2598:τ
2594:⋯
2589:∗
2583:−
2562:−
2553:−
2541:≤
2537:τ
2531:∗
2510:−
2495:≤
2491:τ
2437:−
2426:∖
2398:−
2387:∖
2179:−
2088:→
1918:Δ
1879:−
1843:−
1837:−
1826:∖
1818:−
1797:−
1793:σ
1779:allowable
1755:Δ
1687:−
1644:−
1597:−
1540:−
1451:−
1322:∈
1305:−
1220:≥
1192:→
1184:≥
1174::
991:−
980:∖
921:−
910:∖
880:−
861:−
798:×
783:≅
750:−
744:−
707:⊂
679:−
668:∖
642:For each
603:⊂
600:⋯
597:⊂
584:⊂
571:⊂
563:−
552:∅
535:Hausdorff
491:−
477:- and an
362:−
314:−
219:≅
189:→
164:−
153:×
95:connected
5564:Topology
5549:(2010),
5340:See also
5316:takes IC
2960:Examples
1745:-simplex
537:) space
103:manifold
91:oriented
62:and the
29:topology
5633:, 2001
5589:0696691
956:stratum
816:. Here
382:perfect
87:compact
5601:
5532:
5485:
5475:
3892:, the
3489:. For
3443:where
3078:, the
2827:, and
2686:where
2286:where
1626:, the
1019:is an
348:- and
5469:(PDF)
5322:to IC
5286:with
5192:with
3843:is a
3759:with
3580:colim
2782:into
1133:.) A
387:When
85:of a
5599:ISBN
5530:ISBN
5509:link
5505:link
5483:OCLC
5473:ISBN
3635:for
3276:and
1356:The
391:has
81:The
51:and
5568:doi
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5033:= â
4259:are
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726:in
722:of
650:of
634:of
469:of
27:In
5660::
5629:,
5623:,
5586:MR
5580:,
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5497:{{
5481:.
5025:+
2914:.
2065:A
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