714:
2707:
2385:
3542:
3848:
is a version of
Poincaré duality which provides an isomorphism between the homology of an abelian covering space of a manifold and the corresponding cohomology with compact supports. It is used to get basic structural results about the
3433:
2271:
2080:
3058:
184:, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
2543:
3902:
could be replaced by other theories, once the products on manifolds were constructed; and there are now textbook treatments in generality. More specifically, there is a general
Poincaré duality theorem for a
2554:
2146:
is smooth, the intersection product is computed by perturbing the homology classes to be transverse and computing their oriented intersection number. For the torsion linking form, one computes the pairing of
572:
666:
355:
3144:
3330:
1992:
3673:
2964:
1062:
3789:
1543:
1410:
480:
Here, homology and cohomology are integral, but the isomorphism remains valid over any coefficient ring. In the case where an oriented manifold is not compact, one has to replace homology by
759:
2118:
is the quotient of the rationals by the integers, taken as an additive group. Notice that in the torsion linking form, there is a −1 in the dimension, so the paired dimensions add up to
1908:
2116:
175:
2446:
469:
Homology and cohomology groups are defined to be zero for negative degrees, so
Poincaré duality in particular implies that the homology and cohomology groups of orientable closed
429:
1114:
676:
Given a triangulated manifold, there is a corresponding dual polyhedral decomposition. The dual polyhedral decomposition is a cell decomposition of the manifold such that the
3979:
of a manifold, notably satisfying
Poincaré duality on its homology groups, with respect to a distinguished element (corresponding to the fundamental class). These are used in
2861:
1254:
3709:
2282:
1452:
872:
464:
3900:
2816:
1798:
3598:
2783:
3810:, given a Künneth theorem and a Thom isomorphism for that homology theory. A Thom isomorphism theorem for a homology theory is now viewed as the generalized notion of
3440:
1217:
1181:
2744:
1326:
912:
892:
835:
803:
1835:
3568:
1698:
1600:
1573:
1289:
1141:
990:
704:
3215:
1746:
1718:
1671:
1643:
1620:
932:
387:
3340:
1219:
are the cellular homologies and cohomologies of the dual polyhedral/CW decomposition the manifold respectively. The fact that this is an isomorphism of
3831:
2184:
2003:
4274:
2990:
3911:. The Thom isomorphism theorem in this regard can be considered as the germinal idea for Poincaré duality for generalized homology theories.
4214:
3987:
is one whose singular chain complex is a
Poincaré complex. These are not all manifolds, but their failure to be manifolds can be measured by
2702:{\displaystyle \tau H^{n-i}M\equiv \mathrm {Ext} (H_{n-i-1}M;\mathbb {Z} )\equiv \mathrm {Hom} (\tau H_{n-i-1}M;\mathbb {Q} /\mathbb {Z} )}
2462:
489:
586:
280:
4132:
4048:
3934:
3080:
578:
3261:
1931:
3605:
3149:
The resulting groups, while not a single group with a bilinear form, are a simple chain complex and are studied in algebraic
1223:
is a proof of
Poincaré duality. Roughly speaking, this amounts to the fact that the boundary relation for the triangulation
2909:
1007:
2453:
3945:, such as real or complex algebraic varieties, precisely so as to generalise Poincaré duality to such stratified spaces.
3720:
1479:
1338:
3965:
3907:
which requires a notion of orientation with respect to a homology theory, and is formulated in terms of a generalized
3904:
3807:
3839:
4160:
Blanchfield, Richard C. (1957), "Intersection theory of manifolds with operators with applications to knot theory",
4202:
4040:
719:
4269:
2163:. The form then takes the value equal to the fraction whose numerator is the transverse intersection number of
1843:
481:
4259:
2089:
113:
2399:
3976:
1416:
226:
43:
392:
4162:
3930:
1292:
1067:
4072:
3162:
2380:{\displaystyle \tau H_{i}M\to \mathrm {Hom} _{\mathbb {Z} }(\tau H_{n-i-1}M,\mathbb {Q} /\mathbb {Z} )}
4000:
3926:
3850:
3834:
is a generalisation for manifolds with boundary. In the non-orientable case, taking into account the
3240:
3182:
3165:
and Akira
Yasuhara to give an elementary homotopy and diffeomorphism classification of 3-dimensional
3063:
However, there is also a pairing between the free part of the homology in the lower middle dimension
2821:
1230:
3682:
1425:
848:
437:
4077:
3972:
3854:
3835:
2883:
2134:
1761:
245:
Poincaré duality did not take on its modern form until the advent of cohomology in the 1930s, when
231:
50:
4264:
4179:
4102:
3988:
3949:
3918:
3537:{\displaystyle H_{*}\left(M\times M,(M\times M)\setminus V\right)\to H_{*}(\nu M,\partial \nu M)}
2788:
1770:
181:
3796:
3333:
269:
The modern statement of the
Poincaré duality theorem is in terms of homology and cohomology: if
3806:
This formulation of
Poincaré duality has become popular as it defines Poincaré duality for any
3577:
2749:
4210:
4198:
4128:
4044:
4005:
3984:
3957:
3953:
1918:
1804:
1329:
1186:
1150:
362:
193:
85:
39:
2716:
1298:
897:
877:
820:
788:
4171:
4086:
3942:
3875:
3544:
1810:
4224:
4191:
4142:
4098:
4058:
3550:
1917:
part – all homology groups taken with integer coefficients in this section. Then there are
1676:
1578:
1551:
1267:
1119:
4220:
4187:
4138:
4094:
4054:
3914:
3861:
3838:
of local orientations, one can give a statement that is independent of orientability: see
1922:
1465:-manifolds which is compatible with orientation, i.e. which maps the fundamental class of
707:
250:
69:
1227:
is the incidence relation for the dual polyhedral decomposition under the correspondence
969:
683:
238:
led him to realize that his proof was seriously flawed. In the first two complements to
3980:
3922:
3815:
3200:
3185:
zero, which in turn gives that any manifold that bounds has even Euler characteristic.
2785:
are isomorphic, although there is no natural map giving the isomorphism, and similarly
1731:
1703:
1656:
1628:
1605:
1458:
917:
372:
246:
4034:
3795:, generalizing the intersection product discussed above. A similar argument with the
3428:{\displaystyle H_{*}(M\times M)\to H_{*}\left(M\times M,(M\times M)\setminus V\right)}
3177:
An immediate result from
Poincaré duality is that any closed odd-dimensional manifold
680:-cells of the dual polyhedral decomposition are in bijective correspondence with the (
4253:
4124:
4030:
3961:
3811:
3571:
2876:
1220:
235:
73:
66:
4237:
4106:
3938:
197:
2178:
The statement that the pairings are duality pairings means that the adjoint maps
4075:; Yasuhara, Akira (2003), "Symmetry of links and classification of lens spaces",
2980:, which is less commonly discussed, it is most simply the lower middle dimension
2266:{\displaystyle fH_{i}M\to \mathrm {Hom} _{\mathbb {Z} }(fH_{n-i}M,\mathbb {Z} )}
432:
258:
254:
224:
concept was at that time about 40 years from being clarified. In his 1895 paper
31:
17:
4243:
2075:{\displaystyle \tau H_{i}M\otimes \tau H_{n-i-1}M\to \mathbb {Q} /\mathbb {Z} }
4090:
4010:
3908:
3676:
3194:
3166:
1914:
47:
2984:, and there is a form on the torsion part of the homology in that dimension:
216:
th Betti numbers of a closed (i.e., compact and without boundary) orientable
3053:{\displaystyle \tau H_{k}M\otimes \tau H_{k}M\to \mathbb {Q} /\mathbb {Z} .}
2889:
What is meant by "middle dimension" depends on parity. For even dimension
3865:
3819:
3150:
54:
2875:
between different homology groups, in the middle dimension it induces a
761:– a picture of the parts of the dual-cells in a top-dimensional simplex.
4183:
713:
2538:{\displaystyle fH^{n-i}M\equiv \mathrm {Hom} (H_{n-i}M;\mathbb {Z} )}
27:
Connects homology and cohomology groups for oriented closed manifolds
4175:
389:
is oriented. Then the isomorphism is defined by mapping an element
712:
567:{\displaystyle H^{i}(X){\stackrel {\cong }{\to }}H_{n-i}^{BM}(X),}
3679:. This map is well-defined as there is a standard identification
661:{\displaystyle H_{c}^{i}(X){\stackrel {\cong }{\to }}H_{n-i}(X).}
350:{\displaystyle H^{k}(M,\mathbb {Z} )\to H_{n-k}(M,\mathbb {Z} )}
2903:, and there is a form on the free part of the middle homology:
2899:, which is more common, this is literally the middle dimension
2871:
While for most dimensions, Poincaré duality induces a bilinear
192:
A form of Poincaré duality was first stated, without proof, by
3711:
which is an oriented bundle, so the Thom isomorphism applies.
242:, Poincaré gave a new proof in terms of dual triangulations.
3872:
theories from about 1955, it was realised that the homology
1653:. Naturality does not hold for an arbitrary continuous map
277:-manifold, then there is a canonically defined isomorphism
3818:
on a manifold is a precise analog of an orientation within
4123:. Springer Monographs in Mathematics. With a foreword by
3139:{\displaystyle fH_{k}M\otimes fH_{k+1}M\to \mathbb {Z} .}
706:)-cells of the triangulation, generalizing the notion of
3325:{\displaystyle H_{*}M\otimes H_{*}M\to H_{*}(M\times M)}
1987:{\displaystyle fH_{i}M\otimes fH_{n-i}M\to \mathbb {Z} }
1720:
is a covering map then it maps the fundamental class of
230:, Poincaré tried to prove the theorem using topological
3668:{\displaystyle H_{*}(\nu M,\partial \nu M)\to H_{*-n}M}
2959:{\displaystyle fH_{k}M\otimes fH_{k}M\to \mathbb {Z} }
1057:{\displaystyle C_{i}M\otimes C_{n-i}M\to \mathbb {Z} }
361:. To define such an isomorphism, one chooses a fixed
3975:, which is an algebraic object that behaves like the
3971:
More algebraically, one can abstract the notion of a
3878:
3723:
3685:
3608:
3580:
3553:
3443:
3343:
3264:
3203:
3083:
2993:
2912:
2824:
2791:
2752:
2719:
2557:
2465:
2402:
2285:
2187:
2092:
2006:
1934:
1846:
1813:
1773:
1734:
1706:
1679:
1659:
1631:
1608:
1581:
1554:
1482:
1428:
1341:
1301:
1270:
1233:
1189:
1153:
1122:
1070:
1064:
given by taking intersections induces an isomorphism
1010:
972:
920:
900:
894:
of the barycentres of all subsets of the vertices of
880:
851:
823:
791:
722:
686:
589:
492:
440:
395:
375:
283:
116:
1700:
is not an injection on cohomology. For example, if
261:
and formulated Poincaré duality in these new terms.
3948:There are many other forms of geometric duality in
3894:
3784:{\displaystyle H_{i}M\otimes H_{j}M\to H_{i+j-n}M}
3783:
3703:
3667:
3592:
3562:
3536:
3427:
3324:
3209:
3138:
3052:
2958:
2855:
2810:
2777:
2738:
2701:
2537:
2440:
2393:This result is an application of Poincaré duality
2379:
2265:
2110:
2074:
1986:
1902:
1829:
1792:
1740:
1712:
1692:
1665:
1637:
1614:
1594:
1567:
1538:{\displaystyle D_{N}=f_{*}\circ D_{M}\circ f^{*},}
1537:
1446:
1405:{\displaystyle D_{M}\colon H^{k}(M)\to H_{n-k}(M)}
1404:
1320:
1283:
1248:
1211:
1175:
1135:
1108:
1056:
984:
926:
906:
886:
866:
829:
797:
753:
698:
660:
566:
458:
423:
381:
349:
234:, which he had invented. Criticism of his work by
169:
1625:Note the very strong and crucial hypothesis that
958:-dimensional cell. Moreover, the dual cells to
3917:is the appropriate generalization to (possibly
3161:This approach to Poincaré duality was used by
1143:is the cellular homology of the triangulation
4121:On Thom spectra, orientability, and cobordism
3983:to algebraicize questions about manifolds. A
8:
754:{\displaystyle \cup _{S\in T}\Delta \cap DS}
473:-manifolds are zero for degrees bigger than
42:, is a basic result on the structure of the
4209:, Wiley Classics Library, New York: Wiley,
3193:Poincaré duality is closely related to the
2886:is a very important topological invariant.
2863:are also isomorphic, though not naturally.
992:)-dimensional dual cell that intersects an
180:Poincaré duality holds for any coefficient
2882:on a single homology group. The resulting
1903:{\displaystyle fH_{i}M=H_{i}M/\tau H_{i}M}
1728:. This multiple is the degree of the map
1724:to a multiple of the fundamental class of
1622:in homology and cohomology, respectively.
3883:
3877:
3760:
3744:
3728:
3722:
3684:
3650:
3613:
3607:
3579:
3552:
3504:
3448:
3442:
3376:
3348:
3342:
3301:
3285:
3269:
3263:
3202:
3129:
3128:
3110:
3091:
3082:
3043:
3042:
3037:
3033:
3032:
3020:
3001:
2992:
2952:
2951:
2939:
2920:
2911:
2832:
2823:
2799:
2790:
2760:
2751:
2727:
2718:
2692:
2691:
2686:
2682:
2681:
2657:
2636:
2626:
2625:
2601:
2583:
2565:
2556:
2528:
2527:
2509:
2491:
2473:
2464:
2423:
2407:
2401:
2370:
2369:
2364:
2360:
2359:
2335:
2319:
2318:
2317:
2306:
2293:
2284:
2256:
2255:
2237:
2221:
2220:
2219:
2208:
2195:
2186:
2111:{\displaystyle \mathbb {Q} /\mathbb {Z} }
2104:
2103:
2098:
2094:
2093:
2091:
2068:
2067:
2062:
2058:
2057:
2033:
2014:
2005:
1980:
1979:
1961:
1942:
1933:
1891:
1879:
1870:
1854:
1845:
1818:
1812:
1781:
1772:
1733:
1705:
1684:
1678:
1658:
1630:
1607:
1586:
1580:
1559:
1553:
1526:
1513:
1500:
1487:
1481:
1427:
1381:
1359:
1346:
1340:
1306:
1300:
1275:
1269:
1232:
1194:
1188:
1158:
1152:
1127:
1121:
1091:
1075:
1069:
1050:
1049:
1031:
1015:
1009:
971:
919:
899:
879:
850:
822:
790:
727:
721:
685:
634:
622:
617:
615:
614:
599:
594:
588:
543:
532:
520:
515:
513:
512:
497:
491:
439:
406:
394:
374:
340:
339:
318:
304:
303:
288:
282:
170:{\displaystyle H^{k}(M)\cong H_{n-k}(M).}
143:
121:
115:
4022:
3486:
3414:
2132:The first form is typically called the
2441:{\displaystyle H_{i}M\simeq H^{n-i}M}
7:
3217:be a compact, boundaryless oriented
3173:Application to Euler Characteristics
3826:Generalizations and related results
424:{\displaystyle \alpha \in H^{k}(M)}
196:in 1893. It was stated in terms of
3832:Poincaré–Lefschetz duality theorem
3631:
3522:
3067:and in the upper middle dimension
2643:
2640:
2637:
2590:
2587:
2584:
2498:
2495:
2492:
2313:
2310:
2307:
2215:
2212:
2209:
1109:{\displaystyle C_{i}M\to C^{n-i}M}
901:
881:
852:
824:
792:
739:
25:
3814:for that theory. For example, a
2713:Thus, Poincaré duality says that
2456:, which gives an identification
4207:Principles of algebraic geometry
805:be a top-dimensional simplex of
76:and without boundary), then the
2969:By contrast, for odd dimension
2856:{\displaystyle \tau H_{n-i-1}M}
1249:{\displaystyle S\longmapsto DS}
817:as a subset of the vertices of
579:cohomology with compact support
4275:Theorems in algebraic geometry
3853:and can be used to define the
3753:
3704:{\displaystyle \nu M\equiv TM}
3643:
3640:
3619:
3531:
3510:
3497:
3483:
3471:
3411:
3399:
3369:
3366:
3354:
3319:
3307:
3294:
3125:
3029:
2948:
2696:
2647:
2630:
2594:
2532:
2502:
2374:
2325:
2302:
2260:
2227:
2204:
2159:as the boundary of some class
2054:
1976:
1760:is compact, boundaryless, and
1645:maps the fundamental class of
1447:{\displaystyle f\colon M\to N}
1438:
1399:
1393:
1374:
1371:
1365:
1237:
1084:
1046:
867:{\displaystyle \Delta \cap DS}
652:
646:
618:
611:
605:
558:
552:
516:
509:
503:
459:{\displaystyle \frown \alpha }
447:
441:
418:
412:
344:
330:
311:
308:
294:
161:
155:
133:
127:
1:
3921:) geometric objects, such as
2454:universal coefficient theorem
1752:Bilinear pairings formulation
1332:. The family of isomorphisms
3820:complex topological k-theory
3189:Thom isomorphism formulation
2390:are isomorphisms of groups.
1649:to the fundamental class of
1469:to the fundamental class of
4127:. Berlin: Springer-Verlag.
4039:(1st ed.). Cambridge:
3905:generalized homology theory
3808:generalized homology theory
3717:Combined, this gives a map
2811:{\displaystyle \tau H_{i}M}
2171:, and whose denominator is
1793:{\displaystyle \tau H_{i}M}
1419:in the following sense: if
962:form a CW-decomposition of
4291:
4041:Cambridge University Press
3593:{\displaystyle M\times M}
2778:{\displaystyle fH_{n-i}M}
2142:. Assuming the manifold
934:. One can check that if
769:be a triangulation of an
577:or replace cohomology by
220:-manifold are equal. The
3909:Thom isomorphism theorem
3860:With the development of
3840:twisted Poincaré duality
3195:Thom isomorphism theorem
1602:are the maps induced by
1212:{\displaystyle C^{n-i}M}
1176:{\displaystyle C_{n-i}M}
4091:10.1023/A:1024008222682
2739:{\displaystyle fH_{i}M}
1321:{\displaystyle H_{n-k}}
907:{\displaystyle \Delta }
887:{\displaystyle \Delta }
837:. Define the dual cell
830:{\displaystyle \Delta }
798:{\displaystyle \Delta }
80:th cohomology group of
3977:singular chain complex
3896:
3895:{\displaystyle H'_{*}}
3785:
3705:
3669:
3594:
3564:
3538:
3429:
3334:Homology cross product
3326:
3211:
3140:
3054:
2960:
2857:
2812:
2779:
2740:
2703:
2539:
2442:
2381:
2267:
2112:
2076:
1988:
1904:
1831:
1830:{\displaystyle H_{i}M}
1794:
1756:Assuming the manifold
1742:
1714:
1694:
1667:
1639:
1616:
1596:
1569:
1539:
1448:
1406:
1322:
1285:
1250:
1213:
1177:
1137:
1110:
1058:
986:
928:
908:
888:
874:is the convex hull in
868:
831:
799:
762:
755:
700:
662:
568:
460:
425:
383:
369:, which will exist if
351:
171:
4246:at the Manifold Atlas
4240:at the Manifold Atlas
4163:Annals of Mathematics
4119:Rudyak, Yuli (1998).
3931:intersection homology
3897:
3786:
3706:
3670:
3595:
3565:
3563:{\displaystyle \nu M}
3539:
3430:
3327:
3253:. Consider the maps:
3241:tubular neighbourhood
3212:
3141:
3055:
2961:
2858:
2813:
2780:
2741:
2704:
2540:
2443:
2382:
2268:
2113:
2077:
1989:
1905:
1832:
1795:
1743:
1715:
1695:
1693:{\displaystyle f^{*}}
1668:
1640:
1617:
1597:
1595:{\displaystyle f^{*}}
1570:
1568:{\displaystyle f_{*}}
1540:
1461:between two oriented
1449:
1407:
1323:
1293:contravariant functor
1286:
1284:{\displaystyle H^{k}}
1251:
1214:
1178:
1138:
1136:{\displaystyle C_{i}}
1111:
1059:
987:
929:
909:
889:
869:
832:
813:, so we can think of
800:
756:
716:
701:
663:
569:
461:
426:
384:
352:
273:is a closed oriented
172:
100:th homology group of
57:. It states that if
38:theorem, named after
4001:Bruhat decomposition
3876:
3855:signatures of a knot
3801:torsion linking form
3793:intersection product
3721:
3683:
3606:
3578:
3551:
3441:
3341:
3262:
3201:
3183:Euler characteristic
3081:
2991:
2910:
2822:
2789:
2750:
2717:
2555:
2463:
2400:
2283:
2185:
2140:torsion linking form
2135:intersection product
2090:
2004:
1932:
1844:
1811:
1771:
1732:
1704:
1677:
1657:
1629:
1606:
1579:
1552:
1480:
1426:
1339:
1299:
1268:
1231:
1187:
1151:
1120:
1068:
1008:
1004:. Thus the pairing
970:
918:
898:
878:
849:
821:
789:
720:
684:
672:Dual cell structures
587:
490:
482:Borel–Moore homology
438:
393:
373:
281:
114:
4078:Geometriae Dedicata
4073:Przytycki, Józef H.
3891:
3846:Blanchfield duality
3574:of the diagonal in
3243:of the diagonal in
1925:(explained below).
1673:, since in general
985:{\displaystyle n-i}
942:-dimensional, then
699:{\displaystyle n-k}
604:
551:
232:intersection theory
104:, for all integers
4199:Griffiths, Phillip
4036:Algebraic Topology
3989:obstruction theory
3950:algebraic topology
3892:
3879:
3781:
3701:
3665:
3590:
3572:normal disc bundle
3560:
3534:
3425:
3322:
3207:
3136:
3050:
2956:
2853:
2808:
2775:
2736:
2699:
2535:
2452:together with the
2438:
2377:
2263:
2108:
2072:
1984:
1900:
1827:
1790:
1738:
1710:
1690:
1663:
1635:
1612:
1592:
1565:
1535:
1444:
1402:
1318:
1281:
1246:
1209:
1173:
1133:
1106:
1054:
982:
924:
904:
884:
864:
827:
795:
763:
751:
696:
658:
590:
564:
528:
456:
421:
379:
347:
265:Modern formulation
167:
4238:Intersection form
4216:978-0-471-05059-9
4006:Fundamental class
3958:Alexander duality
3954:Lefschetz duality
3943:stratified spaces
3935:Robert MacPherson
3933:was developed by
3235:with itself. Let
3210:{\displaystyle M}
2884:intersection form
2125:, rather than to
1741:{\displaystyle f}
1713:{\displaystyle f}
1666:{\displaystyle f}
1638:{\displaystyle f}
1615:{\displaystyle f}
927:{\displaystyle S}
841:corresponding to
627:
525:
382:{\displaystyle M}
363:fundamental class
16:(Redirected from
4282:
4270:Duality theories
4227:
4194:
4147:
4146:
4116:
4110:
4109:
4069:
4063:
4062:
4027:
3973:Poincaré complex
3901:
3899:
3898:
3893:
3887:
3851:Alexander module
3790:
3788:
3787:
3782:
3777:
3776:
3749:
3748:
3733:
3732:
3710:
3708:
3707:
3702:
3677:Thom isomorphism
3674:
3672:
3671:
3666:
3661:
3660:
3618:
3617:
3599:
3597:
3596:
3591:
3569:
3567:
3566:
3561:
3543:
3541:
3540:
3535:
3509:
3508:
3496:
3492:
3453:
3452:
3434:
3432:
3431:
3426:
3424:
3420:
3381:
3380:
3353:
3352:
3331:
3329:
3328:
3323:
3306:
3305:
3290:
3289:
3274:
3273:
3252:
3230:
3216:
3214:
3213:
3208:
3145:
3143:
3142:
3137:
3132:
3121:
3120:
3096:
3095:
3073:
3059:
3057:
3056:
3051:
3046:
3041:
3036:
3025:
3024:
3006:
3005:
2979:
2965:
2963:
2962:
2957:
2955:
2944:
2943:
2925:
2924:
2898:
2867:Middle dimension
2862:
2860:
2859:
2854:
2849:
2848:
2817:
2815:
2814:
2809:
2804:
2803:
2784:
2782:
2781:
2776:
2771:
2770:
2745:
2743:
2742:
2737:
2732:
2731:
2708:
2706:
2705:
2700:
2695:
2690:
2685:
2674:
2673:
2646:
2629:
2618:
2617:
2593:
2576:
2575:
2544:
2542:
2541:
2536:
2531:
2520:
2519:
2501:
2484:
2483:
2447:
2445:
2444:
2439:
2434:
2433:
2412:
2411:
2386:
2384:
2383:
2378:
2373:
2368:
2363:
2352:
2351:
2324:
2323:
2322:
2316:
2298:
2297:
2272:
2270:
2269:
2264:
2259:
2248:
2247:
2226:
2225:
2224:
2218:
2200:
2199:
2138:and the 2nd the
2124:
2117:
2115:
2114:
2109:
2107:
2102:
2097:
2081:
2079:
2078:
2073:
2071:
2066:
2061:
2050:
2049:
2019:
2018:
1993:
1991:
1990:
1985:
1983:
1972:
1971:
1947:
1946:
1923:duality pairings
1909:
1907:
1906:
1901:
1896:
1895:
1883:
1875:
1874:
1859:
1858:
1836:
1834:
1833:
1828:
1823:
1822:
1799:
1797:
1796:
1791:
1786:
1785:
1747:
1745:
1744:
1739:
1719:
1717:
1716:
1711:
1699:
1697:
1696:
1691:
1689:
1688:
1672:
1670:
1669:
1664:
1644:
1642:
1641:
1636:
1621:
1619:
1618:
1613:
1601:
1599:
1598:
1593:
1591:
1590:
1574:
1572:
1571:
1566:
1564:
1563:
1544:
1542:
1541:
1536:
1531:
1530:
1518:
1517:
1505:
1504:
1492:
1491:
1453:
1451:
1450:
1445:
1411:
1409:
1408:
1403:
1392:
1391:
1364:
1363:
1351:
1350:
1327:
1325:
1324:
1319:
1317:
1316:
1290:
1288:
1287:
1282:
1280:
1279:
1255:
1253:
1252:
1247:
1218:
1216:
1215:
1210:
1205:
1204:
1182:
1180:
1179:
1174:
1169:
1168:
1142:
1140:
1139:
1134:
1132:
1131:
1115:
1113:
1112:
1107:
1102:
1101:
1080:
1079:
1063:
1061:
1060:
1055:
1053:
1042:
1041:
1020:
1019:
991:
989:
988:
983:
966:, and the only (
957:
933:
931:
930:
925:
913:
911:
910:
905:
893:
891:
890:
885:
873:
871:
870:
865:
836:
834:
833:
828:
804:
802:
801:
796:
781:be a simplex of
760:
758:
757:
752:
738:
737:
705:
703:
702:
697:
667:
665:
664:
659:
645:
644:
629:
628:
626:
621:
616:
603:
598:
573:
571:
570:
565:
550:
542:
527:
526:
524:
519:
514:
502:
501:
465:
463:
462:
457:
430:
428:
427:
422:
411:
410:
388:
386:
385:
380:
357:for any integer
356:
354:
353:
348:
343:
329:
328:
307:
293:
292:
215:
176:
174:
173:
168:
154:
153:
126:
125:
99:
36:Poincaré duality
21:
18:Poincare duality
4290:
4289:
4285:
4284:
4283:
4281:
4280:
4279:
4260:Homology theory
4250:
4249:
4234:
4217:
4197:
4176:10.2307/1969966
4159:
4156:
4154:Further reading
4151:
4150:
4135:
4118:
4117:
4113:
4071:
4070:
4066:
4051:
4029:
4028:
4024:
4019:
3997:
3923:analytic spaces
3915:Verdier duality
3874:
3873:
3862:homology theory
3828:
3797:Künneth theorem
3791:, which is the
3756:
3740:
3724:
3719:
3718:
3681:
3680:
3646:
3609:
3604:
3603:
3576:
3575:
3549:
3548:
3500:
3458:
3454:
3444:
3439:
3438:
3386:
3382:
3372:
3344:
3339:
3338:
3297:
3281:
3265:
3260:
3259:
3244:
3231:the product of
3222:
3221:-manifold, and
3199:
3198:
3191:
3175:
3163:Józef Przytycki
3106:
3087:
3079:
3078:
3068:
3016:
2997:
2989:
2988:
2970:
2935:
2916:
2908:
2907:
2890:
2828:
2820:
2819:
2795:
2787:
2786:
2756:
2748:
2747:
2723:
2715:
2714:
2653:
2597:
2561:
2553:
2552:
2505:
2469:
2461:
2460:
2419:
2403:
2398:
2397:
2331:
2305:
2289:
2281:
2280:
2233:
2207:
2191:
2183:
2182:
2119:
2088:
2087:
2029:
2010:
2002:
2001:
1957:
1938:
1930:
1929:
1887:
1866:
1850:
1842:
1841:
1814:
1809:
1808:
1777:
1769:
1768:
1754:
1730:
1729:
1702:
1701:
1680:
1675:
1674:
1655:
1654:
1627:
1626:
1604:
1603:
1582:
1577:
1576:
1555:
1550:
1549:
1522:
1509:
1496:
1483:
1478:
1477:
1424:
1423:
1377:
1355:
1342:
1337:
1336:
1302:
1297:
1296:
1271:
1266:
1265:
1262:
1229:
1228:
1221:chain complexes
1190:
1185:
1184:
1154:
1149:
1148:
1123:
1118:
1117:
1087:
1071:
1066:
1065:
1027:
1011:
1006:
1005:
968:
967:
947:
916:
915:
896:
895:
876:
875:
847:
846:
819:
818:
787:
786:
765:Precisely, let
723:
718:
717:
682:
681:
674:
630:
585:
584:
493:
488:
487:
436:
435:
402:
391:
390:
371:
370:
314:
284:
279:
278:
267:
251:Hassler Whitney
205:
190:
139:
117:
112:
111:
89:
70:closed manifold
28:
23:
22:
15:
12:
11:
5:
4288:
4286:
4278:
4277:
4272:
4267:
4262:
4252:
4251:
4248:
4247:
4241:
4233:
4232:External links
4230:
4229:
4228:
4215:
4203:Harris, Joseph
4195:
4170:(2): 340–356,
4155:
4152:
4149:
4148:
4133:
4111:
4064:
4049:
4031:Hatcher, Allen
4021:
4020:
4018:
4015:
4014:
4013:
4008:
4003:
3996:
3993:
3985:Poincaré space
3981:surgery theory
3890:
3886:
3882:
3827:
3824:
3816:spin-structure
3780:
3775:
3772:
3769:
3766:
3763:
3759:
3755:
3752:
3747:
3743:
3739:
3736:
3731:
3727:
3715:
3714:
3713:
3712:
3700:
3697:
3694:
3691:
3688:
3664:
3659:
3656:
3653:
3649:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3616:
3612:
3601:
3589:
3586:
3583:
3559:
3556:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3507:
3503:
3499:
3495:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3457:
3451:
3447:
3436:
3423:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3385:
3379:
3375:
3371:
3368:
3365:
3362:
3359:
3356:
3351:
3347:
3336:
3321:
3318:
3315:
3312:
3309:
3304:
3300:
3296:
3293:
3288:
3284:
3280:
3277:
3272:
3268:
3206:
3190:
3187:
3174:
3171:
3159:
3158:
3147:
3146:
3135:
3131:
3127:
3124:
3119:
3116:
3113:
3109:
3105:
3102:
3099:
3094:
3090:
3086:
3061:
3060:
3049:
3045:
3040:
3035:
3031:
3028:
3023:
3019:
3015:
3012:
3009:
3004:
3000:
2996:
2967:
2966:
2954:
2950:
2947:
2942:
2938:
2934:
2931:
2928:
2923:
2919:
2915:
2869:
2868:
2852:
2847:
2844:
2841:
2838:
2835:
2831:
2827:
2807:
2802:
2798:
2794:
2774:
2769:
2766:
2763:
2759:
2755:
2735:
2730:
2726:
2722:
2711:
2710:
2698:
2694:
2689:
2684:
2680:
2677:
2672:
2669:
2666:
2663:
2660:
2656:
2652:
2649:
2645:
2642:
2639:
2635:
2632:
2628:
2624:
2621:
2616:
2613:
2610:
2607:
2604:
2600:
2596:
2592:
2589:
2586:
2582:
2579:
2574:
2571:
2568:
2564:
2560:
2546:
2545:
2534:
2530:
2526:
2523:
2518:
2515:
2512:
2508:
2504:
2500:
2497:
2494:
2490:
2487:
2482:
2479:
2476:
2472:
2468:
2450:
2449:
2437:
2432:
2429:
2426:
2422:
2418:
2415:
2410:
2406:
2388:
2387:
2376:
2372:
2367:
2362:
2358:
2355:
2350:
2347:
2344:
2341:
2338:
2334:
2330:
2327:
2321:
2315:
2312:
2309:
2304:
2301:
2296:
2292:
2288:
2274:
2273:
2262:
2258:
2254:
2251:
2246:
2243:
2240:
2236:
2232:
2229:
2223:
2217:
2214:
2211:
2206:
2203:
2198:
2194:
2190:
2106:
2101:
2096:
2084:
2083:
2070:
2065:
2060:
2056:
2053:
2048:
2045:
2042:
2039:
2036:
2032:
2028:
2025:
2022:
2017:
2013:
2009:
1995:
1994:
1982:
1978:
1975:
1970:
1967:
1964:
1960:
1956:
1953:
1950:
1945:
1941:
1937:
1911:
1910:
1899:
1894:
1890:
1886:
1882:
1878:
1873:
1869:
1865:
1862:
1857:
1853:
1849:
1826:
1821:
1817:
1801:
1800:
1789:
1784:
1780:
1776:
1753:
1750:
1737:
1709:
1687:
1683:
1662:
1634:
1611:
1589:
1585:
1562:
1558:
1546:
1545:
1534:
1529:
1525:
1521:
1516:
1512:
1508:
1503:
1499:
1495:
1490:
1486:
1459:continuous map
1455:
1454:
1443:
1440:
1437:
1434:
1431:
1413:
1412:
1401:
1398:
1395:
1390:
1387:
1384:
1380:
1376:
1373:
1370:
1367:
1362:
1358:
1354:
1349:
1345:
1315:
1312:
1309:
1305:
1278:
1274:
1261:
1258:
1245:
1242:
1239:
1236:
1208:
1203:
1200:
1197:
1193:
1172:
1167:
1164:
1161:
1157:
1130:
1126:
1105:
1100:
1097:
1094:
1090:
1086:
1083:
1078:
1074:
1052:
1048:
1045:
1040:
1037:
1034:
1030:
1026:
1023:
1018:
1014:
981:
978:
975:
923:
903:
883:
863:
860:
857:
854:
826:
794:
750:
747:
744:
741:
736:
733:
730:
726:
708:dual polyhedra
695:
692:
689:
673:
670:
669:
668:
657:
654:
651:
648:
643:
640:
637:
633:
625:
620:
613:
610:
607:
602:
597:
593:
575:
574:
563:
560:
557:
554:
549:
546:
541:
538:
535:
531:
523:
518:
511:
508:
505:
500:
496:
455:
452:
449:
446:
443:
420:
417:
414:
409:
405:
401:
398:
378:
346:
342:
338:
335:
332:
327:
324:
321:
317:
313:
310:
306:
302:
299:
296:
291:
287:
266:
263:
240:Analysis Situs
227:Analysis Situs
194:Henri Poincaré
189:
186:
178:
177:
166:
163:
160:
157:
152:
149:
146:
142:
138:
135:
132:
129:
124:
120:
40:Henri Poincaré
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4287:
4276:
4273:
4271:
4268:
4266:
4263:
4261:
4258:
4257:
4255:
4245:
4242:
4239:
4236:
4235:
4231:
4226:
4222:
4218:
4212:
4208:
4204:
4200:
4196:
4193:
4189:
4185:
4181:
4177:
4173:
4169:
4165:
4164:
4158:
4157:
4153:
4144:
4140:
4136:
4134:3-540-62043-5
4130:
4126:
4125:Haynes Miller
4122:
4115:
4112:
4108:
4104:
4100:
4096:
4092:
4088:
4084:
4080:
4079:
4074:
4068:
4065:
4060:
4056:
4052:
4050:9780521795401
4046:
4042:
4038:
4037:
4032:
4026:
4023:
4016:
4012:
4009:
4007:
4004:
4002:
3999:
3998:
3994:
3992:
3990:
3986:
3982:
3978:
3974:
3969:
3967:
3963:
3962:Hodge duality
3959:
3955:
3951:
3946:
3944:
3940:
3936:
3932:
3928:
3924:
3920:
3916:
3912:
3910:
3906:
3888:
3884:
3880:
3871:
3870:extraordinary
3867:
3863:
3858:
3856:
3852:
3847:
3843:
3841:
3837:
3833:
3825:
3823:
3821:
3817:
3813:
3812:orientability
3809:
3804:
3802:
3798:
3794:
3778:
3773:
3770:
3767:
3764:
3761:
3757:
3750:
3745:
3741:
3737:
3734:
3729:
3725:
3698:
3695:
3692:
3689:
3686:
3678:
3662:
3657:
3654:
3651:
3647:
3637:
3634:
3628:
3625:
3622:
3614:
3610:
3602:
3587:
3584:
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3573:
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2279:
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2238:
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2230:
2201:
2196:
2192:
2188:
2181:
2180:
2179:
2176:
2174:
2170:
2166:
2162:
2158:
2155:by realizing
2154:
2150:
2145:
2141:
2137:
2136:
2130:
2128:
2122:
2099:
2063:
2051:
2046:
2043:
2040:
2037:
2034:
2030:
2026:
2023:
2020:
2015:
2011:
2007:
2000:
1999:
1998:
1973:
1968:
1965:
1962:
1958:
1954:
1951:
1948:
1943:
1939:
1935:
1928:
1927:
1926:
1924:
1920:
1919:bilinear maps
1916:
1897:
1892:
1888:
1884:
1880:
1876:
1871:
1867:
1863:
1860:
1855:
1851:
1847:
1840:
1839:
1838:
1824:
1819:
1815:
1806:
1787:
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1778:
1774:
1767:
1766:
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1763:
1759:
1751:
1749:
1735:
1727:
1723:
1707:
1685:
1681:
1660:
1652:
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1587:
1583:
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1532:
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1514:
1510:
1506:
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1497:
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1472:
1468:
1464:
1460:
1441:
1435:
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1422:
1421:
1420:
1418:
1396:
1388:
1385:
1382:
1378:
1368:
1360:
1356:
1352:
1347:
1343:
1335:
1334:
1333:
1331:
1313:
1310:
1307:
1303:
1294:
1276:
1272:
1259:
1257:
1243:
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1234:
1226:
1222:
1206:
1201:
1198:
1195:
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1170:
1165:
1162:
1159:
1155:
1146:
1128:
1124:
1103:
1098:
1095:
1092:
1088:
1081:
1076:
1072:
1043:
1038:
1035:
1032:
1028:
1024:
1021:
1016:
1012:
1003:
999:
995:
979:
976:
973:
965:
961:
955:
951:
945:
941:
937:
921:
914:that contain
861:
858:
855:
844:
840:
816:
812:
808:
784:
780:
776:
772:
768:
748:
745:
742:
734:
731:
728:
724:
715:
711:
709:
693:
690:
687:
679:
671:
655:
649:
641:
638:
635:
631:
623:
608:
600:
595:
591:
583:
582:
581:
580:
561:
555:
547:
544:
539:
536:
533:
529:
521:
506:
498:
494:
486:
485:
484:
483:
478:
476:
472:
467:
453:
450:
444:
434:
415:
407:
403:
399:
396:
376:
368:
364:
360:
336:
333:
325:
322:
319:
315:
300:
297:
289:
285:
276:
272:
264:
262:
260:
256:
253:invented the
252:
248:
243:
241:
237:
236:Poul Heegaard
233:
229:
228:
223:
219:
213:
209:
203:
199:
198:Betti numbers
195:
187:
185:
183:
164:
158:
150:
147:
144:
140:
136:
130:
122:
118:
110:
109:
108:
107:
103:
97:
93:
87:
83:
79:
75:
71:
68:
65:-dimensional
64:
60:
56:
52:
49:
45:
41:
37:
33:
19:
4244:Linking form
4206:
4167:
4161:
4120:
4114:
4085:(1): 57–61,
4082:
4076:
4067:
4035:
4025:
3970:
3952:, including
3947:
3939:Mark Goresky
3913:
3869:
3859:
3845:
3844:
3829:
3805:
3800:
3792:
3716:
3545:excision map
3249:
3245:
3236:
3232:
3227:
3223:
3218:
3192:
3178:
3176:
3160:
3157:Applications
3148:
3069:
3064:
3062:
2981:
2975:
2971:
2968:
2900:
2895:
2891:
2888:
2878:
2872:
2870:
2712:
2547:
2451:
2392:
2389:
2275:
2177:
2172:
2168:
2164:
2160:
2156:
2152:
2148:
2143:
2139:
2133:
2131:
2126:
2120:
2085:
1996:
1912:
1807:subgroup of
1802:
1757:
1755:
1725:
1721:
1650:
1646:
1624:
1547:
1470:
1466:
1462:
1456:
1414:
1263:
1224:
1144:
1001:
997:
993:
963:
959:
953:
949:
943:
939:
935:
842:
838:
814:
810:
806:
782:
778:
774:
770:
766:
764:
677:
675:
576:
479:
474:
470:
468:
366:
358:
274:
270:
268:
259:cap products
244:
239:
225:
221:
217:
211:
207:
201:
191:
179:
105:
101:
95:
91:
81:
77:
62:
58:
35:
29:
3864:to include
3239:be an open
3167:lens spaces
1803:denote the
809:containing
433:cap product
247:Eduard Čech
32:mathematics
4254:Categories
4017:References
4011:Weyl group
3868:and other
3799:gives the
3435:inclusion.
1921:which are
1762:orientable
1264:Note that
1260:Naturality
773:-manifold
222:cohomology
86:isomorphic
48:cohomology
4265:Manifolds
3966:S-duality
3885:∗
3771:−
3754:→
3738:⊗
3693:≡
3687:ν
3655:−
3652:∗
3644:→
3635:ν
3632:∂
3623:ν
3615:∗
3585:×
3555:ν
3526:ν
3523:∂
3514:ν
3506:∗
3498:→
3487:∖
3478:×
3463:×
3450:∗
3415:∖
3406:×
3391:×
3378:∗
3370:→
3361:×
3350:∗
3314:×
3303:∗
3295:→
3287:∗
3279:⊗
3271:∗
3126:→
3101:⊗
3030:→
3014:τ
3011:⊗
2995:τ
2949:→
2930:⊗
2877:bilinear
2843:−
2837:−
2826:τ
2793:τ
2765:−
2668:−
2662:−
2651:τ
2634:≡
2612:−
2606:−
2581:≡
2570:−
2559:τ
2514:−
2489:≡
2478:−
2428:−
2417:≃
2346:−
2340:−
2329:τ
2303:→
2287:τ
2242:−
2205:→
2055:→
2044:−
2038:−
2027:τ
2024:⊗
2008:τ
1977:→
1966:−
1952:⊗
1885:τ
1775:τ
1686:∗
1588:∗
1561:∗
1528:∗
1520:∘
1507:∘
1502:∗
1439:→
1433::
1386:−
1375:→
1353::
1330:covariant
1311:−
1238:⟼
1199:−
1163:−
1096:−
1085:→
1047:→
1036:−
1025:⊗
977:−
902:Δ
882:Δ
856:∩
853:Δ
825:Δ
793:Δ
743:∩
740:Δ
732:∈
725:∪
691:−
639:−
624:≅
619:→
537:−
522:≅
517:→
454:α
451:⌢
400:∈
397:α
323:−
312:→
148:−
137:≅
55:manifolds
4205:(1994),
4107:14601373
4033:(2002).
3995:See also
3929:, while
3919:singular
3889:′
3866:K-theory
3151:L-theory
1837:and let
1116:, where
845:so that
67:oriented
44:homology
4225:1288523
4192:0085512
4184:1969966
4143:1627486
4099:1988423
4059:1867354
3927:schemes
3570:is the
2873:pairing
1913:be the
1805:torsion
1473:, then
1417:natural
785:. Let
777:. Let
431:to the
204:th and
188:History
88:to the
74:compact
4223:
4213:
4190:
4182:
4141:
4131:
4105:
4097:
4057:
4047:
3964:, and
3547:where
3197:. Let
1764:, let
1548:where
1295:while
1147:, and
996:-cell
946:is an
200:: The
61:is an
51:groups
34:, the
4180:JSTOR
4103:S2CID
3836:sheaf
2548:and
2167:with
2086:Here
1457:is a
1291:is a
4211:ISBN
4129:ISBN
4045:ISBN
3941:for
3937:and
3830:The
3675:the
3332:the
3181:has
2879:form
2818:and
2746:and
2276:and
2151:and
1997:and
1915:free
1575:and
1183:and
257:and
249:and
182:ring
46:and
4172:doi
4087:doi
3925:or
3072:+ 1
2978:+ 1
2974:= 2
2894:= 2
2123:− 1
1415:is
1328:is
1000:is
938:is
365:of
255:cup
84:is
53:of
30:In
4256::
4221:MR
4219:,
4201:;
4188:MR
4186:,
4178:,
4168:65
4166:,
4139:MR
4137:.
4101:,
4095:MR
4093:,
4083:98
4081:,
4055:MR
4053:.
4043:.
3991:.
3968:.
3960:,
3956:,
3857:.
3842:.
3822:.
3803:.
3248:×
3226:×
3169:.
3153:.
3074::
2175:.
2157:nx
2129:.
1748:.
1256:.
1002:DS
952:−
944:DS
839:DS
710:.
477:.
466:.
210:−
94:−
4174::
4145:.
4089::
4061:.
3881:H
3779:M
3774:n
3768:j
3765:+
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3620:(
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3490:V
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3472:(
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3403:M
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3367:)
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3355:(
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3219:n
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3134:.
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3039:/
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2762:n
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2754:f
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2709:.
2697:)
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2688:/
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2679:;
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2671:1
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2648:(
2644:m
2641:o
2638:H
2631:)
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2595:(
2591:t
2588:x
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2522:M
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2503:(
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2414:M
2409:i
2405:H
2375:)
2371:Z
2366:/
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2357:,
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2349:1
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2333:H
2326:(
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2314:m
2311:o
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2300:M
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2261:)
2257:Z
2253:,
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2245:i
2239:n
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2231:f
2228:(
2222:Z
2216:m
2213:o
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2202:M
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2173:n
2169:y
2165:z
2161:z
2153:y
2149:x
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2127:n
2121:n
2105:Z
2100:/
2095:Q
2082:.
2069:Z
2064:/
2059:Q
2052:M
2047:1
2041:i
2035:n
2031:H
2021:M
2016:i
2012:H
1981:Z
1974:M
1969:i
1963:n
1959:H
1955:f
1949:M
1944:i
1940:H
1936:f
1898:M
1893:i
1889:H
1881:/
1877:M
1872:i
1868:H
1864:=
1861:M
1856:i
1852:H
1848:f
1825:M
1820:i
1816:H
1788:M
1783:i
1779:H
1758:M
1736:f
1726:N
1722:M
1708:f
1682:f
1661:f
1651:N
1647:M
1633:f
1610:f
1584:f
1557:f
1533:,
1524:f
1515:M
1511:D
1498:f
1494:=
1489:N
1485:D
1471:N
1467:M
1463:n
1442:N
1436:M
1430:f
1400:)
1397:M
1394:(
1389:k
1383:n
1379:H
1372:)
1369:M
1366:(
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1277:k
1273:H
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1235:S
1225:T
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1196:n
1192:C
1171:M
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1160:n
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1129:i
1125:C
1104:M
1099:i
1093:n
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1077:i
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1051:Z
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1022:M
1017:i
1013:C
998:S
994:i
980:i
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960:T
956:)
954:i
950:n
948:(
940:i
936:S
922:S
862:S
859:D
843:S
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807:T
783:T
779:S
775:M
771:n
767:T
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735:T
729:S
694:k
688:n
678:k
656:.
653:)
650:X
647:(
642:i
636:n
632:H
612:)
609:X
606:(
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562:,
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556:X
553:(
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545:B
540:i
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504:(
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96:k
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90:(
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72:(
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20:)
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