4691:
4085:
4686:{\displaystyle {\begin{matrix}\cdots &H_{n+1}(X_{1})&\longrightarrow &H_{n}(A_{1}\cap B_{1})&\longrightarrow &H_{n}(A_{1})\oplus H_{n}(B_{1})&\longrightarrow &H_{n}(X_{1})&\longrightarrow &H_{n-1}(A_{1}\cap B_{1})&\longrightarrow &\cdots \\&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }&&f_{*}{\Bigg \downarrow }\\\cdots &H_{n+1}(X_{2})&\longrightarrow &H_{n}(A_{2}\cap B_{2})&\longrightarrow &H_{n}(A_{2})\oplus H_{n}(B_{2})&\longrightarrow &H_{n}(X_{2})&\longrightarrow &H_{n-1}(A_{2}\cap B_{2})&\longrightarrow &\cdots \\\end{matrix}}}
2227:
901:
3078:
153:, many other important (co)homology theories, especially singular (co)homology, are not computable directly from their definition for nontrivial spaces. For singular (co)homology, the singular (co)chains and (co)cycles groups are often too big to handle directly. More subtle and indirect approaches become necessary. The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection.
611:
1792:
2623:
3067:
3606:
2026:
325:
2612:
5535:
2423:
1404:
4012:
747:
5112:
2197:
2890:
124:
are constructed by piecing together very simple patches. Carefully choosing the two covering subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the
3329:
1878:
606:{\displaystyle \cdots \to H_{n+1}(X)\,\xrightarrow {\partial _{*}} \,H_{n}(A\cap B)\,\xrightarrow {\left({\begin{smallmatrix}i_{*}\\j_{*}\end{smallmatrix}}\right)} \,H_{n}(A)\oplus H_{n}(B)\,\xrightarrow {k_{*}-l_{*}} \,H_{n}(X)\,\xrightarrow {\partial _{*}} \,H_{n-1}(A\cap B)\to \cdots }
2443:
5933:
5327:
2297:
4895:
1185:
3833:
617:
6178:
4929:
2053:
3062:{\displaystyle {\tilde {H}}_{n}\left(S^{2}\vee S^{2}\right)\cong \delta _{2n}\,(\mathbb {Z} \oplus \mathbb {Z} )=\left\{{\begin{matrix}\mathbb {Z} \oplus \mathbb {Z} &{\mbox{if }}n=2,\\0&{\mbox{if }}n\neq 2.\end{matrix}}\right.}
2859:
830:
120:(co)homology. Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in
211:
considered as the union of two cylinders. Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence. The concept of an exact sequence only appeared in print in the 1952 book
1574:
3601:{\displaystyle \cdots \to H_{n}(A\cap B,C\cap D)\,{\xrightarrow {(i_{*},j_{*})}}\,H_{n}(A,C)\oplus H_{n}(B,D)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{n}(X,Y)\,{\xrightarrow {\partial _{*}}}\,H_{n-1}(A\cap B,C\cap D)\to \cdots }
2428:
and the trivial part implies vanishing homology for dimensions greater than 2. The central map α sends 1 to (2, −2) since the boundary circle of a Möbius band wraps twice around the core circle. In particular α is
2021:{\displaystyle \cdots \longrightarrow 0\longrightarrow {\tilde {H}}_{n}\!\left(S^{k}\right)\,{\xrightarrow {\overset {}{\partial _{*}}}}\,{\tilde {H}}_{n-1}\!\left(S^{k-1}\right)\longrightarrow 0\longrightarrow \cdots }
5695:
1671:
2607:{\displaystyle {\tilde {H}}_{n}\left(X\right)\cong \delta _{1n}\,(\mathbb {Z} \oplus \mathbb {Z} _{2})={\begin{cases}\mathbb {Z} \oplus \mathbb {Z} _{2}&{\mbox{if }}n=1,\\0&{\mbox{if }}n\neq 1.\end{cases}}}
4900:
where the dimension preserving maps are restriction maps induced from inclusions, and the (co-)boundary maps are defined in a similar fashion to the homological version. There is also a relative formulation.
3253:
5790:
5530:{\displaystyle \cdots \to H_{c}^{n}(U\cap V)\,\xrightarrow {\delta } \,H_{c}^{n}(U)\oplus H_{c}^{n}(V)\,\xrightarrow {\Sigma } \,H_{c}^{n}(X)\,\xrightarrow {d^{*}} \,H_{c}^{n+1}(U\cap V)\to \cdots }
3757:
3825:
2418:{\displaystyle 0\rightarrow {\tilde {H}}_{2}(X)\rightarrow \mathbb {Z} \ {\xrightarrow {\overset {}{\alpha }}}\ \mathbb {Z} \oplus \mathbb {Z} \rightarrow \,{\tilde {H}}_{1}(X)\rightarrow 0}
1399:{\displaystyle \cdots \to {\tilde {H}}_{0}(A\cap B)\,{\xrightarrow {(i_{*},j_{*})}}\,{\tilde {H}}_{0}(A)\oplus {\tilde {H}}_{0}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,{\tilde {H}}_{0}(X)\to 0.}
4007:{\displaystyle {\begin{matrix}X_{1}=A_{1}\cup B_{1}\\X_{2}=A_{2}\cup B_{2}\end{matrix}}\qquad {\text{and}}\qquad {\begin{matrix}f(A_{1})\subset A_{2}\\f(B_{1})\subset B_{2}\end{matrix}}}
4722:
74:, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a
5254:
742:{\displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdots \to H_{0}(A)\oplus H_{0}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{0}(X)\to 0.}
862:
6916:, Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs, vol. 230 (Translated from the 2002 Japanese edition by Tamaki ed.), Providence, RI:
3664:
6063:
5107:{\displaystyle \cdots \to H^{n}(X)\,{\xrightarrow {\rho }}\,H^{n}(U)\oplus H^{n}(V)\,{\xrightarrow {\Delta }}\,H^{n}(U\cap V)\,{\xrightarrow {d^{*}}}\,H^{n+1}(X)\to \cdots }
4046:
5193:
2192:{\displaystyle {\tilde {H}}_{n}\!\left(S^{k}\right)\cong \delta _{kn}\,\mathbb {Z} ={\begin{cases}\mathbb {Z} &{\mbox{if }}n=k,\\0&{\mbox{if }}n\neq k,\end{cases}}}
1753:
184:
of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are
1714:
5762:
5618:
1442:
886:
5722:
5166:
4073:
6593:
2751:
5742:
5598:
5578:
5558:
1773:
755:
196:, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability.
7142:
1454:
5623:
1585:
7079:
6958:
6925:
6894:
6848:
6820:
6783:
6699:
6569:
6254:) in the case where the open cover used to compute the Čech cohomology consists of two open sets. This spectral sequence exists in arbitrary
5196:
3170:
7029:
6749:
6721:
6215:
149:
of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of
7063:
7021:
6950:
6691:
7056:
Séminaire de Géométrie Algébrique du Bois Marie – 1963–64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – Tome 2
199:
Mayer was introduced to topology by his colleague
Vietoris when attending his lectures in 1926 and 1927 at a local university in
5928:{\displaystyle 0\to C_{n}(A\cap B)\,{\xrightarrow {\alpha }}\,C_{n}(A)\oplus C_{n}(B)\,{\xrightarrow {\beta }}\,C_{n}(A+B)\to 0}
6195:
1415:
126:
3667:
6187:
109:
7093:
6973:
6917:
6882:
7059:
3677:
889:
82:
6251:
6211:
3762:
2671:
2246:
A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the
900:
6840:
6812:
4890:{\displaystyle \cdots \to H^{n}(X;G)\to H^{n}(A;G)\oplus H^{n}(B;G)\to H^{n}(A\cap B;G)\to H^{n+1}(X;G)\to \cdots }
2226:
2207:
3077:
2206:. Such a complete understanding of the homology groups for spheres is in stark contrast with current knowledge of
3617:
2268:
301:
86:
75:
1414:
There is an analogy between the Mayer–Vietoris sequence (especially for homology groups of dimension 1) and the
7174:
161:
5219:
6886:
3121:
956:
7047:
3671:
2044:
835:
4713:
3102:
6173:{\displaystyle 0\to \Omega ^{n}(X)\to \Omega ^{n}(U)\oplus \Omega ^{n}(V)\to \Omega ^{n}(U\cap V)\to 0}
6206:
The derivation of the Mayer–Vietoris sequence from the
Eilenberg–Steenrod axioms does not require the
5321:
For de Rham cohomology with compact supports, there exists a "flipped" version of the above sequence:
3623:
203:. He was told about the conjectured result and a way to its solution, and solved the question for the
6866:
6219:
5777:
185:
6741:
6735:
4021:
2528:
2433:
so homology of dimension 2 also vanishes. Finally, choosing (1, 0) and (1, −1) as a basis for
2125:
6878:
5773:
5171:
4706:
4702:
4076:
2667:
2231:
1833:
1677:
281:
249:
177:
165:
105:
101:
78:
51:
7110:
7051:
7039:
6998:
6587:
5257:
4920:
2430:
1722:
319:
For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:
189:
173:
35:
6239:
7151:
7075:
7025:
6990:
6954:
6942:
6921:
6900:
6890:
6844:
6816:
6779:
6745:
6731:
6717:
6695:
6575:
6565:
6247:
6243:
6055:
3296:
1717:
1683:
265:
237:
156:
The most natural and convenient way to express the relation involves the algebraic concept of
142:
130:
117:
93:
71:
55:
5747:
5603:
1421:
871:
7102:
7088:
7067:
6982:
6800:
6771:
6267:
6235:
2854:{\displaystyle {\tilde {H}}_{n}(K\vee L)\cong {\tilde {H}}_{n}(K)\oplus {\tilde {H}}_{n}(L)}
2036:
1869:
1164:
217:
181:
113:
67:
6935:
6858:
6793:
5700:
5195:
from above. It can be briefly described as follows. For a cohomology class represented by
5144:
4051:
6931:
6854:
6804:
6789:
6767:
6759:
6207:
6191:
4916:
2203:
825:{\displaystyle i:A\cap B\hookrightarrow A,j:A\cap B\hookrightarrow B,k:A\hookrightarrow X}
221:
97:
39:
2258:
3759:
such that the composition of pushforwards is the pushforward of a composition: that is,
5727:
5583:
5563:
5543:
1758:
1445:
1139:). Notice that the maps in the Mayer–Vietoris sequence depend on choosing an order for
157:
150:
146:
17:
7134:
7168:
7114:
7043:
7013:
7002:
6968:
6830:
6272:
5781:
2663:
1865:
865:
63:
6870:
2742:
2247:
1853:
1845:
204:
1791:
6186:
From a formal point of view, the Mayer–Vietoris sequence can be derived from the
1569:{\displaystyle H_{1}(X)\cong (H_{1}(A)\oplus H_{1}(B))/{\text{Ker}}(k_{*}-l_{*})}
4909:
2745:
by construction. The reduced version of the sequence then yields (by exactness)
416:
85:
of the (co)homology groups of the subspaces, and the (co)homology groups of the
31:
4430:
4410:
4390:
4370:
4350:
6775:
6683:
6054:
The same computation applied to the short exact sequences of vector spaces of
5130:
2622:
193:
7155:
6994:
6904:
1179:
intersection. The sequence is identical for positive dimensions and ends as:
1105:
lie in the same homology class; nor does choosing a different representative
224:, where the results of Mayer and Vietoris were expressed in the modern form.
6834:
6709:
6223:
2651:
1176:
6579:
5690:{\displaystyle \delta :\omega \mapsto (i_{*}^{U}\omega ,-i_{*}^{V}\omega )}
284:
relating the singular homology groups (with coefficient group the integers
2288:
2040:
1807:
169:
121:
6559:
3128:
of the top and bottom 'vertices' of the double cone, respectively. Then
1680:
statement of the
Seifert–van Kampen theorem. Compare with the fact that
1666:{\displaystyle {\text{Ker}}(k_{*}-l_{*})\cong {\text{Im}}(i_{*},j_{*}).}
7106:
7071:
6986:
6250:
that relates Čech cohomology to sheaf cohomology (sometimes called the
4915:
and the underlying topological space has the additional structure of a
2234:
with appropriate edge identifications) decomposed as two Möbius strips
59:
47:
920:
is the sum of two 1-chains whose boundary lies in the intersection of
7091:(1930), "Über die Homologiegruppen der Vereinigung zweier Komplexe",
200:
5886:
5831:
5471:
5435:
5372:
5058:
5019:
4964:
3534:
3475:
3382:
2351:
1938:
1334:
1235:
686:
552:
501:
409:
365:
81:, whose entries are the (co)homology groups of the whole space, the
3274:− 1)-sphere, it is easy to derive the homology groups of the
1167:
there is also a Mayer–Vietoris sequence, under the assumption that
108:. In general, the sequence holds for those theories satisfying the
6255:
3076:
2621:
2271:
along their boundary circle (see illustration on the right). Then
2225:
1790:
899:
208:
6051:). This gives the Mayer–Vietoris sequence for singular homology.
3827:
The Mayer–Vietoris sequence is also natural in the sense that if
3248:{\displaystyle {\tilde {H}}_{n}(SY)\cong {\tilde {H}}_{n-1}(Y).}
936:
lowering the dimension may be defined as follows. An element in
133:, and a precise relation exists for homology of dimension one.
4018:
then the connecting morphism of the Mayer–Vietoris sequence,
1448:, the reduced Mayer–Vietoris sequence yields the isomorphism
27:
Algebraic tool for computing invariants of topological spaces
6183:
yields the Mayer–Vietoris sequence for de Rham cohomology.
3056:
2600:
2185:
6737:
A History of
Algebraic and Differential Topology 1900–1960
2291:
to circles, so the nontrivial part of the sequence yields
1011:. This implies that the images of both these boundary (
6714:
Modern
Algebra and the Rise of Mathematical Structures
4090:
3929:
3838:
3037:
3009:
2990:
2582:
2554:
2164:
2136:
264:
need not be disjoint.) The Mayer–Vietoris sequence in
6066:
5984:) is the chain group consisting of sums of chains in
5793:
5750:
5730:
5703:
5626:
5606:
5586:
5566:
5546:
5330:
5222:
5174:
5147:
4932:
4725:
4088:
4054:
4024:
3836:
3765:
3680:
3626:
3332:
3173:
2893:
2754:
2446:
2300:
2056:
1881:
1761:
1725:
1686:
1588:
1457:
1424:
1188:
874:
838:
758:
620:
328:
3299:
form of the Mayer–Vietoris sequence also exists. If
3160:. Hence the Mayer–Vietoris sequence yields, for all
1840:− 1)-dimensional equatorial sphere. Since the
6397:
6395:
92:The Mayer–Vietoris sequence holds for a variety of
6172:
5927:
5756:
5736:
5716:
5689:
5612:
5592:
5572:
5552:
5529:
5248:
5187:
5160:
5106:
4889:
4685:
4067:
4040:
4006:
3819:
3752:{\displaystyle f_{*}:H_{k}(X_{1})\to H_{k}(X_{2})}
3751:
3658:
3600:
3247:
3061:
2853:
2606:
2417:
2191:
2020:
1767:
1747:
1708:
1665:
1568:
1436:
1398:
1147:. In particular, the boundary map changes sign if
880:
856:
824:
741:
605:
6533:
6371:
6359:
5724:extends a form with compact support to a form on
4716:to the homological version. It is the following:
3258:The illustration on the right shows the 1-sphere
2076:
1981:
1913:
7062:(in French), vol. 270, Berlin; Heidelberg:
3820:{\displaystyle (g\circ h)_{*}=g_{*}\circ h_{*}.}
70:. The method consists of splitting a space into
311:. There is an unreduced and a reduced version.
6877:, Israel Mathematical Conference Proceedings,
2662:, and suppose furthermore that the identified
1015:− 1)-cycles are contained in the intersection
959:for example, can be written as the sum of two
8:
7143:Notices of the American Mathematical Society
6382:
6380:
6238:, the Mayer–Vietoris sequence is related to
2031:Exactness immediately implies that the map ∂
125:Mayer–Vietoris sequence is analogous to the
4701:The Mayer–Vietoris long exact sequence for
2880:. For this specific case, using the result
1410:Analogy with the Seifert–van Kampen theorem
6642:
6592:: CS1 maint: location missing publisher (
6290:
4079:(the horizontal maps are the usual ones):
1806:To completely compute the homology of the
7042:(1972), "Cohomologie dans les topos", in
6312:
6143:
6121:
6099:
6077:
6065:
5898:
5893:
5881:
5880:
5865:
5843:
5838:
5826:
5825:
5804:
5792:
5749:
5729:
5708:
5702:
5675:
5670:
5651:
5646:
5625:
5605:
5585:
5565:
5545:
5494:
5489:
5484:
5476:
5466:
5451:
5446:
5441:
5430:
5415:
5410:
5388:
5383:
5378:
5367:
5346:
5341:
5329:
5240:
5227:
5221:
5179:
5173:
5152:
5146:
5077:
5072:
5063:
5053:
5052:
5031:
5026:
5014:
5013:
4998:
4976:
4971:
4959:
4958:
4943:
4931:
4854:
4820:
4792:
4764:
4736:
4724:
4660:
4647:
4628:
4608:
4595:
4575:
4562:
4546:
4533:
4513:
4500:
4487:
4467:
4448:
4429:
4428:
4422:
4409:
4408:
4402:
4389:
4388:
4382:
4369:
4368:
4362:
4349:
4348:
4342:
4314:
4301:
4282:
4262:
4249:
4229:
4216:
4200:
4187:
4167:
4154:
4141:
4121:
4102:
4089:
4087:
4059:
4053:
4029:
4023:
3994:
3978:
3958:
3942:
3928:
3922:
3911:
3898:
3885:
3871:
3858:
3845:
3837:
3835:
3808:
3795:
3782:
3764:
3740:
3727:
3711:
3698:
3685:
3679:
3650:
3637:
3625:
3553:
3548:
3539:
3529:
3528:
3507:
3502:
3493:
3480:
3470:
3469:
3448:
3420:
3415:
3403:
3390:
3377:
3376:
3343:
3331:
3221:
3210:
3209:
3187:
3176:
3175:
3172:
3036:
3008:
3002:
3001:
2994:
2993:
2989:
2975:
2974:
2967:
2966:
2962:
2953:
2935:
2922:
2907:
2896:
2895:
2892:
2836:
2825:
2824:
2805:
2794:
2793:
2768:
2757:
2756:
2753:
2581:
2553:
2545:
2541:
2540:
2532:
2531:
2523:
2511:
2507:
2506:
2498:
2497:
2493:
2484:
2460:
2449:
2448:
2445:
2394:
2383:
2382:
2380:
2373:
2372:
2365:
2364:
2346:
2339:
2338:
2320:
2309:
2308:
2299:
2163:
2135:
2129:
2128:
2120:
2113:
2112:
2111:
2102:
2085:
2070:
2059:
2058:
2055:
1990:
1969:
1958:
1957:
1955:
1944:
1933:
1932:
1922:
1907:
1896:
1895:
1880:
1760:
1730:
1724:
1691:
1685:
1651:
1638:
1626:
1614:
1601:
1589:
1587:
1557:
1544:
1532:
1527:
1509:
1487:
1462:
1456:
1423:
1375:
1364:
1363:
1361:
1352:
1339:
1329:
1328:
1313:
1302:
1301:
1282:
1271:
1270:
1268:
1256:
1243:
1230:
1229:
1208:
1197:
1196:
1187:
873:
837:
757:
718:
713:
704:
691:
681:
680:
665:
643:
619:
570:
565:
557:
547:
532:
527:
519:
506:
496:
481:
459:
454:
437:
423:
414:
404:
383:
378:
370:
360:
339:
327:
6869:(1999), "Emmy Noether and Topology", in
6688:Differential Forms in Algebraic Topology
6561:Differential forms in algebraic topology
6336:
6666:
6630:
6606:
6545:
6509:
6497:
6485:
6473:
6461:
6449:
6437:
6413:
6401:
6386:
6283:
5780:of chain groups (constituent groups of
5272:, for example. The exterior derivative
5249:{\displaystyle \omega _{U}-\omega _{V}}
207:in 1929. He applied his results to the
6618:
6585:
6521:
6425:
2868:. The illustration on the right shows
1027:() can be defined to be the class of ∂
6654:
6347:
6324:
6301:
6000:whose images are contained in either
3307:and is the union of the interiors of
3148:contractible. Also, the intersection
3081:This decomposition of the suspension
7:
6971:(1929), "Über abstrakte Topologie",
2626:This decomposition of the wedge sum
857:{\displaystyle l:B\hookrightarrow X}
228:Basic versions for singular homology
6947:Algebraic Topology: An Introduction
6242:. Specifically, it arises from the
6008:generate all of the homology group
137:Background, motivation, and history
6912:Kōno, Akira; Tamaki, Dai (2006) ,
6140:
6118:
6096:
6074:
5751:
5436:
5176:
5020:
4919:, the Mayer–Vietoris sequence for
4904:As an important special case when
4026:
3536:
3270:-sphere is the suspension of the (
3262:as the suspension of the 0-sphere
3089:yields all the homology groups of
2638:yields all the homology groups of
1941:
1868:. The Mayer–Vietoris sequence for
1049:). Choosing another decomposition
904:Illustration of the boundary map ∂
554:
367:
25:
6809:Foundations of Algebraic Topology
6216:extraordinary cohomology theories
5992:. It is a fact that the singular
5774:long exact sequence associated to
5296:and therefore together define an
5137:denotes the restriction map, and
4075:. That is, the following diagram
2253:. One uses the decomposition of
415:
214:Foundations of Algebraic Topology
112:, and it has variations for both
6252:Mayer–Vietoris spectral sequence
6210:, so in addition to existing in
5168:is defined similarly as the map
3659:{\displaystyle f:X_{1}\to X_{2}}
6196:long exact sequence in homology
3927:
3921:
3670:map, then there is a canonical
908:on the torus where the 1-cycle
632:
631:
630:
629:
628:
627:
626:
625:
624:
623:
622:
621:
7135:"Leopold Vietoris (1891–2002)"
6164:
6161:
6149:
6136:
6133:
6127:
6111:
6105:
6092:
6089:
6083:
6070:
5919:
5916:
5904:
5877:
5871:
5855:
5849:
5822:
5810:
5797:
5684:
5639:
5636:
5521:
5518:
5506:
5463:
5457:
5427:
5421:
5400:
5394:
5364:
5352:
5334:
5260:subordinate to the open cover
5098:
5095:
5089:
5049:
5037:
5010:
5004:
4988:
4982:
4955:
4949:
4936:
4881:
4878:
4866:
4847:
4844:
4826:
4813:
4810:
4798:
4782:
4770:
4757:
4754:
4742:
4729:
4671:
4666:
4640:
4619:
4614:
4601:
4586:
4581:
4568:
4552:
4539:
4524:
4519:
4493:
4478:
4473:
4460:
4325:
4320:
4294:
4273:
4268:
4255:
4240:
4235:
4222:
4206:
4193:
4178:
4173:
4147:
4132:
4127:
4114:
4041:{\displaystyle \partial _{*},}
3984:
3971:
3948:
3935:
3779:
3766:
3746:
3733:
3720:
3717:
3704:
3643:
3592:
3589:
3565:
3525:
3513:
3466:
3454:
3438:
3426:
3409:
3383:
3373:
3349:
3336:
3323:, then the exact sequence is:
3239:
3233:
3215:
3202:
3193:
3181:
2979:
2963:
2901:
2848:
2842:
2830:
2817:
2811:
2799:
2786:
2774:
2762:
2517:
2494:
2454:
2409:
2406:
2400:
2388:
2377:
2335:
2332:
2326:
2314:
2304:
2064:
2012:
2006:
1963:
1901:
1891:
1885:
1742:
1736:
1703:
1697:
1657:
1631:
1620:
1594:
1563:
1537:
1524:
1521:
1515:
1499:
1493:
1480:
1474:
1468:
1390:
1387:
1381:
1369:
1325:
1319:
1307:
1294:
1288:
1276:
1262:
1236:
1226:
1214:
1202:
1192:
947:) is the homology class of an
848:
816:
798:
774:
733:
730:
724:
677:
671:
655:
649:
636:
597:
594:
582:
544:
538:
493:
487:
471:
465:
401:
389:
357:
351:
332:
1:
7133:Reitberger, Heinrich (2002),
6918:American Mathematical Society
6883:American Mathematical Society
6534:Eilenberg & Steenrod 1952
6372:Eilenberg & Steenrod 1952
6360:Eilenberg & Steenrod 1952
5188:{\displaystyle \partial _{*}}
3279:
3266:. Noting in general that the
2881:
2218:about which little is known.
2035:is an isomorphism. Using the
1844:-dimensional hemispheres are
1716:is the abelianization of the
176:) between them such that the
7060:Lecture Notes in Mathematics
6875:The Heritage of Emmy Noether
6212:ordinary cohomology theories
5620:is the signed inclusion map
2872:as the sum of two 2-spheres
890:direct sum of abelian groups
6716:, Birkhäuser, p. 345,
6558:Bott, Raoul (16 May 1995).
5141:is the difference. The map
1748:{\displaystyle \pi _{1}(X)}
971:whose images lie wholly in
180:of one morphism equals the
58:. The result is due to two
7191:
7094:Monatshefte für Mathematik
6974:Monatshefte für Mathematik
6841:Cambridge University Press
6813:Princeton University Press
6609:, Proposition 2.21, p. 119
6564:. Tu, Loring W. New York.
6234:From the point of view of
3156:is homotopy equivalent to
2210:, especially for the case
2208:homotopy groups of spheres
1856:, the homology groups for
1416:Seifert–van Kampen theorem
127:Seifert–van Kampen theorem
6776:10.1007/978-3-642-18868-8
6500:, Exercise 32 on page 158
6488:, Exercise 31 on page 158
6188:Eilenberg–Steenrod axioms
5216:as a difference of forms
1061:does not affect , since ∂
110:Eilenberg–Steenrod axioms
6766:, Universitext, Berlin:
6686:; Tu, Loring W. (1982),
4705:groups with coefficient
3616:The homology groups are
1709:{\displaystyle H_{1}(X)}
1003:is a cycle, ∂x = 0, so ∂
7048:Grothendieck, Alexander
6887:Oxford University Press
6202:Other homology theories
5757:{\displaystyle \Sigma }
5613:{\displaystyle \delta }
3674:map of homology groups
2884:for 2-spheres, one has
2279:and their intersection
1437:{\displaystyle A\cap B}
957:barycentric subdivision
881:{\displaystyle \oplus }
248:be two subspaces whose
44:Mayer–Vietoris sequence
18:Mayer-Vietoris sequence
6914:Generalized cohomology
6740:, Birkhäuser, p.
6643:Kōno & Tamaki 2006
6452:, Example 2.46, p. 150
6440:, Theorem 2A.1, p. 166
6174:
5929:
5758:
5738:
5718:
5691:
5614:
5594:
5574:
5554:
5531:
5250:
5189:
5162:
5108:
4891:
4697:Cohomological versions
4687:
4069:
4042:
4008:
3821:
3753:
3660:
3602:
3278:-sphere by induction,
3249:
3094:
3063:
2855:
2643:
2608:
2419:
2243:
2193:
2022:
1828:be two hemispheres of
1803:
1795:The decomposition for
1769:
1749:
1710:
1676:This is precisely the
1667:
1570:
1438:
1400:
979:, respectively. Thus ∂
929:
882:
858:
826:
743:
607:
6867:Hirzebruch, Friedrich
6175:
5930:
5778:short exact sequences
5759:
5739:
5719:
5717:{\displaystyle i^{U}}
5692:
5615:
5595:
5575:
5555:
5532:
5251:
5190:
5163:
5161:{\displaystyle d^{*}}
5109:
4892:
4688:
4070:
4068:{\displaystyle f_{*}}
4043:
4009:
3822:
3754:
3661:
3620:in the sense that if
3603:
3250:
3080:
3064:
2856:
2625:
2609:
2420:
2229:
2194:
2023:
1794:
1770:
1750:
1711:
1668:
1579:where, by exactness,
1571:
1439:
1401:
903:
883:
859:
827:
744:
608:
186:topological manifolds
50:tool to help compute
6690:, Berlin, New York:
6220:topological K-theory
6064:
5791:
5748:
5728:
5701:
5624:
5604:
5584:
5564:
5544:
5328:
5220:
5172:
5145:
4930:
4723:
4086:
4052:
4022:
3834:
3763:
3678:
3624:
3330:
3171:
2891:
2752:
2444:
2298:
2257:as the union of two
2054:
1879:
1759:
1723:
1684:
1586:
1455:
1422:
1186:
872:
836:
756:
618:
326:
256:. (The interiors of
190:simplicial complexes
52:algebraic invariants
7052:Verdier, Jean-Louis
7040:Verdier, Jean-Louis
6879:Bar-Ilan University
6764:Sheaves in topology
6038:) is isomorphic to
6021:). In other words,
5890:
5835:
5680:
5656:
5505:
5482:
5456:
5439:
5420:
5393:
5376:
5351:
5069:
5023:
4968:
4703:singular cohomology
3545:
3499:
3412:
2864:for all dimensions
2668:deformation retract
2358:
2289:homotopy equivalent
2232:fundamental polygon
1952:
1872:groups then yields
1834:homotopy equivalent
1775:is path-connected.
1358:
1265:
932:The boundary maps ∂
710:
563:
525:
452:
376:
282:long exact sequence
174:group homomorphisms
106:singular cohomology
102:simplicial homology
79:long exact sequence
7107:10.1007/BF01696765
7072:10.1007/BFb0061320
7018:Algebraic Topology
6987:10.1007/BF02307601
6889:, pp. 61–63,
6836:Algebraic Topology
6619:Bott & Tu 1982
6170:
6056:differential forms
5925:
5754:
5734:
5714:
5687:
5666:
5642:
5610:
5590:
5570:
5550:
5527:
5485:
5442:
5406:
5379:
5337:
5258:partition of unity
5246:
5185:
5158:
5104:
4921:de Rham cohomology
4887:
4683:
4681:
4065:
4038:
4004:
4002:
3919:
3817:
3749:
3656:
3598:
3286:Further discussion
3245:
3095:
3059:
3054:
3041:
3013:
2851:
2672:open neighborhoods
2644:
2604:
2599:
2586:
2558:
2415:
2244:
2230:The Klein bottle (
2189:
2184:
2168:
2140:
2043:(two points) as a
2018:
1852:-discs, which are
1832:with intersection
1804:
1779:Basic applications
1765:
1745:
1706:
1663:
1566:
1434:
1396:
1097:), and therefore ∂
930:
878:
854:
822:
739:
603:
446:
445:
89:of the subspaces.
56:topological spaces
36:algebraic topology
7089:Vietoris, Leopold
7081:978-3-540-06012-3
6960:978-0-387-90271-5
6927:978-0-8218-3514-2
6896:978-0-19-851045-1
6850:978-0-521-79540-1
6822:978-0-691-07965-3
6801:Eilenberg, Samuel
6785:978-3-540-20665-1
6701:978-0-387-90613-3
6571:978-0-387-90613-3
6248:spectral sequence
6192:homology theories
5891:
5836:
5737:{\displaystyle U}
5593:{\displaystyle X}
5573:{\displaystyle V}
5553:{\displaystyle U}
5483:
5440:
5377:
5070:
5024:
4969:
3925:
3546:
3500:
3413:
3218:
3184:
3040:
3012:
2904:
2833:
2802:
2765:
2630:of two 2-spheres
2585:
2557:
2457:
2391:
2363:
2359:
2357:
2356:
2345:
2317:
2167:
2139:
2067:
1966:
1953:
1951:
1950:
1904:
1768:{\displaystyle X}
1718:fundamental group
1629:
1592:
1535:
1372:
1359:
1310:
1279:
1266:
1205:
1081:, which implies ∂
711:
564:
526:
453:
377:
315:Unreduced version
266:singular homology
238:topological space
143:fundamental group
141:Similarly to the
131:fundamental group
98:homology theories
16:(Redirected from
7182:
7158:
7139:
7121:
7117:
7084:
7034:
7009:
7005:
6963:
6938:
6907:
6861:
6825:
6805:Steenrod, Norman
6796:
6760:Dimca, Alexandru
6754:
6726:
6704:
6670:
6664:
6658:
6657:, pp. 35–36
6652:
6646:
6645:, pp. 25–26
6640:
6634:
6628:
6622:
6616:
6610:
6604:
6598:
6597:
6591:
6583:
6555:
6549:
6543:
6537:
6531:
6525:
6519:
6513:
6507:
6501:
6495:
6489:
6483:
6477:
6471:
6465:
6459:
6453:
6447:
6441:
6435:
6429:
6423:
6417:
6411:
6405:
6399:
6390:
6384:
6375:
6369:
6363:
6357:
6351:
6345:
6339:
6334:
6328:
6322:
6316:
6310:
6304:
6299:
6293:
6288:
6268:Excision theorem
6236:sheaf cohomology
6230:Sheaf cohomology
6179:
6177:
6176:
6171:
6148:
6147:
6126:
6125:
6104:
6103:
6082:
6081:
5934:
5932:
5931:
5926:
5903:
5902:
5892:
5882:
5870:
5869:
5848:
5847:
5837:
5827:
5809:
5808:
5763:
5761:
5760:
5755:
5743:
5741:
5740:
5735:
5723:
5721:
5720:
5715:
5713:
5712:
5696:
5694:
5693:
5688:
5679:
5674:
5655:
5650:
5619:
5617:
5616:
5611:
5599:
5597:
5596:
5591:
5579:
5577:
5576:
5571:
5559:
5557:
5556:
5551:
5536:
5534:
5533:
5528:
5504:
5493:
5481:
5480:
5467:
5455:
5450:
5431:
5419:
5414:
5392:
5387:
5368:
5350:
5345:
5317:
5310:
5306:
5302:
5295:
5285:
5278:
5271:
5255:
5253:
5252:
5247:
5245:
5244:
5232:
5231:
5215:
5211:
5201:
5194:
5192:
5191:
5186:
5184:
5183:
5167:
5165:
5164:
5159:
5157:
5156:
5140:
5136:
5128:
5113:
5111:
5110:
5105:
5088:
5087:
5071:
5068:
5067:
5054:
5036:
5035:
5025:
5015:
5003:
5002:
4981:
4980:
4970:
4960:
4948:
4947:
4908:is the group of
4896:
4894:
4893:
4888:
4865:
4864:
4825:
4824:
4797:
4796:
4769:
4768:
4741:
4740:
4692:
4690:
4689:
4684:
4682:
4665:
4664:
4652:
4651:
4639:
4638:
4613:
4612:
4600:
4599:
4580:
4579:
4567:
4566:
4551:
4550:
4538:
4537:
4518:
4517:
4505:
4504:
4492:
4491:
4472:
4471:
4459:
4458:
4434:
4433:
4427:
4426:
4416:
4414:
4413:
4407:
4406:
4396:
4394:
4393:
4387:
4386:
4376:
4374:
4373:
4367:
4366:
4356:
4354:
4353:
4347:
4346:
4336:
4319:
4318:
4306:
4305:
4293:
4292:
4267:
4266:
4254:
4253:
4234:
4233:
4221:
4220:
4205:
4204:
4192:
4191:
4172:
4171:
4159:
4158:
4146:
4145:
4126:
4125:
4113:
4112:
4074:
4072:
4071:
4066:
4064:
4063:
4047:
4045:
4044:
4039:
4034:
4033:
4013:
4011:
4010:
4005:
4003:
3999:
3998:
3983:
3982:
3963:
3962:
3947:
3946:
3926:
3923:
3920:
3916:
3915:
3903:
3902:
3890:
3889:
3876:
3875:
3863:
3862:
3850:
3849:
3826:
3824:
3823:
3818:
3813:
3812:
3800:
3799:
3787:
3786:
3758:
3756:
3755:
3750:
3745:
3744:
3732:
3731:
3716:
3715:
3703:
3702:
3690:
3689:
3665:
3663:
3662:
3657:
3655:
3654:
3642:
3641:
3607:
3605:
3604:
3599:
3564:
3563:
3547:
3544:
3543:
3530:
3512:
3511:
3501:
3498:
3497:
3485:
3484:
3471:
3453:
3452:
3425:
3424:
3414:
3408:
3407:
3395:
3394:
3378:
3348:
3347:
3254:
3252:
3251:
3246:
3232:
3231:
3220:
3219:
3211:
3192:
3191:
3186:
3185:
3177:
3085:of the 0-sphere
3068:
3066:
3065:
3060:
3058:
3055:
3042:
3038:
3014:
3010:
3005:
2997:
2978:
2970:
2961:
2960:
2945:
2941:
2940:
2939:
2927:
2926:
2912:
2911:
2906:
2905:
2897:
2860:
2858:
2857:
2852:
2841:
2840:
2835:
2834:
2826:
2810:
2809:
2804:
2803:
2795:
2773:
2772:
2767:
2766:
2758:
2713:it follows that
2613:
2611:
2610:
2605:
2603:
2602:
2587:
2583:
2559:
2555:
2550:
2549:
2544:
2535:
2516:
2515:
2510:
2501:
2492:
2491:
2476:
2465:
2464:
2459:
2458:
2450:
2424:
2422:
2421:
2416:
2399:
2398:
2393:
2392:
2384:
2376:
2368:
2361:
2360:
2352:
2347:
2343:
2342:
2325:
2324:
2319:
2318:
2310:
2198:
2196:
2195:
2190:
2188:
2187:
2169:
2165:
2141:
2137:
2132:
2116:
2110:
2109:
2094:
2090:
2089:
2075:
2074:
2069:
2068:
2060:
2037:reduced homology
2027:
2025:
2024:
2019:
2005:
2001:
2000:
1980:
1979:
1968:
1967:
1959:
1954:
1949:
1948:
1939:
1934:
1931:
1927:
1926:
1912:
1911:
1906:
1905:
1897:
1870:reduced homology
1774:
1772:
1771:
1766:
1754:
1752:
1751:
1746:
1735:
1734:
1715:
1713:
1712:
1707:
1696:
1695:
1672:
1670:
1669:
1664:
1656:
1655:
1643:
1642:
1630:
1627:
1619:
1618:
1606:
1605:
1593:
1590:
1575:
1573:
1572:
1567:
1562:
1561:
1549:
1548:
1536:
1533:
1531:
1514:
1513:
1492:
1491:
1467:
1466:
1443:
1441:
1440:
1435:
1405:
1403:
1402:
1397:
1380:
1379:
1374:
1373:
1365:
1360:
1357:
1356:
1344:
1343:
1330:
1318:
1317:
1312:
1311:
1303:
1287:
1286:
1281:
1280:
1272:
1267:
1261:
1260:
1248:
1247:
1231:
1213:
1212:
1207:
1206:
1198:
1165:reduced homology
887:
885:
884:
879:
863:
861:
860:
855:
831:
829:
828:
823:
748:
746:
745:
740:
723:
722:
712:
709:
708:
696:
695:
682:
670:
669:
648:
647:
612:
610:
609:
604:
581:
580:
562:
561:
548:
537:
536:
524:
523:
511:
510:
497:
486:
485:
464:
463:
451:
447:
442:
441:
428:
427:
405:
388:
387:
375:
374:
361:
350:
349:
288:) of the spaces
218:Samuel Eilenberg
68:Leopold Vietoris
62:mathematicians,
21:
7190:
7189:
7185:
7184:
7183:
7181:
7180:
7179:
7175:Homology theory
7165:
7164:
7137:
7132:
7129:
7127:Further reading
7124:
7119:
7087:
7082:
7064:Springer-Verlag
7038:
7032:
7022:Springer-Verlag
7012:
7007:
6967:
6961:
6951:Springer-Verlag
6943:Massey, William
6941:
6928:
6911:
6897:
6865:
6851:
6829:
6823:
6799:
6786:
6768:Springer-Verlag
6758:
6752:
6732:Dieudonné, Jean
6730:
6724:
6708:
6702:
6692:Springer-Verlag
6682:
6678:
6673:
6665:
6661:
6653:
6649:
6641:
6637:
6629:
6625:
6617:
6613:
6605:
6601:
6584:
6572:
6557:
6556:
6552:
6544:
6540:
6532:
6528:
6520:
6516:
6508:
6504:
6496:
6492:
6484:
6480:
6472:
6468:
6460:
6456:
6448:
6444:
6436:
6432:
6424:
6420:
6412:
6408:
6400:
6393:
6385:
6378:
6370:
6366:
6358:
6354:
6346:
6342:
6335:
6331:
6323:
6319:
6311:
6307:
6300:
6296:
6291:Hirzebruch 1999
6289:
6285:
6281:
6264:
6240:Čech cohomology
6232:
6208:dimension axiom
6204:
6139:
6117:
6095:
6073:
6062:
6061:
6046:
6029:
6016:
5975:
5894:
5861:
5839:
5800:
5789:
5788:
5782:chain complexes
5770:
5746:
5745:
5726:
5725:
5704:
5699:
5698:
5622:
5621:
5602:
5601:
5582:
5581:
5562:
5561:
5542:
5541:
5472:
5326:
5325:
5312:
5311:. One then has
5308:
5304:
5297:
5287:
5284:
5280:
5277:
5273:
5261:
5236:
5223:
5218:
5217:
5213:
5203:
5199:
5175:
5170:
5169:
5148:
5143:
5142:
5138:
5134:
5118:
5073:
5059:
5027:
4994:
4972:
4939:
4928:
4927:
4917:smooth manifold
4850:
4816:
4788:
4760:
4732:
4721:
4720:
4699:
4680:
4679:
4674:
4669:
4656:
4643:
4624:
4622:
4617:
4604:
4591:
4589:
4584:
4571:
4558:
4542:
4529:
4527:
4522:
4509:
4496:
4483:
4481:
4476:
4463:
4444:
4442:
4436:
4435:
4418:
4415:
4398:
4395:
4378:
4375:
4358:
4355:
4338:
4334:
4333:
4328:
4323:
4310:
4297:
4278:
4276:
4271:
4258:
4245:
4243:
4238:
4225:
4212:
4196:
4183:
4181:
4176:
4163:
4150:
4137:
4135:
4130:
4117:
4098:
4096:
4084:
4083:
4055:
4050:
4049:
4025:
4020:
4019:
4001:
4000:
3990:
3974:
3965:
3964:
3954:
3938:
3918:
3917:
3907:
3894:
3881:
3878:
3877:
3867:
3854:
3841:
3832:
3831:
3804:
3791:
3778:
3761:
3760:
3736:
3723:
3707:
3694:
3681:
3676:
3675:
3646:
3633:
3622:
3621:
3614:
3549:
3535:
3503:
3489:
3476:
3444:
3416:
3399:
3386:
3339:
3328:
3327:
3293:
3288:
3208:
3174:
3169:
3168:
3075:
3053:
3052:
3034:
3028:
3027:
3006:
2985:
2949:
2931:
2918:
2917:
2913:
2894:
2889:
2888:
2823:
2792:
2755:
2750:
2749:
2620:
2598:
2597:
2579:
2573:
2572:
2551:
2539:
2524:
2505:
2480:
2466:
2447:
2442:
2441:
2381:
2307:
2296:
2295:
2224:
2204:Kronecker delta
2202:where δ is the
2183:
2182:
2161:
2155:
2154:
2133:
2121:
2098:
2081:
2077:
2057:
2052:
2051:
2034:
1986:
1982:
1956:
1940:
1918:
1914:
1894:
1877:
1876:
1789:
1781:
1757:
1756:
1726:
1721:
1720:
1687:
1682:
1681:
1647:
1634:
1610:
1597:
1584:
1583:
1553:
1540:
1505:
1483:
1458:
1453:
1452:
1420:
1419:
1412:
1362:
1348:
1335:
1300:
1269:
1252:
1239:
1195:
1184:
1183:
1161:
1159:Reduced version
1134:
1040:
1026:
941:
935:
907:
898:
870:
869:
834:
833:
754:
753:
714:
700:
687:
661:
639:
616:
615:
566:
553:
528:
515:
502:
477:
455:
444:
443:
433:
430:
429:
419:
410:
379:
366:
335:
324:
323:
317:
268:for the triad (
230:
222:Norman Steenrod
160:: sequences of
158:exact sequences
147:homotopy groups
139:
40:homology theory
34:, particularly
28:
23:
22:
15:
12:
11:
5:
7188:
7186:
7178:
7177:
7167:
7166:
7161:
7160:
7128:
7125:
7123:
7122:
7085:
7080:
7044:Artin, Michael
7036:
7030:
7014:Spanier, Edwin
7010:
6969:Mayer, Walther
6965:
6959:
6939:
6926:
6909:
6895:
6863:
6849:
6831:Hatcher, Allen
6827:
6821:
6797:
6784:
6756:
6750:
6728:
6722:
6706:
6700:
6679:
6677:
6674:
6672:
6671:
6659:
6647:
6635:
6623:
6611:
6599:
6570:
6550:
6538:
6536:, Theorem 15.4
6526:
6514:
6502:
6490:
6478:
6466:
6454:
6442:
6430:
6418:
6406:
6391:
6376:
6364:
6362:, Theorem 15.3
6352:
6340:
6329:
6317:
6313:Dieudonné 1989
6305:
6294:
6282:
6280:
6277:
6276:
6275:
6270:
6263:
6260:
6231:
6228:
6214:, it holds in
6203:
6200:
6181:
6180:
6169:
6166:
6163:
6160:
6157:
6154:
6151:
6146:
6142:
6138:
6135:
6132:
6129:
6124:
6120:
6116:
6113:
6110:
6107:
6102:
6098:
6094:
6091:
6088:
6085:
6080:
6076:
6072:
6069:
6042:
6025:
6012:
5996:-simplices of
5988:and chains in
5971:
5937:
5936:
5924:
5921:
5918:
5915:
5912:
5909:
5906:
5901:
5897:
5889:
5885:
5879:
5876:
5873:
5868:
5864:
5860:
5857:
5854:
5851:
5846:
5842:
5834:
5830:
5824:
5821:
5818:
5815:
5812:
5807:
5803:
5799:
5796:
5769:
5766:
5753:
5733:
5711:
5707:
5686:
5683:
5678:
5673:
5669:
5665:
5662:
5659:
5654:
5649:
5645:
5641:
5638:
5635:
5632:
5629:
5609:
5600:are as above,
5589:
5569:
5549:
5538:
5537:
5526:
5523:
5520:
5517:
5514:
5511:
5508:
5503:
5500:
5497:
5492:
5488:
5479:
5475:
5470:
5465:
5462:
5459:
5454:
5449:
5445:
5438:
5434:
5429:
5426:
5423:
5418:
5413:
5409:
5405:
5402:
5399:
5396:
5391:
5386:
5382:
5375:
5371:
5366:
5363:
5360:
5357:
5354:
5349:
5344:
5340:
5336:
5333:
5282:
5275:
5243:
5239:
5235:
5230:
5226:
5182:
5178:
5155:
5151:
5115:
5114:
5103:
5100:
5097:
5094:
5091:
5086:
5083:
5080:
5076:
5066:
5062:
5057:
5051:
5048:
5045:
5042:
5039:
5034:
5030:
5022:
5018:
5012:
5009:
5006:
5001:
4997:
4993:
4990:
4987:
4984:
4979:
4975:
4967:
4963:
4957:
4954:
4951:
4946:
4942:
4938:
4935:
4898:
4897:
4886:
4883:
4880:
4877:
4874:
4871:
4868:
4863:
4860:
4857:
4853:
4849:
4846:
4843:
4840:
4837:
4834:
4831:
4828:
4823:
4819:
4815:
4812:
4809:
4806:
4803:
4800:
4795:
4791:
4787:
4784:
4781:
4778:
4775:
4772:
4767:
4763:
4759:
4756:
4753:
4750:
4747:
4744:
4739:
4735:
4731:
4728:
4698:
4695:
4694:
4693:
4678:
4675:
4673:
4670:
4668:
4663:
4659:
4655:
4650:
4646:
4642:
4637:
4634:
4631:
4627:
4623:
4621:
4618:
4616:
4611:
4607:
4603:
4598:
4594:
4590:
4588:
4585:
4583:
4578:
4574:
4570:
4565:
4561:
4557:
4554:
4549:
4545:
4541:
4536:
4532:
4528:
4526:
4523:
4521:
4516:
4512:
4508:
4503:
4499:
4495:
4490:
4486:
4482:
4480:
4477:
4475:
4470:
4466:
4462:
4457:
4454:
4451:
4447:
4443:
4441:
4438:
4437:
4432:
4425:
4421:
4417:
4412:
4405:
4401:
4397:
4392:
4385:
4381:
4377:
4372:
4365:
4361:
4357:
4352:
4345:
4341:
4337:
4335:
4332:
4329:
4327:
4324:
4322:
4317:
4313:
4309:
4304:
4300:
4296:
4291:
4288:
4285:
4281:
4277:
4275:
4272:
4270:
4265:
4261:
4257:
4252:
4248:
4244:
4242:
4239:
4237:
4232:
4228:
4224:
4219:
4215:
4211:
4208:
4203:
4199:
4195:
4190:
4186:
4182:
4180:
4177:
4175:
4170:
4166:
4162:
4157:
4153:
4149:
4144:
4140:
4136:
4134:
4131:
4129:
4124:
4120:
4116:
4111:
4108:
4105:
4101:
4097:
4095:
4092:
4091:
4062:
4058:
4048:commutes with
4037:
4032:
4028:
4016:
4015:
3997:
3993:
3989:
3986:
3981:
3977:
3973:
3970:
3967:
3966:
3961:
3957:
3953:
3950:
3945:
3941:
3937:
3934:
3931:
3930:
3914:
3910:
3906:
3901:
3897:
3893:
3888:
3884:
3880:
3879:
3874:
3870:
3866:
3861:
3857:
3853:
3848:
3844:
3840:
3839:
3816:
3811:
3807:
3803:
3798:
3794:
3790:
3785:
3781:
3777:
3774:
3771:
3768:
3748:
3743:
3739:
3735:
3730:
3726:
3722:
3719:
3714:
3710:
3706:
3701:
3697:
3693:
3688:
3684:
3653:
3649:
3645:
3640:
3636:
3632:
3629:
3613:
3610:
3609:
3608:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3562:
3559:
3556:
3552:
3542:
3538:
3533:
3527:
3524:
3521:
3518:
3515:
3510:
3506:
3496:
3492:
3488:
3483:
3479:
3474:
3468:
3465:
3462:
3459:
3456:
3451:
3447:
3443:
3440:
3437:
3434:
3431:
3428:
3423:
3419:
3411:
3406:
3402:
3398:
3393:
3389:
3385:
3381:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3346:
3342:
3338:
3335:
3292:
3289:
3287:
3284:
3256:
3255:
3244:
3241:
3238:
3235:
3230:
3227:
3224:
3217:
3214:
3207:
3204:
3201:
3198:
3195:
3190:
3183:
3180:
3074:
3071:
3070:
3069:
3057:
3051:
3048:
3045:
3035:
3033:
3030:
3029:
3026:
3023:
3020:
3017:
3007:
3004:
3000:
2996:
2992:
2991:
2988:
2984:
2981:
2977:
2973:
2969:
2965:
2959:
2956:
2952:
2948:
2944:
2938:
2934:
2930:
2925:
2921:
2916:
2910:
2903:
2900:
2862:
2861:
2850:
2847:
2844:
2839:
2832:
2829:
2822:
2819:
2816:
2813:
2808:
2801:
2798:
2791:
2788:
2785:
2782:
2779:
2776:
2771:
2764:
2761:
2654:of two spaces
2619:
2616:
2615:
2614:
2601:
2596:
2593:
2590:
2580:
2578:
2575:
2574:
2571:
2568:
2565:
2562:
2552:
2548:
2543:
2538:
2534:
2530:
2529:
2527:
2522:
2519:
2514:
2509:
2504:
2500:
2496:
2490:
2487:
2483:
2479:
2475:
2472:
2469:
2463:
2456:
2453:
2426:
2425:
2414:
2411:
2408:
2405:
2402:
2397:
2390:
2387:
2379:
2375:
2371:
2367:
2355:
2350:
2341:
2337:
2334:
2331:
2328:
2323:
2316:
2313:
2306:
2303:
2238:(in blue) and
2223:
2220:
2200:
2199:
2186:
2181:
2178:
2175:
2172:
2162:
2160:
2157:
2156:
2153:
2150:
2147:
2144:
2134:
2131:
2127:
2126:
2124:
2119:
2115:
2108:
2105:
2101:
2097:
2093:
2088:
2084:
2080:
2073:
2066:
2063:
2032:
2029:
2028:
2017:
2014:
2011:
2008:
2004:
1999:
1996:
1993:
1989:
1985:
1978:
1975:
1972:
1965:
1962:
1947:
1943:
1937:
1930:
1925:
1921:
1917:
1910:
1903:
1900:
1893:
1890:
1887:
1884:
1788:
1782:
1780:
1777:
1764:
1744:
1741:
1738:
1733:
1729:
1705:
1702:
1699:
1694:
1690:
1674:
1673:
1662:
1659:
1654:
1650:
1646:
1641:
1637:
1633:
1625:
1622:
1617:
1613:
1609:
1604:
1600:
1596:
1577:
1576:
1565:
1560:
1556:
1552:
1547:
1543:
1539:
1530:
1526:
1523:
1520:
1517:
1512:
1508:
1504:
1501:
1498:
1495:
1490:
1486:
1482:
1479:
1476:
1473:
1470:
1465:
1461:
1446:path-connected
1433:
1430:
1427:
1411:
1408:
1407:
1406:
1395:
1392:
1389:
1386:
1383:
1378:
1371:
1368:
1355:
1351:
1347:
1342:
1338:
1333:
1327:
1324:
1321:
1316:
1309:
1306:
1299:
1296:
1293:
1290:
1285:
1278:
1275:
1264:
1259:
1255:
1251:
1246:
1242:
1238:
1234:
1228:
1225:
1222:
1219:
1216:
1211:
1204:
1201:
1194:
1191:
1160:
1157:
1129:
1035:
1024:
939:
933:
905:
897:
894:
877:
866:inclusion maps
853:
850:
847:
844:
841:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
776:
773:
770:
767:
764:
761:
750:
749:
738:
735:
732:
729:
726:
721:
717:
707:
703:
699:
694:
690:
685:
679:
676:
673:
668:
664:
660:
657:
654:
651:
646:
642:
638:
635:
613:
602:
599:
596:
593:
590:
587:
584:
579:
576:
573:
569:
560:
556:
551:
546:
543:
540:
535:
531:
522:
518:
514:
509:
505:
500:
495:
492:
489:
484:
480:
476:
473:
470:
467:
462:
458:
450:
440:
436:
432:
431:
426:
422:
418:
417:
413:
408:
403:
400:
397:
394:
391:
386:
382:
373:
369:
364:
359:
356:
353:
348:
345:
342:
338:
334:
331:
316:
313:
229:
226:
172:(in this case
164:(in this case
151:linear algebra
145:or the higher
138:
135:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7187:
7176:
7173:
7172:
7170:
7163:
7157:
7153:
7149:
7145:
7144:
7136:
7131:
7130:
7126:
7116:
7112:
7108:
7104:
7100:
7096:
7095:
7090:
7086:
7083:
7077:
7073:
7069:
7066:, p. 1,
7065:
7061:
7057:
7053:
7049:
7045:
7041:
7037:
7033:
7031:0-387-94426-5
7027:
7023:
7019:
7015:
7011:
7004:
7000:
6996:
6992:
6988:
6984:
6980:
6976:
6975:
6970:
6966:
6962:
6956:
6952:
6948:
6944:
6940:
6937:
6933:
6929:
6923:
6919:
6915:
6910:
6906:
6902:
6898:
6892:
6888:
6884:
6880:
6876:
6872:
6868:
6864:
6860:
6856:
6852:
6846:
6842:
6838:
6837:
6832:
6828:
6824:
6818:
6814:
6810:
6806:
6802:
6798:
6795:
6791:
6787:
6781:
6777:
6773:
6769:
6765:
6761:
6757:
6753:
6751:0-8176-3388-X
6747:
6743:
6739:
6738:
6733:
6729:
6725:
6723:3-7643-7002-5
6719:
6715:
6711:
6707:
6703:
6697:
6693:
6689:
6685:
6681:
6680:
6675:
6668:
6663:
6660:
6656:
6651:
6648:
6644:
6639:
6636:
6633:, p. 162
6632:
6627:
6624:
6620:
6615:
6612:
6608:
6603:
6600:
6595:
6589:
6581:
6577:
6573:
6567:
6563:
6562:
6554:
6551:
6548:, p. 203
6547:
6542:
6539:
6535:
6530:
6527:
6524:, p. 208
6523:
6518:
6515:
6512:, p. 152
6511:
6506:
6503:
6499:
6494:
6491:
6487:
6482:
6479:
6476:, p. 151
6475:
6470:
6467:
6464:, p. 384
6463:
6458:
6455:
6451:
6446:
6443:
6439:
6434:
6431:
6428:, p. 240
6427:
6422:
6419:
6416:, p. 187
6415:
6410:
6407:
6404:, p. 150
6403:
6398:
6396:
6392:
6389:, p. 149
6388:
6383:
6381:
6377:
6373:
6368:
6365:
6361:
6356:
6353:
6350:, p. 345
6349:
6344:
6341:
6338:
6337:Vietoris 1930
6333:
6330:
6326:
6321:
6318:
6314:
6309:
6306:
6303:
6298:
6295:
6292:
6287:
6284:
6278:
6274:
6273:Zig-zag lemma
6271:
6269:
6266:
6265:
6261:
6259:
6257:
6253:
6249:
6245:
6241:
6237:
6229:
6227:
6225:
6221:
6217:
6213:
6209:
6201:
6199:
6197:
6193:
6189:
6184:
6167:
6158:
6155:
6152:
6144:
6130:
6122:
6114:
6108:
6100:
6086:
6078:
6067:
6060:
6059:
6058:
6057:
6052:
6050:
6045:
6041:
6037:
6033:
6028:
6024:
6020:
6015:
6011:
6007:
6003:
5999:
5995:
5991:
5987:
5983:
5979:
5974:
5970:
5966:
5962:
5958:
5954:
5950:
5946:
5942:
5922:
5913:
5910:
5907:
5899:
5895:
5887:
5883:
5874:
5866:
5862:
5858:
5852:
5844:
5840:
5832:
5828:
5819:
5816:
5813:
5805:
5801:
5794:
5787:
5786:
5785:
5783:
5779:
5775:
5772:Consider the
5767:
5765:
5744:by zero, and
5731:
5709:
5705:
5681:
5676:
5671:
5667:
5663:
5660:
5657:
5652:
5647:
5643:
5633:
5630:
5627:
5607:
5587:
5567:
5547:
5524:
5515:
5512:
5509:
5501:
5498:
5495:
5490:
5486:
5477:
5473:
5468:
5460:
5452:
5447:
5443:
5432:
5424:
5416:
5411:
5407:
5403:
5397:
5389:
5384:
5380:
5373:
5369:
5361:
5358:
5355:
5347:
5342:
5338:
5331:
5324:
5323:
5322:
5319:
5315:
5300:
5294:
5290:
5269:
5265:
5259:
5241:
5237:
5233:
5228:
5224:
5210:
5206:
5198:
5180:
5153:
5149:
5132:
5126:
5122:
5101:
5092:
5084:
5081:
5078:
5074:
5064:
5060:
5055:
5046:
5043:
5040:
5032:
5028:
5016:
5007:
4999:
4995:
4991:
4985:
4977:
4973:
4965:
4961:
4952:
4944:
4940:
4933:
4926:
4925:
4924:
4922:
4918:
4914:
4911:
4907:
4902:
4884:
4875:
4872:
4869:
4861:
4858:
4855:
4851:
4841:
4838:
4835:
4832:
4829:
4821:
4817:
4807:
4804:
4801:
4793:
4789:
4785:
4779:
4776:
4773:
4765:
4761:
4751:
4748:
4745:
4737:
4733:
4726:
4719:
4718:
4717:
4715:
4711:
4708:
4704:
4696:
4676:
4661:
4657:
4653:
4648:
4644:
4635:
4632:
4629:
4625:
4609:
4605:
4596:
4592:
4576:
4572:
4563:
4559:
4555:
4547:
4543:
4534:
4530:
4514:
4510:
4506:
4501:
4497:
4488:
4484:
4468:
4464:
4455:
4452:
4449:
4445:
4439:
4423:
4419:
4403:
4399:
4383:
4379:
4363:
4359:
4343:
4339:
4330:
4315:
4311:
4307:
4302:
4298:
4289:
4286:
4283:
4279:
4263:
4259:
4250:
4246:
4230:
4226:
4217:
4213:
4209:
4201:
4197:
4188:
4184:
4168:
4164:
4160:
4155:
4151:
4142:
4138:
4122:
4118:
4109:
4106:
4103:
4099:
4093:
4082:
4081:
4080:
4078:
4060:
4056:
4035:
4030:
3995:
3991:
3987:
3979:
3975:
3968:
3959:
3955:
3951:
3943:
3939:
3932:
3912:
3908:
3904:
3899:
3895:
3891:
3886:
3882:
3872:
3868:
3864:
3859:
3855:
3851:
3846:
3842:
3830:
3829:
3828:
3814:
3809:
3805:
3801:
3796:
3792:
3788:
3783:
3775:
3772:
3769:
3741:
3737:
3728:
3724:
3712:
3708:
3699:
3695:
3691:
3686:
3682:
3673:
3669:
3651:
3647:
3638:
3634:
3630:
3627:
3619:
3611:
3595:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3560:
3557:
3554:
3550:
3540:
3531:
3522:
3519:
3516:
3508:
3504:
3494:
3490:
3486:
3481:
3477:
3472:
3463:
3460:
3457:
3449:
3445:
3441:
3435:
3432:
3429:
3421:
3417:
3404:
3400:
3396:
3391:
3387:
3379:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3344:
3340:
3333:
3326:
3325:
3324:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3291:Relative form
3290:
3285:
3283:
3281:
3277:
3273:
3269:
3265:
3261:
3242:
3236:
3228:
3225:
3222:
3212:
3205:
3199:
3196:
3188:
3178:
3167:
3166:
3165:
3163:
3159:
3155:
3151:
3147:
3143:
3139:
3135:
3132:is the union
3131:
3127:
3123:
3119:
3115:
3111:
3107:
3104:
3100:
3092:
3088:
3084:
3079:
3072:
3049:
3046:
3043:
3031:
3024:
3021:
3018:
3015:
2998:
2986:
2982:
2971:
2957:
2954:
2950:
2946:
2942:
2936:
2932:
2928:
2923:
2919:
2914:
2908:
2898:
2887:
2886:
2885:
2883:
2879:
2875:
2871:
2867:
2845:
2837:
2827:
2820:
2814:
2806:
2796:
2789:
2783:
2780:
2777:
2769:
2759:
2748:
2747:
2746:
2744:
2740:
2736:
2732:
2728:
2724:
2720:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2688:
2684:
2680:
2676:
2673:
2669:
2665:
2661:
2657:
2653:
2649:
2641:
2637:
2633:
2629:
2624:
2617:
2594:
2591:
2588:
2576:
2569:
2566:
2563:
2560:
2546:
2536:
2525:
2520:
2512:
2502:
2488:
2485:
2481:
2477:
2473:
2470:
2467:
2461:
2451:
2440:
2439:
2438:
2437:, it follows
2436:
2432:
2412:
2403:
2395:
2385:
2369:
2353:
2348:
2329:
2321:
2311:
2301:
2294:
2293:
2292:
2290:
2286:
2282:
2278:
2274:
2270:
2267:
2263:
2260:
2259:Möbius strips
2256:
2252:
2249:
2241:
2237:
2233:
2228:
2221:
2219:
2217:
2213:
2209:
2205:
2179:
2176:
2173:
2170:
2158:
2151:
2148:
2145:
2142:
2122:
2117:
2106:
2103:
2099:
2095:
2091:
2086:
2082:
2078:
2071:
2061:
2050:
2049:
2048:
2047:, it follows
2046:
2042:
2038:
2015:
2009:
2002:
1997:
1994:
1991:
1987:
1983:
1976:
1973:
1970:
1960:
1945:
1935:
1928:
1923:
1919:
1915:
1908:
1898:
1888:
1882:
1875:
1874:
1873:
1871:
1867:
1863:
1859:
1855:
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1812:
1810:
1802:
1798:
1793:
1786:
1783:
1778:
1776:
1762:
1739:
1731:
1727:
1719:
1700:
1692:
1688:
1679:
1660:
1652:
1648:
1644:
1639:
1635:
1623:
1615:
1611:
1607:
1602:
1598:
1582:
1581:
1580:
1558:
1554:
1550:
1545:
1541:
1528:
1518:
1510:
1506:
1502:
1496:
1488:
1484:
1477:
1471:
1463:
1459:
1451:
1450:
1449:
1447:
1431:
1428:
1425:
1417:
1409:
1393:
1384:
1376:
1366:
1353:
1349:
1345:
1340:
1336:
1331:
1322:
1314:
1304:
1297:
1291:
1283:
1273:
1257:
1253:
1249:
1244:
1240:
1232:
1223:
1220:
1217:
1209:
1199:
1189:
1182:
1181:
1180:
1178:
1174:
1170:
1166:
1158:
1156:
1155:are swapped.
1154:
1150:
1146:
1142:
1138:
1132:
1128:
1124:
1120:
1116:
1112:
1109:, since then
1108:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1038:
1034:
1030:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
982:
978:
974:
970:
966:
962:
958:
954:
950:
946:
942:
927:
923:
919:
915:
911:
902:
895:
893:
891:
875:
867:
851:
845:
842:
839:
819:
813:
810:
807:
804:
801:
795:
792:
789:
786:
783:
780:
777:
771:
768:
765:
762:
759:
736:
727:
719:
715:
705:
701:
697:
692:
688:
683:
674:
666:
662:
658:
652:
644:
640:
633:
614:
600:
591:
588:
585:
577:
574:
571:
567:
558:
549:
541:
533:
529:
520:
516:
512:
507:
503:
498:
490:
482:
478:
474:
468:
460:
456:
448:
438:
434:
424:
420:
411:
406:
398:
395:
392:
384:
380:
371:
362:
354:
346:
343:
340:
336:
329:
322:
321:
320:
314:
312:
310:
306:
303:
299:
295:
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
251:
247:
243:
239:
235:
227:
225:
223:
219:
215:
210:
206:
205:Betti numbers
202:
197:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
154:
152:
148:
144:
136:
134:
132:
128:
123:
119:
115:
111:
107:
103:
99:
95:
90:
88:
84:
80:
77:
73:
69:
65:
64:Walther Mayer
61:
57:
53:
49:
45:
41:
37:
33:
19:
7162:
7147:
7141:
7098:
7092:
7055:
7017:
6978:
6972:
6946:
6913:
6874:
6835:
6808:
6763:
6736:
6713:
6687:
6667:Verdier 1972
6662:
6650:
6638:
6631:Hatcher 2002
6626:
6614:
6607:Hatcher 2002
6602:
6560:
6553:
6546:Hatcher 2002
6541:
6529:
6517:
6510:Hatcher 2002
6505:
6498:Hatcher 2002
6493:
6486:Hatcher 2002
6481:
6474:Hatcher 2002
6469:
6462:Hatcher 2002
6457:
6450:Hatcher 2002
6445:
6438:Hatcher 2002
6433:
6421:
6414:Spanier 1966
6409:
6402:Hatcher 2002
6387:Hatcher 2002
6367:
6355:
6343:
6332:
6327:, p. 41
6320:
6315:, p. 39
6308:
6297:
6286:
6244:degeneration
6233:
6205:
6185:
6182:
6053:
6048:
6043:
6039:
6035:
6031:
6026:
6022:
6018:
6013:
6009:
6005:
6001:
5997:
5993:
5989:
5985:
5981:
5977:
5972:
5968:
5964:
5960:
5956:
5952:
5948:
5944:
5940:
5938:
5771:
5764:is the sum.
5539:
5320:
5313:
5298:
5292:
5288:
5267:
5263:
5208:
5204:
5124:
5120:
5116:
4912:
4910:real numbers
4905:
4903:
4899:
4709:
4700:
4017:
3615:
3320:
3316:
3312:
3308:
3304:
3300:
3294:
3275:
3271:
3267:
3263:
3259:
3257:
3161:
3157:
3153:
3149:
3145:
3141:
3137:
3133:
3129:
3125:
3117:
3113:
3109:
3105:
3098:
3096:
3090:
3086:
3082:
2877:
2873:
2869:
2865:
2863:
2743:contractible
2738:
2734:
2730:
2726:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2659:
2655:
2647:
2645:
2639:
2635:
2631:
2627:
2434:
2427:
2284:
2280:
2276:
2272:
2265:
2261:
2254:
2250:
2248:Klein bottle
2245:
2239:
2235:
2222:Klein bottle
2215:
2211:
2201:
2030:
1861:
1857:
1854:contractible
1849:
1846:homeomorphic
1841:
1837:
1829:
1825:
1821:
1817:
1813:
1808:
1805:
1800:
1796:
1784:
1675:
1578:
1413:
1172:
1168:
1162:
1152:
1148:
1144:
1140:
1136:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1046:
1042:
1036:
1032:
1028:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
980:
976:
972:
968:
964:
960:
952:
948:
944:
937:
931:
925:
921:
917:
913:
909:
896:Boundary map
888:denotes the
751:
318:
308:
304:
302:intersection
297:
293:
289:
285:
277:
273:
269:
261:
257:
253:
245:
241:
233:
231:
213:
198:
194:CW complexes
155:
140:
100:, including
91:
87:intersection
43:
29:
7120:(in German)
7008:(in German)
6981:(1): 1–42,
6871:Teicher, M.
6684:Bott, Raoul
6669:(SGA 4.V.3)
6522:Massey 1984
6426:Massey 1984
5197:closed form
3672:pushforward
3122:complements
3108:of a space
3073:Suspensions
2741:, which is
1678:abelianized
1418:. Whenever
32:mathematics
7101:: 159–62,
6710:Corry, Leo
6676:References
6655:Dimca 2004
6348:Corry 2004
6325:Mayer 1929
6302:Mayer 1929
6194:using the
5768:Derivation
5212:, express
5131:open cover
3668:continuous
3612:Naturality
3103:suspension
2882:from above
2689:. Letting
2618:Wedge sums
955:which, by
300:, and the
94:cohomology
83:direct sum
7156:0002-9920
7115:121151421
7003:120803366
6995:0026-9255
6905:223099225
6588:cite book
6224:cobordism
6218:(such as
6165:→
6156:∩
6141:Ω
6137:→
6119:Ω
6115:⊕
6097:Ω
6093:→
6075:Ω
6071:→
5920:→
5888:β
5859:⊕
5833:α
5817:∩
5798:→
5752:Σ
5682:ω
5672:∗
5664:−
5658:ω
5648:∗
5637:↦
5634:ω
5628:δ
5608:δ
5525:⋯
5522:→
5513:∩
5478:∗
5437:Σ
5404:⊕
5374:δ
5359:∩
5335:→
5332:⋯
5286:agree on
5238:ω
5234:−
5225:ω
5181:∗
5177:∂
5154:∗
5102:⋯
5099:→
5065:∗
5044:∩
5021:Δ
4992:⊕
4966:ρ
4937:→
4934:⋯
4885:⋯
4882:→
4848:→
4833:∩
4814:→
4786:⊕
4758:→
4730:→
4727:⋯
4677:⋯
4672:⟶
4654:∩
4633:−
4620:⟶
4587:⟶
4556:⊕
4525:⟶
4507:∩
4479:⟶
4440:⋯
4424:∗
4404:∗
4384:∗
4364:∗
4344:∗
4331:⋯
4326:⟶
4308:∩
4287:−
4274:⟶
4241:⟶
4210:⊕
4179:⟶
4161:∩
4133:⟶
4094:⋯
4061:∗
4031:∗
4027:∂
3988:⊂
3952:⊂
3905:∪
3865:∪
3810:∗
3802:∘
3797:∗
3784:∗
3773:∘
3721:→
3687:∗
3644:→
3596:⋯
3593:→
3584:∩
3572:∩
3558:−
3541:∗
3537:∂
3495:∗
3487:−
3482:∗
3442:⊕
3405:∗
3392:∗
3368:∩
3356:∩
3337:→
3334:⋯
3226:−
3216:~
3206:≅
3182:~
3047:≠
2999:⊕
2972:⊕
2951:δ
2947:≅
2929:∨
2902:~
2831:~
2821:⊕
2800:~
2790:≅
2781:∨
2763:~
2664:basepoint
2652:wedge sum
2592:≠
2537:⊕
2503:⊕
2482:δ
2478:≅
2455:~
2431:injective
2410:→
2389:~
2378:→
2370:⊕
2354:α
2336:→
2315:~
2305:→
2242:(in red).
2174:≠
2100:δ
2096:≅
2065:~
2045:base case
2016:⋯
2013:⟶
2007:⟶
1995:−
1974:−
1964:~
1946:∗
1942:∂
1902:~
1892:⟶
1886:⟶
1883:⋯
1728:π
1653:∗
1640:∗
1624:≅
1616:∗
1608:−
1603:∗
1559:∗
1551:−
1546:∗
1503:⊕
1478:≅
1429:∩
1391:→
1370:~
1354:∗
1346:−
1341:∗
1308:~
1298:⊕
1277:~
1258:∗
1245:∗
1221:∩
1203:~
1193:→
1190:⋯
1177:non-empty
1121:for some
1085:− ∂
876:⊕
849:↪
817:↪
799:↪
793:∩
775:↪
769:∩
734:→
706:∗
698:−
693:∗
659:⊕
637:→
634:⋯
601:⋯
598:→
589:∩
575:−
559:∗
555:∂
521:∗
513:−
508:∗
475:⊕
439:∗
425:∗
396:∩
372:∗
368:∂
333:→
330:⋯
250:interiors
170:morphisms
72:subspaces
48:algebraic
7169:Category
7054:(eds.),
7016:(1966),
6945:(1984),
6833:(2002),
6807:(1952),
6762:(2004),
6734:(1989),
6712:(2004),
6262:See also
5939:where α(
5884:→
5829:→
5469:→
5433:→
5370:→
5056:→
5017:→
4962:→
4431:↓
4411:↓
4391:↓
4371:↓
4351:↓
4077:commutes
3532:→
3473:→
3380:→
3297:relative
3280:as above
3039:if
3011:if
2584:if
2556:if
2349:→
2166:if
2138:if
2041:0-sphere
1936:→
1332:→
1233:→
1093:−
1039:−1
1023:. Then ∂
999:. Since
963:-chains
684:→
550:→
499:→
407:→
363:→
129:for the
122:topology
118:relative
60:Austrian
6936:2225848
6873:(ed.),
6859:1867354
6794:2050072
6580:7597142
6246:of the
3618:natural
3140:, with
3120:be the
3101:is the
2650:be the
2039:of the
1866:trivial
1811:-sphere
1787:-sphere
1025:∗
951:-cycle
934:∗
906:∗
280:) is a
162:objects
114:reduced
76:natural
7154:
7150:(20),
7113:
7078:
7028:
7001:
6993:
6957:
6934:
6924:
6903:
6893:
6857:
6847:
6819:
6792:
6782:
6748:
6720:
6698:
6621:, §I.2
6578:
6568:
5967:, and
5697:where
5540:where
5256:via a
5129:is an
5117:where
3112:, let
2362:
2344:
1836:to a (
1820:, let
832:, and
252:cover
201:Vienna
182:kernel
168:) and
166:groups
46:is an
42:, the
7138:(PDF)
7111:S2CID
6999:S2CID
6374:, §15
6279:Notes
6256:topoi
5951:), β(
5943:) = (
5316:() =
5303:form
4707:group
3666:is a
2666:is a
2269:glued
2214:>
1755:when
1175:have
1101:and ∂
991:) = ∂
752:Here
236:be a
209:torus
192:, or
178:image
7152:ISSN
7076:ISBN
7026:ISBN
6991:ISSN
6955:ISBN
6922:ISBN
6901:OCLC
6891:ISBN
6845:ISBN
6817:ISBN
6780:ISBN
6746:ISBN
6718:ISBN
6696:ISBN
6594:link
6576:OCLC
6566:ISBN
6222:and
6190:for
5959:) =
5776:the
5279:and
5135:X, ρ
4714:dual
3315:and
3144:and
3116:and
2876:and
2725:and
2701:and
2681:and
2658:and
2646:Let
2634:and
2287:are
2264:and
1864:are
1860:and
1824:and
1171:and
1163:For
1151:and
1143:and
1089:= ∂(
1007:= −∂
983:= ∂(
975:and
967:and
924:and
868:and
864:are
260:and
240:and
232:Let
220:and
116:and
104:and
96:and
66:and
38:and
7103:doi
7068:doi
6983:doi
6772:doi
6226:).
6004:or
5947:, −
5307:on
5301:+ 1
5202:in
5133:of
4923:is
4712:is
3924:and
3124:in
3097:If
2670:of
1848:to
1591:Ker
1534:Ker
1444:is
1125:in
1117:= ∂
1077:+ ∂
1073:= ∂
1069:= ∂
1065:+ ∂
1031:in
995:+ ∂
216:by
54:of
30:In
7171::
7148:49
7146:,
7140:,
7118:.
7109:,
7099:37
7097:,
7074:,
7058:,
7050:;
7046:;
7024:,
7020:,
7006:.
6997:,
6989:,
6979:36
6977:,
6953:,
6949:,
6932:MR
6930:,
6920:,
6899:,
6855:MR
6853:,
6843:,
6839:,
6815:,
6811:,
6803:;
6790:MR
6788:,
6778:,
6770:,
6744:,
6742:39
6694:,
6590:}}
6586:{{
6574:.
6394:^
6379:^
6258:.
6198:.
6034:+
5980:+
5963:+
5955:,
5784:)
5318:.
5281:dω
5274:dω
5266:,
5123:,
3319:⊂
3311:⊂
3303:⊂
3295:A
3282:.
3164:,
3106:SY
3050:2.
2737:∪
2733:=
2729:∩
2721:=
2717:∪
2709:∪
2705:=
2697:∪
2693:=
2685:⊆
2677:⊆
2595:1.
2275:,
1816:=
1799:=
1628:Im
1394:0.
1133:+1
1113:-
1111:x′
1107:x′
1103:u′
1091:v′
1087:u′
1079:v′
1075:u′
1059:v′
1057:+
1055:u′
1053:=
987:+
916:+
912:=
892:.
737:0.
296:,
292:,
276:,
272:,
244:,
188:,
7159:.
7105::
7070::
7035:.
6985::
6964:.
6908:.
6885:/
6881:/
6862:.
6826:.
6774::
6755:.
6727:.
6705:.
6596:)
6582:.
6168:0
6162:)
6159:V
6153:U
6150:(
6145:n
6134:)
6131:V
6128:(
6123:n
6112:)
6109:U
6106:(
6101:n
6090:)
6087:X
6084:(
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6068:0
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6047:(
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6040:H
6036:B
6032:A
6030:(
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6023:H
6019:X
6017:(
6014:n
6010:H
6006:B
6002:A
5998:X
5994:n
5990:B
5986:A
5982:B
5978:A
5976:(
5973:n
5969:C
5965:y
5961:x
5957:y
5953:x
5949:x
5945:x
5941:x
5935:,
5923:0
5917:)
5914:B
5911:+
5908:A
5905:(
5900:n
5896:C
5878:)
5875:B
5872:(
5867:n
5863:C
5856:)
5853:A
5850:(
5845:n
5841:C
5823:)
5820:B
5814:A
5811:(
5806:n
5802:C
5795:0
5732:U
5710:U
5706:i
5685:)
5677:V
5668:i
5661:,
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5644:i
5640:(
5631::
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5507:(
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5496:n
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5474:d
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5461:X
5458:(
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5428:)
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5395:(
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5381:H
5365:)
5362:V
5356:U
5353:(
5348:n
5343:c
5339:H
5314:d
5309:X
5305:σ
5299:n
5293:V
5291:∩
5289:U
5283:V
5276:U
5270:}
5268:V
5264:U
5262:{
5242:V
5229:U
5214:ω
5209:V
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5205:U
5200:ω
5150:d
5139:Δ
5127:}
5125:V
5121:U
5119:{
5096:)
5093:X
5090:(
5085:1
5082:+
5079:n
5075:H
5061:d
5050:)
5047:V
5041:U
5038:(
5033:n
5029:H
5011:)
5008:V
5005:(
5000:n
4996:H
4989:)
4986:U
4983:(
4978:n
4974:H
4956:)
4953:X
4950:(
4945:n
4941:H
4913:R
4906:G
4879:)
4876:G
4873:;
4870:X
4867:(
4862:1
4859:+
4856:n
4852:H
4845:)
4842:G
4839:;
4836:B
4830:A
4827:(
4822:n
4818:H
4811:)
4808:G
4805:;
4802:B
4799:(
4794:n
4790:H
4783:)
4780:G
4777:;
4774:A
4771:(
4766:n
4762:H
4755:)
4752:G
4749:;
4746:X
4743:(
4738:n
4734:H
4710:G
4667:)
4662:2
4658:B
4649:2
4645:A
4641:(
4636:1
4630:n
4626:H
4615:)
4610:2
4606:X
4602:(
4597:n
4593:H
4582:)
4577:2
4573:B
4569:(
4564:n
4560:H
4553:)
4548:2
4544:A
4540:(
4535:n
4531:H
4520:)
4515:2
4511:B
4502:2
4498:A
4494:(
4489:n
4485:H
4474:)
4469:2
4465:X
4461:(
4456:1
4453:+
4450:n
4446:H
4420:f
4400:f
4380:f
4360:f
4340:f
4321:)
4316:1
4312:B
4303:1
4299:A
4295:(
4290:1
4284:n
4280:H
4269:)
4264:1
4260:X
4256:(
4251:n
4247:H
4236:)
4231:1
4227:B
4223:(
4218:n
4214:H
4207:)
4202:1
4198:A
4194:(
4189:n
4185:H
4174:)
4169:1
4165:B
4156:1
4152:A
4148:(
4143:n
4139:H
4128:)
4123:1
4119:X
4115:(
4110:1
4107:+
4104:n
4100:H
4057:f
4036:,
4014:,
3996:2
3992:B
3985:)
3980:1
3976:B
3972:(
3969:f
3960:2
3956:A
3949:)
3944:1
3940:A
3936:(
3933:f
3913:2
3909:B
3900:2
3896:A
3892:=
3887:2
3883:X
3873:1
3869:B
3860:1
3856:A
3852:=
3847:1
3843:X
3815:.
3806:h
3793:g
3789:=
3780:)
3776:h
3770:g
3767:(
3747:)
3742:2
3738:X
3734:(
3729:k
3725:H
3718:)
3713:1
3709:X
3705:(
3700:k
3696:H
3692::
3683:f
3652:2
3648:X
3639:1
3635:X
3631::
3628:f
3590:)
3587:D
3581:C
3578:,
3575:B
3569:A
3566:(
3561:1
3555:n
3551:H
3526:)
3523:Y
3520:,
3517:X
3514:(
3509:n
3505:H
3491:l
3478:k
3467:)
3464:D
3461:,
3458:B
3455:(
3450:n
3446:H
3439:)
3436:C
3433:,
3430:A
3427:(
3422:n
3418:H
3410:)
3401:j
3397:,
3388:i
3384:(
3374:)
3371:D
3365:C
3362:,
3359:B
3353:A
3350:(
3345:n
3341:H
3321:B
3317:D
3313:A
3309:C
3305:X
3301:Y
3276:k
3272:k
3268:k
3264:Y
3260:X
3243:.
3240:)
3237:Y
3234:(
3229:1
3223:n
3213:H
3203:)
3200:Y
3197:S
3194:(
3189:n
3179:H
3162:n
3158:Y
3154:B
3152:∩
3150:A
3146:B
3142:A
3138:B
3136:∪
3134:A
3130:X
3126:X
3118:B
3114:A
3110:Y
3099:X
3093:.
3091:X
3087:Y
3083:X
3044:n
3032:0
3025:,
3022:2
3019:=
3016:n
3003:Z
2995:Z
2987:{
2983:=
2980:)
2976:Z
2968:Z
2964:(
2958:n
2955:2
2943:)
2937:2
2933:S
2924:2
2920:S
2915:(
2909:n
2899:H
2878:L
2874:K
2870:X
2866:n
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2846:L
2843:(
2838:n
2828:H
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2815:K
2812:(
2807:n
2797:H
2787:)
2784:L
2778:K
2775:(
2770:n
2760:H
2739:V
2735:U
2731:B
2727:A
2723:X
2719:B
2715:A
2711:L
2707:U
2703:B
2699:V
2695:K
2691:A
2687:L
2683:V
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2675:U
2660:L
2656:K
2648:X
2642:.
2640:X
2636:L
2632:K
2628:X
2589:n
2577:0
2570:,
2567:1
2564:=
2561:n
2547:2
2542:Z
2533:Z
2526:{
2521:=
2518:)
2513:2
2508:Z
2499:Z
2495:(
2489:n
2486:1
2474:)
2471:X
2468:(
2462:n
2452:H
2435:Z
2413:0
2407:)
2404:X
2401:(
2396:1
2386:H
2374:Z
2366:Z
2340:Z
2333:)
2330:X
2327:(
2322:2
2312:H
2302:0
2285:B
2283:∩
2281:A
2277:B
2273:A
2266:B
2262:A
2255:X
2251:X
2240:B
2236:A
2216:k
2212:n
2180:,
2177:k
2171:n
2159:0
2152:,
2149:k
2146:=
2143:n
2130:Z
2123:{
2118:=
2114:Z
2107:n
2104:k
2092:)
2087:k
2083:S
2079:(
2072:n
2062:H
2033:*
2010:0
2003:)
1998:1
1992:k
1988:S
1984:(
1977:1
1971:n
1961:H
1929:)
1924:k
1920:S
1916:(
1909:n
1899:H
1889:0
1862:B
1858:A
1850:k
1842:k
1838:k
1830:X
1826:B
1822:A
1818:S
1814:X
1809:k
1801:S
1797:X
1785:k
1763:X
1743:)
1740:X
1737:(
1732:1
1704:)
1701:X
1698:(
1693:1
1689:H
1661:.
1658:)
1649:j
1645:,
1636:i
1632:(
1621:)
1612:l
1599:k
1595:(
1564:)
1555:l
1542:k
1538:(
1529:/
1525:)
1522:)
1519:B
1516:(
1511:1
1507:H
1500:)
1497:A
1494:(
1489:1
1485:H
1481:(
1475:)
1472:X
1469:(
1464:1
1460:H
1432:B
1426:A
1388:)
1385:X
1382:(
1377:0
1367:H
1350:l
1337:k
1326:)
1323:B
1320:(
1315:0
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1295:)
1292:A
1289:(
1284:0
1274:H
1263:)
1254:j
1250:,
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1237:(
1227:)
1224:B
1218:A
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1200:H
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1131:n
1127:H
1123:φ
1119:φ
1115:x
1099:u
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1047:B
1045:∩
1043:A
1041:(
1037:n
1033:H
1029:u
1021:B
1019:∩
1017:A
1013:n
1009:v
1005:u
1001:x
997:v
993:u
989:v
985:u
981:x
977:B
973:A
969:v
965:u
961:n
953:x
949:n
945:X
943:(
940:n
938:H
928:.
926:B
922:A
918:v
914:u
910:x
852:X
846:B
843::
840:l
820:X
814:A
811::
808:k
805:,
802:B
796:B
790:A
787::
784:j
781:,
778:A
772:B
766:A
763::
760:i
731:)
728:X
725:(
720:0
716:H
702:l
689:k
678:)
675:B
672:(
667:0
663:H
656:)
653:A
650:(
645:0
641:H
595:)
592:B
586:A
583:(
578:1
572:n
568:H
545:)
542:X
539:(
534:n
530:H
517:l
504:k
494:)
491:B
488:(
483:n
479:H
472:)
469:A
466:(
461:n
457:H
449:)
435:j
421:i
412:(
402:)
399:B
393:A
390:(
385:n
381:H
358:)
355:X
352:(
347:1
344:+
341:n
337:H
309:B
307:∩
305:A
298:B
294:A
290:X
286:Z
278:B
274:A
270:X
262:B
258:A
254:X
246:B
242:A
234:X
20:)
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