31:
7715:
3970:
7498:
1379:
7736:
7704:
3627:
7773:
7746:
7726:
4960:
1162:
5393:
2751:
5907:
5508:
4312:
3965:{\displaystyle {\begin{aligned}\nabla \times {\vec {A}}&=\left(\partial _{y}A_{z}-\partial _{z}A_{y}\right){\hat {i}}+\left(\partial _{z}A_{x}-\partial _{x}A_{z}\right){\hat {j}}+\left(\partial _{x}A_{y}-\partial _{y}A_{x}\right){\hat {k}}\\&=F_{x}{\hat {i}}+F_{y}{\hat {j}}+F_{z}{\hat {k}}={\vec {F}}\end{aligned}}}
4679:
281:
1146:
2771:
Too much use of "we", "note". Also, this section is too technical for most readers of this article: it should be reduced to the definitions that are needed for understanding the statement (exactness of a sequence). The proof and the technical details do not belong to this article, but should appear
1374:{\displaystyle {\begin{aligned}0\rightarrow K_{1}\rightarrow {}&A_{1}\rightarrow K_{2}\rightarrow 0,\\0\rightarrow K_{2}\rightarrow {}&A_{2}\rightarrow K_{3}\rightarrow 0,\\&\ \,\vdots \\0\rightarrow K_{n-1}\rightarrow {}&A_{n-1}\rightarrow K_{n}\rightarrow 0,\\\end{aligned}}}
5289:
1859:
1581:
4496:
5686:
3311:
5767:
1735:
5423:
4087:
5256:. Since definitionally we have landed on a space of integrable functions, any such function can (at least formally) be integrated in order to produce a vector field which divergence is that function — so the image of the divergence is the entirety of
5073:
4955:{\displaystyle {\begin{aligned}-\partial _{x}\int F_{x}dz+\partial _{x}{\vec {C_{2}}}(x,y)-\partial _{y}\int F_{y}dz&=F_{z}\\-\int \left(\partial _{x}F_{x}+\partial _{y}F_{y}\right)dz+\partial _{x}{\vec {C_{2}}}(x,y)&=F_{z}\end{aligned}}}
3106:
6582:
6072:
118:
5759:
988:
7230:
If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a
3027:
6197:
5182:
2899:
6671:(This is true for a number of interesting categories, including any abelian category such as the abelian groups; but it is not true for all categories that allow exact sequences, and in particular is not true for the
3550:
810:
6746:
5388:{\displaystyle 0\to L^{2}\mathrel {\xrightarrow {\operatorname {grad} } } \mathbb {H} _{3}\mathrel {\xrightarrow {\operatorname {curl} } } \mathbb {H} _{3}\mathrel {\xrightarrow {\operatorname {div} } } L^{2}\to 0}
6666:
1789:
6454:
6360:
The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence
2735:
875:
672:
4017:
1520:
4323:
5581:
3413:
5912:
we see that if a function is an eigenfunction of the vector
Laplacian, its divergence must be an eigenfunction of the scalar Laplacian with the same eigenvalue. Then we can build our inverse function
5902:{\displaystyle {\begin{aligned}\nabla \cdot \nabla ^{2}{\vec {A}}&=\nabla \cdot \nabla \left(\nabla \cdot {\vec {A}}\right)\\&=\nabla ^{2}\left(\nabla \cdot {\vec {A}}\right)\\\end{aligned}}}
391:
5573:
3229:
2289:
7058:
5772:
4684:
4328:
3632:
1167:
2118:
1436:
3363:
5503:{\displaystyle 0\to L^{2}\mathrel {\xrightarrow {\operatorname {grad} } } \mathbb {H} _{3}\mathrel {\xrightarrow {\operatorname {curl} } } \operatorname {im} (\operatorname {curl} )\to 0.}
4557:
2619:
6987:
1683:
6005:
3591:
4307:{\displaystyle -\partial _{z}A_{y}{\hat {i}}+\partial _{z}A_{x}{\hat {j}}+\left(\partial _{x}A_{y}-\partial _{y}A_{x}\right){\hat {k}}=F_{x}{\hat {i}}+F_{y}{\hat {j}}+F_{z}{\hat {k}}}
2014:
2938:
2336:
1671:
966:
6922:
6843:
6105:
5969:
4968:
3218:
2967:
2471:
1643:
506:. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).
4671:
5940:
454:
6320:
For non-commutative groups, the splitting lemma does not apply, and one has only the equivalence between the two last conditions, with "the direct sum" replaced with "a
4075:
4046:
3620:
3442:
3177:
3148:
2533:
2433:
4317:
Then by simply integrating the first two components, and noting that the 'constant' of integration may still depend on any variable not integrated over, we find that
6876:
2368:
4604:
2186:
502:
To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with the
5281:
5254:
5216:
4635:
3046:
2846:
2505:
2402:
2148:
1487:
422:
321:
62:
276:{\displaystyle G_{0}\;{\xrightarrow {\ f_{1}\ }}\;G_{1}\;{\xrightarrow {\ f_{2}\ }}\;G_{2}\;{\xrightarrow {\ f_{3}\ }}\;\cdots \;{\xrightarrow {\ f_{n}\ }}\;G_{n}}
6472:
6017:
2561:
6797:
2040:
7776:
1456:
1141:{\displaystyle A_{0}\;\xrightarrow {\ f_{1}\ } \;A_{1}\;\xrightarrow {\ f_{2}\ } \;A_{2}\;\xrightarrow {\ f_{3}\ } \;\cdots \;\xrightarrow {\ f_{n}\ } \;A_{n},}
5694:
5517:, and since the Hilbert space of square-integrable functions can be spanned by the eigenfunctions of the Laplacian, we already see that some inverse mapping
5229:
Having thus proved that the image of the curl is precisely the kernel of the divergence, this morphism in turn takes us back to the space we started from
5410:
can be written as a curl of another field. (Reasoning in this direction thus makes use of the fact that 3-dimensional space is topologically trivial.)
2972:
6125:
5081:
6766:
2851:
3451:
7131:
3191:
is then precisely the kernel of the curl, and so we can then take the curl to be our next morphism, taking us again to a (different) subset of
743:
6698:
1854:{\displaystyle 0\to \mathbf {Z} \mathrel {\overset {2\times }{\longrightarrow }} \mathbf {Z} \longrightarrow \mathbf {Z} /2\mathbf {Z} \to 0}
6594:
3121:
is a subset of the kernel of the curl. To prove that they are in fact the same set, prove the converse: that if the curl of a vector field
6367:
1576:{\displaystyle \mathbf {Z} \mathrel {\overset {2\times }{\,\hookrightarrow }} \mathbf {Z} \twoheadrightarrow \mathbf {Z} /2\mathbf {Z} }
4491:{\displaystyle {\begin{aligned}A_{x}&=\int F_{y}dz+{\vec {C_{1}}}(x,y)\\A_{y}&=-\int F_{x}dz+{\vec {C_{2}}}(x,y)\end{aligned}}}
2634:
1748:
and although it looks like an identity function, it is not onto (that is, not an epimorphism) because the odd numbers don't belong to 2
821:
7410:
619:
5681:{\displaystyle \nabla ^{2}{\vec {A}}=\nabla \left(\nabla \cdot {\vec {A}}\right)+\nabla \times \left(\nabla \times {\vec {A}}\right)}
3978:
1772:
used in the previous sequence. This latter sequence does differ in the concrete nature of its first object from the previous one as 2
7764:
7759:
7379:
7349:
6588:
Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:
2794:
1957:
to be mapped by inclusion (that is, by a monomorphism) as a proper subgroup of itself. Instead the sequence that emerges from the
5971:
into the vector-Laplacian eigenbasis, scaling each by the inverse of their eigenvalue, and taking the divergence; the action of
3368:
7754:
7087:
Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.
5188:
3306:{\displaystyle \operatorname {div} \left(\operatorname {curl} {\vec {v}}\right)\equiv \nabla \cdot \nabla \times {\vec {v}}=0,}
2776:
326:
7227:
Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.
5691:
Since we are trying to construct an identity mapping by composing some function with the gradient, we know that in our case
5520:
2232:
7656:
6992:
7802:
7797:
6353:
gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the
2055:
7248:
2766:
983:
A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence
1389:
4610:. This permits us to eliminate either of the terms in favor of the other, without spoiling our earlier work that set
3333:
1730:{\displaystyle 2\mathbf {Z} \mathrel {\,\hookrightarrow } \mathbf {Z} \twoheadrightarrow \mathbf {Z} /2\mathbf {Z} }
7664:
4504:
2574:
30:
7095:
In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about
6927:
5403:
1958:
7735:
7463:
7135:
5974:
3555:
483:
5575:
must exist. To explicitly construct such an inverse, we can start from the definition of the vector
Laplacian
1677:
of the monomorphism is the kernel of the epimorphism. Essentially "the same" sequence can also be written as
1967:
518:. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (from
7749:
5414:
5407:
2908:
2568:
These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence
7684:
7679:
7605:
7482:
7470:
7443:
7403:
7212:
2303:
479:
5068:{\displaystyle \operatorname {div} {\vec {F}}=\partial _{x}F_{x}+\partial _{y}F_{y}+\partial _{z}F_{z}=0}
1656:
7526:
7453:
2808:
7714:
1783:
The first sequence may also be written without using special symbols for monomorphism and epimorphism:
936:
6881:
6081:
5945:
3194:
3029:— specifically, the set of such functions that represent conservative vector fields. (The generalized
2943:
2438:
1628:
7674:
7626:
7600:
7448:
7251:
is another example. Long exact sequences induced by short exact sequences are also characteristic of
6802:
882:
89:
4640:
2761:
7725:
7521:
7371:
6339:
2217:
475:
463:
97:
81:
35:
5915:
427:
7719:
7669:
7590:
7580:
7458:
7438:
7240:
6672:
6321:
3184:
3037:
1899:
is {0}, and the kernel of multiplication by 2 is also {0}, so the sequence is exact at the first
1888:
491:
287:
85:
7689:
6760:).) Then we obtain a commutative diagram in which all the diagonals are short exact sequences:
4051:
4022:
3596:
3418:
3180:
3153:
3124:
3030:
5402:
space, a curl-free vector field (a field in the kernel of the curl) can always be written as a
2512:
2412:
7707:
7573:
7531:
7396:
7375:
7345:
7341:
7244:
7235:(that is, an exact sequence indexed by the natural numbers) on homology by application of the
7103:
6753:
3321:
3101:{\displaystyle \operatorname {curl} (\operatorname {grad} f)\equiv \nabla \times (\nabla f)=0}
2812:
6848:
2347:
1949:
The first and third sequences are somewhat of a special case owing to the infinite nature of
7487:
7433:
7363:
7333:
6765:
6577:{\displaystyle C_{k}\cong \ker(A_{k}\to A_{k+1})\cong \operatorname {im} (A_{k-1}\to A_{k})}
6354:
6067:{\displaystyle \mathbb {H} _{3}\cong L^{2}\oplus \operatorname {im} (\operatorname {curl} )}
5399:
4574:
2161:
597:
101:
93:
5259:
5232:
5194:
4613:
2824:
2478:
2375:
2126:
1465:
400:
299:
40:
7546:
7541:
7287:
7252:
7220:
7069:
6116:
6008:
2625:
722:
7364:
6107:
can be broken into the sum of a gradient and a curl — which is what we set out to prove.
2540:
7311:
2022:
1673:
indicates an epimorphism (the map mod 2). This is an exact sequence because the image 2
7636:
7568:
7359:
6775:
5754:{\displaystyle \nabla \times {\vec {A}}=\operatorname {curl} \left({\vec {A}}\right)=0}
2189:
1441:
6772:
The only portion of this diagram that depends on the cokernel condition is the object
7791:
7646:
7556:
7536:
7334:
7329:
7258:
7236:
7180:
4048:
without changing the curl. We can use this gauge freedom to set any one component of
2819:
2043:
911:
545:. Therefore the sequence is exact if and only if the image of the leftmost map (from
503:
65:
7739:
7631:
7551:
7497:
5413:
This short exact sequence also permits a much shorter proof of the validity of the
2969:, the space of vector valued, still square-integrable functions on the same domain
2151:
1954:
1877:
527:
467:
17:
7060:
is ensured. Again taking the example of the category of groups, the fact that im(
3022:{\displaystyle \left\lbrace f:\mathbb {R} ^{3}\to \mathbb {R} ^{3}\right\rbrace }
7729:
7641:
6335:
6192:{\displaystyle 0\to A\;\xrightarrow {\ f\ } \;B\;\xrightarrow {\ g\ } \;C\to 0,}
5177:{\displaystyle \int \partial _{z}F_{z}dz+\partial _{x}{\vec {C_{2}}}(x,y)=F_{z}}
585:
558:
7585:
7516:
7475:
7118:?" In the category of groups, this is equivalent to the question, what groups
6350:
6343:
6291:
3317:
3179:
is the gradient of some scalar function. This follows almost immediately from
2407:
919:
581:
471:
456:, i.e., if the image of each homomorphism is equal to the kernel of the next.
2894:{\displaystyle \left\lbrace f:\mathbb {R} ^{3}\to \mathbb {R} \right\rbrace }
7610:
7096:
5514:
5417:
that does not rely on brute-force vector calculus. Consider the subsequence
698:
77:
459:
The sequence of groups and homomorphisms may be either finite or infinite.
3545:{\displaystyle {\vec {F}}=F_{x}{\hat {i}}+F_{y}{\hat {j}}+F_{z}{\hat {k}}}
27:
Sequence of homomorphisms such that each kernel equals the preceding image
7595:
7563:
7512:
7419:
3188:
3118:
2902:
805:{\displaystyle C\cong B/\operatorname {im} (f)=B/\operatorname {ker} (g)}
487:
7114:
of a short exact sequence, what possibilities exist for the middle term
7262:
6741:{\displaystyle H/{\left\langle \operatorname {im} f\right\rangle }^{H}}
5222:, a solution to the above system of equations is guaranteed to exist.
478:. More generally, the notion of an exact sequence makes sense in any
7130:
as the corresponding factor group? This problem is important in the
6661:{\displaystyle C_{k}\cong \operatorname {coker} (A_{k-2}\to A_{k-1})}
6164:
6143:
5472:
5448:
5362:
5338:
5314:
1105:
1077:
1042:
1007:
851:
838:
649:
636:
242:
212:
175:
138:
6449:{\displaystyle A_{1}\to A_{2}\to A_{3}\to A_{4}\to A_{5}\to A_{6}}
2049:
As a more concrete example of an exact sequence on finite groups:
980:, to distinguish from the special case of a short exact sequence.
29:
2848:
of scalar-valued square-integrable functions on three dimensions
2730:{\displaystyle 0\to R/(I\cap J)\to R/I\oplus R/J\to R/(I+J)\to 0}
870:{\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0\,}
1756:
through this monomorphism is however exactly the same subset of
667:{\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0.}
7392:
6342:
with two exact rows gives rise to a longer exact sequence. The
4012:{\displaystyle \operatorname {curl} (\operatorname {grad} f)=0}
7219:
Exact sequences are precisely those chain complexes which are
3320:. The converse is somewhat involved (for the general case see
2744:
6357:
is a special case thereof applying to short exact sequences.
4077:
to zero without changing its curl; choosing arbitrarily the
7388:
7366:
Commutative
Algebra: with a View Toward Algebraic Geometry
3408:{\displaystyle {\vec {F}}=\operatorname {curl} {\vec {A}}}
3316:
so the image of the curl is a subset of the kernel of the
7072:, which coincides with its conjugate closure; thus coker(
6327:
In both cases, one says that such a short exact sequence
5406:(and thus is in the image of the gradient). Similarly, a
4673:
and applying the last component as a constraint, we have
677:
As established above, for any such short exact sequence,
526:) has kernel {0}; that is, if and only if that map is a
386:{\displaystyle \operatorname {im} (f_{i})=\ker(f_{i+1})}
6078:
or equivalently, any square-integrable vector field on
3448:
We shall proceed by construction: given a vector field
6805:
6778:
5568:{\displaystyle \nabla ^{-1}:\mathbb {H} _{3}\to L^{2}}
5398:
Equivalently, we could have reasoned in reverse: in a
3326:
613:
Short exact sequences are exact sequences of the form
7265:
that transform exact sequences into exact sequences.
6995:
6930:
6884:
6851:
6701:
6597:
6475:
6370:
6128:
6084:
6020:
5977:
5948:
5918:
5770:
5697:
5584:
5523:
5426:
5292:
5262:
5235:
5197:
5084:
4971:
4682:
4643:
4616:
4577:
4507:
4326:
4090:
4054:
4025:
3981:
3630:
3599:
3558:
3454:
3421:
3371:
3336:
3232:
3197:
3156:
3127:
3049:
2975:
2946:
2911:
2854:
2827:
2637:
2577:
2543:
2515:
2481:
2441:
2415:
2378:
2350:
2306:
2284:{\displaystyle 0\to I\cap J\to I\oplus J\to I+J\to 0}
2235:
2164:
2129:
2058:
2025:
1970:
1792:
1686:
1659:
1631:
1523:
1468:
1444:
1392:
1165:
991:
939:
824:
746:
622:
430:
403:
329:
302:
121:
43:
4571:, then we can add another gradient of some function
4019:, we can add the gradient of any scalar function to
466:. For example, one could have an exact sequence of
7655:
7619:
7505:
7426:
7215:of this chain complex is trivial. More succinctly:
7053:{\displaystyle 0\to C_{k}\to A_{k}\to C_{k+1}\to 0}
1514:Consider the following sequence of abelian groups:
580:is both a monomorphism and epimorphism (that is, a
7141:Notice that in an exact sequence, the composition
7052:
6981:
6916:
6870:
6837:
6791:
6740:
6660:
6576:
6448:
6191:
6099:
6066:
5999:
5963:
5934:
5901:
5753:
5680:
5567:
5502:
5387:
5275:
5248:
5210:
5176:
5067:
4954:
4665:
4629:
4598:
4551:
4490:
4306:
4069:
4040:
4011:
3964:
3614:
3585:
3544:
3436:
3407:
3357:
3305:
3212:
3171:
3142:
3117:. However, this only proves that the image of the
3100:
3021:
2961:
2932:
2893:
2840:
2729:
2613:
2555:
2527:
2499:
2465:
2427:
2396:
2362:
2330:
2283:
2180:
2142:
2113:{\displaystyle 1\to C_{n}\to D_{2n}\to C_{2}\to 1}
2112:
2034:
2008:
1853:
1729:
1665:
1637:
1575:
1481:
1450:
1430:
1373:
1140:
960:
869:
804:
666:
448:
416:
385:
315:
275:
56:
7106:is essentially the question "Given the end terms
5191:requires that the first term above be precisely
1431:{\displaystyle K_{i}=\operatorname {im} (f_{i})}
5761:. Then if we take the divergence of both sides
1910:, and the kernel of reducing modulo 2 is also 2
1864:Here 0 denotes the trivial group, the map from
1157:2, we can split it up into the short sequences
976:A general exact sequence is sometimes called a
3358:{\displaystyle \operatorname {div} {\vec {F}}}
1780:even though the two are isomorphic as groups.
7404:
4552:{\displaystyle {\vec {C_{1}}},{\vec {C_{2}}}}
2614:{\displaystyle 0\to R\to R\oplus R\to R\to 0}
685:is an epimorphism. Furthermore, the image of
8:
5513:Since the divergence of the gradient is the
1653:is a monomorphism, and the two-headed arrow
1602:. The second homomorphism maps each element
6982:{\displaystyle A_{k-1}\to A_{k}\to A_{k+1}}
2741:Grad, curl and div in differential geometry
1937:, so the sequence is exact at the position
462:A similar definition can be made for other
112:In the context of group theory, a sequence
7772:
7745:
7411:
7397:
7389:
6176:
6159:
6155:
6138:
6007:is thus clearly the identity. Thus by the
4501:Note that since the two integration terms
1124:
1100:
1096:
1072:
1061:
1037:
1026:
1002:
262:
236:
232:
206:
195:
169:
158:
132:
7032:
7019:
7006:
6994:
6967:
6954:
6935:
6929:
6902:
6889:
6883:
6856:
6850:
6823:
6810:
6804:
6783:
6777:
6732:
6711:
6705:
6700:
6643:
6624:
6602:
6596:
6565:
6546:
6515:
6502:
6480:
6474:
6440:
6427:
6414:
6401:
6388:
6375:
6369:
6127:
6119:states that, for a short exact sequence
6091:
6087:
6086:
6083:
6040:
6027:
6023:
6022:
6019:
6000:{\displaystyle \nabla ^{-1}\circ \nabla }
5982:
5976:
5955:
5951:
5950:
5947:
5923:
5917:
5879:
5878:
5861:
5831:
5830:
5792:
5791:
5785:
5771:
5769:
5730:
5729:
5705:
5704:
5696:
5662:
5661:
5625:
5624:
5596:
5595:
5589:
5583:
5559:
5546:
5542:
5541:
5528:
5522:
5467:
5461:
5457:
5456:
5443:
5437:
5425:
5373:
5357:
5351:
5347:
5346:
5333:
5327:
5323:
5322:
5309:
5303:
5291:
5267:
5261:
5240:
5234:
5202:
5196:
5168:
5134:
5128:
5127:
5121:
5102:
5092:
5083:
5053:
5043:
5030:
5020:
5007:
4997:
4979:
4978:
4970:
4942:
4904:
4898:
4897:
4891:
4867:
4857:
4844:
4834:
4809:
4786:
4773:
4739:
4733:
4732:
4726:
4707:
4694:
4683:
4681:
4651:
4645:
4644:
4642:
4621:
4615:
4576:
4537:
4531:
4530:
4515:
4509:
4508:
4506:
4457:
4451:
4450:
4435:
4412:
4377:
4371:
4370:
4355:
4335:
4327:
4325:
4293:
4292:
4286:
4268:
4267:
4261:
4243:
4242:
4236:
4218:
4217:
4206:
4196:
4183:
4173:
4150:
4149:
4143:
4133:
4115:
4114:
4108:
4098:
4089:
4056:
4055:
4053:
4027:
4026:
4024:
3980:
3947:
3946:
3932:
3931:
3925:
3907:
3906:
3900:
3882:
3881:
3875:
3850:
3849:
3838:
3828:
3815:
3805:
3782:
3781:
3770:
3760:
3747:
3737:
3714:
3713:
3702:
3692:
3679:
3669:
3642:
3641:
3631:
3629:
3601:
3600:
3598:
3586:{\displaystyle \nabla \cdot {\vec {F}}=0}
3566:
3565:
3557:
3531:
3530:
3524:
3506:
3505:
3499:
3481:
3480:
3474:
3456:
3455:
3453:
3423:
3422:
3420:
3394:
3393:
3373:
3372:
3370:
3344:
3343:
3335:
3283:
3282:
3251:
3250:
3231:
3204:
3200:
3199:
3196:
3158:
3157:
3155:
3129:
3128:
3126:
3048:
3008:
3004:
3003:
2993:
2989:
2988:
2974:
2953:
2949:
2948:
2945:
2924:
2920:
2919:
2910:
2882:
2881:
2872:
2868:
2867:
2853:
2832:
2826:
2795:Learn how and when to remove this message
2701:
2687:
2673:
2647:
2636:
2576:
2542:
2514:
2480:
2440:
2414:
2377:
2349:
2305:
2234:
2169:
2163:
2134:
2128:
2098:
2082:
2069:
2057:
2024:
1992:
1969:
1929:, and the kernel of the zero map is also
1914:, so the sequence is exact at the second
1872:is multiplication by 2, and the map from
1840:
1832:
1827:
1819:
1804:
1799:
1791:
1722:
1714:
1709:
1701:
1696:
1695:
1690:
1685:
1658:
1630:
1586:The first homomorphism maps each element
1568:
1560:
1555:
1547:
1532:
1529:
1524:
1522:
1473:
1467:
1443:
1419:
1397:
1391:
1352:
1333:
1325:
1310:
1292:
1269:
1256:
1248:
1239:
1210:
1197:
1189:
1180:
1166:
1164:
1129:
1113:
1085:
1066:
1050:
1031:
1015:
996:
990:
938:
866:
823:
782:
756:
745:
621:
572:→ 0 is exact if and only if the map from
557:; that is, if and only if that map is an
429:
408:
402:
368:
343:
328:
307:
301:
267:
250:
237:
220:
207:
200:
183:
170:
163:
146:
133:
126:
120:
48:
42:
7064:) is the kernel of some homomorphism on
6199:the following conditions are equivalent.
4081:-component, we thus require simply that
2300:-modules, where the module homomorphism
1497:is a long exact sequence if and only if
541:→ 0. The kernel of the rightmost map is
7279:
6459:which implies that there exist objects
3975:First, note that since as proved above
2772:in an article of differential geometry.
2042:as these groups are not supposed to be
2009:{\displaystyle 1\to N\to G\to G/N\to 1}
721:as the corresponding factor object (or
2811:, especially relevant for work on the
2933:{\displaystyle f\in \mathbb {H} _{1}}
1906:the image of multiplication by 2 is 2
1891:2. This is indeed an exact sequence:
92:, and, more generally, objects of an
34:Illustration of an exact sequence of
7:
7370:. Springer-Verlag New York. p.
7288:"exact sequence in nLab, Remark 2.3"
5283:, and we can complete our sequence:
2807:Another example can be derived from
2331:{\displaystyle I\cap J\to I\oplus J}
5942:simply by breaking any function in
3040:of all such fields is zero — since
2019:(here the trivial group is denoted
1740:In this case the monomorphism is 2
1666:{\displaystyle \twoheadrightarrow }
1489:'s (regardless of the exactness of
5994:
5979:
5920:
5872:
5858:
5824:
5816:
5810:
5782:
5775:
5698:
5655:
5644:
5618:
5610:
5586:
5525:
5118:
5089:
5040:
5017:
4994:
4888:
4854:
4831:
4770:
4723:
4691:
4193:
4170:
4130:
4095:
3825:
3802:
3757:
3734:
3689:
3666:
3635:
3559:
3276:
3270:
3083:
3074:
1921:the image of reducing modulo 2 is
25:
1458:. By construction, the sequences
961:{\displaystyle B\cong A\oplus C.}
7771:
7744:
7734:
7724:
7713:
7703:
7702:
7496:
6989:is exact, then the exactness of
6917:{\displaystyle A_{k}\to A_{k+1}}
6838:{\textstyle A_{6}\to C_{7}\to 0}
6799:and the final pair of morphisms
6764:
6100:{\displaystyle \mathbb {R} ^{3}}
5964:{\displaystyle \mathbb {H} _{3}}
3213:{\displaystyle \mathbb {H} _{3}}
2962:{\displaystyle \mathbb {H} _{3}}
2749:
2466:{\displaystyle I\oplus J\to I+J}
2196:, which is a non-abelian group.
1841:
1828:
1820:
1800:
1723:
1710:
1702:
1691:
1638:{\displaystyle \hookrightarrow }
1614:in the quotient group; that is,
1569:
1556:
1548:
1525:
7179:, so every exact sequence is a
7091:Applications of exact sequences
5189:fundamental theorem of calculus
4637:to zero. Choosing to eliminate
2200:Intersection and sum of modules
1645:indicates that the map 2× from
1501:are all short exact sequences.
910:. It follows that if these are
886:if there exists a homomorphism
7044:
7025:
7012:
6999:
6960:
6947:
6895:
6845:. If there exists any object
6829:
6816:
6655:
6636:
6617:
6571:
6558:
6539:
6527:
6508:
6495:
6433:
6420:
6407:
6394:
6381:
6180:
6132:
6061:
6055:
5884:
5836:
5797:
5735:
5710:
5667:
5630:
5601:
5552:
5494:
5491:
5485:
5430:
5379:
5296:
5158:
5146:
5140:
4984:
4928:
4916:
4910:
4763:
4751:
4745:
4666:{\displaystyle {\vec {C_{1}}}}
4657:
4593:
4581:
4543:
4521:
4481:
4469:
4463:
4401:
4389:
4383:
4298:
4273:
4248:
4223:
4155:
4120:
4061:
4032:
4000:
3988:
3952:
3937:
3912:
3887:
3855:
3787:
3719:
3647:
3606:
3571:
3536:
3511:
3486:
3461:
3428:
3399:
3378:
3349:
3288:
3256:
3163:
3134:
3089:
3080:
3068:
3056:
3033:has preserved integrability.)
2999:
2878:
2721:
2718:
2706:
2695:
2667:
2664:
2652:
2641:
2628:yields another exact sequence
2605:
2599:
2587:
2581:
2494:
2482:
2451:
2391:
2379:
2316:
2275:
2263:
2251:
2239:
2104:
2091:
2075:
2062:
2000:
1986:
1980:
1974:
1887:is given by reducing integers
1845:
1824:
1806:
1796:
1706:
1697:
1660:
1632:
1552:
1533:
1425:
1412:
1358:
1345:
1322:
1303:
1275:
1262:
1245:
1232:
1216:
1203:
1186:
1173:
860:
828:
799:
793:
773:
767:
658:
626:
380:
361:
349:
336:
80:between objects (for example,
1:
7076:) is isomorphic to the image
5404:gradient of a scalar function
4606:that also does not depend on
7340:. Berlin: Springer. p.
5935:{\displaystyle \nabla ^{-1}}
693:. It is helpful to think of
564:Therefore, the sequence 0 →
494:, where it is widely used.
449:{\displaystyle 1\leq i<n}
2769:. The specific problem is:
1953:. It is not possible for a
533:Consider the dual sequence
530:(injective, or one-to-one).
100:of one morphism equals the
7819:
7665:Banach fixed-point theorem
6466:in the category such that
4070:{\displaystyle {\vec {A}}}
4041:{\displaystyle {\vec {A}}}
3615:{\displaystyle {\vec {A}}}
3437:{\displaystyle {\vec {A}}}
3172:{\displaystyle {\vec {F}}}
3143:{\displaystyle {\vec {F}}}
2765:to meet Knowledge (XXG)'s
898:such that the composition
689:is equal to the kernel of
510:Consider the sequence 0 →
7698:
7494:
7126:as a normal subgroup and
2528:{\displaystyle I\oplus J}
2428:{\displaystyle I\oplus J}
1959:first isomorphism theorem
1895:the image of the map 0 →
815:The short exact sequence
393:. The sequence is called
7136:Outer automorphism group
7132:classification of groups
7084:) of the next morphism.
6270:There exists a morphism
6237:There exists a morphism
6204:There exists a morphism
3223:Similarly, we note that
2940:moves us to a subset of
2294:is an exact sequence of
737:inducing an isomorphism
490:, and more specially in
7249:Mayer–Vietoris sequence
7192:-images of elements of
6871:{\displaystyle A_{k+1}}
5415:Helmholtz decomposition
5408:solenoidal vector field
2435:, and the homomorphism
2363:{\displaystyle I\cap J}
1776:is not the same set as
1590:in the set of integers
906:is the identity map on
397:if it is exact at each
7720:Mathematics portal
7620:Metrics and properties
7606:Second-countable space
7312:"Divergenceless field"
7054:
6983:
6918:
6872:
6839:
6793:
6742:
6662:
6578:
6450:
6193:
6101:
6068:
6001:
5965:
5936:
5903:
5755:
5682:
5569:
5504:
5389:
5277:
5250:
5212:
5178:
5069:
4956:
4667:
4631:
4600:
4599:{\displaystyle f(x,y)}
4553:
4492:
4308:
4071:
4042:
4013:
3966:
3616:
3587:
3546:
3438:
3409:
3359:
3307:
3214:
3173:
3144:
3102:
3023:
2963:
2934:
2895:
2842:
2731:
2615:
2557:
2529:
2501:
2467:
2429:
2398:
2364:
2332:
2285:
2182:
2181:{\displaystyle D_{2n}}
2144:
2114:
2036:
2010:
1855:
1731:
1667:
1639:
1625:. Here the hook arrow
1577:
1483:
1452:
1432:
1375:
1142:
962:
871:
806:
681:is a monomorphism and
668:
596:(this always holds in
561:(surjective, or onto).
450:
418:
387:
317:
277:
69:
58:
7068:implies that it is a
7055:
6984:
6919:
6873:
6840:
6794:
6743:
6663:
6579:
6451:
6194:
6102:
6069:
6002:
5966:
5937:
5904:
5756:
5683:
5570:
5505:
5390:
5278:
5276:{\displaystyle L^{2}}
5251:
5249:{\displaystyle L^{2}}
5213:
5211:{\displaystyle F_{z}}
5179:
5070:
4957:
4668:
4632:
4630:{\displaystyle A_{z}}
4601:
4554:
4493:
4309:
4072:
4043:
4014:
3967:
3617:
3593:, we produce a field
3588:
3547:
3439:
3410:
3360:
3308:
3215:
3174:
3145:
3103:
3024:
2964:
2935:
2896:
2843:
2841:{\displaystyle L^{2}}
2809:differential geometry
2732:
2616:
2558:
2530:
2502:
2500:{\displaystyle (x,y)}
2468:
2430:
2399:
2397:{\displaystyle (x,x)}
2365:
2333:
2286:
2183:
2145:
2143:{\displaystyle C_{n}}
2115:
2037:
2011:
1856:
1732:
1668:
1640:
1578:
1484:
1482:{\displaystyle K_{i}}
1453:
1433:
1376:
1143:
963:
918:is isomorphic to the
872:
807:
669:
584:), and so usually an
451:
419:
417:{\displaystyle G_{i}}
388:
318:
316:{\displaystyle G_{i}}
278:
59:
57:{\displaystyle G_{i}}
33:
7675:Invariance of domain
7627:Euler characteristic
7601:Bundle (mathematics)
7330:Spanier, Edwin Henry
7183:. Furthermore, only
7099:and factor objects.
6993:
6928:
6882:
6849:
6803:
6776:
6699:
6595:
6473:
6368:
6126:
6082:
6018:
5975:
5946:
5916:
5768:
5695:
5582:
5521:
5424:
5290:
5260:
5233:
5195:
5082:
4969:
4680:
4641:
4614:
4575:
4559:both depend only on
4505:
4324:
4088:
4052:
4023:
3979:
3628:
3597:
3556:
3452:
3419:
3369:
3334:
3230:
3195:
3187:.) The image of the
3154:
3125:
3047:
2973:
2944:
2909:
2852:
2825:
2777:improve this section
2635:
2575:
2541:
2513:
2479:
2439:
2413:
2376:
2348:
2304:
2233:
2162:
2127:
2056:
2023:
1968:
1790:
1684:
1657:
1629:
1521:
1466:
1442:
1390:
1163:
989:
937:
822:
744:
620:
609:Short exact sequence
476:module homomorphisms
474:, or of modules and
464:algebraic structures
428:
401:
327:
300:
119:
41:
7803:Additive categories
7798:Homological algebra
7685:Tychonoff's theorem
7680:Poincaré conjecture
7434:General (point-set)
7314:. December 6, 2009.
7233:long exact sequence
7201:are mapped to 0 by
6346:is a special case.
6340:commutative diagram
6261:is the identity on
6228:is the identity on
6174:
6153:
5476:
5452:
5366:
5342:
5318:
5218:plus a constant in
2556:{\displaystyle x-y}
1510:Integers modulo two
1122:
1094:
1059:
1024:
978:long exact sequence
972:Long exact sequence
855:
842:
653:
640:
288:group homomorphisms
259:
229:
192:
155:
18:Long exact sequence
7670:De Rham cohomology
7591:Polyhedral complex
7581:Simplicial complex
7336:Algebraic Topology
7241:algebraic topology
7050:
6979:
6914:
6868:
6835:
6792:{\textstyle C_{7}}
6789:
6748:, the quotient of
6738:
6673:category of groups
6658:
6574:
6446:
6322:semidirect product
6189:
6097:
6064:
5997:
5961:
5932:
5899:
5897:
5751:
5678:
5565:
5500:
5385:
5273:
5246:
5208:
5174:
5065:
4952:
4950:
4663:
4627:
4596:
4549:
4488:
4486:
4304:
4067:
4038:
4009:
3962:
3960:
3612:
3583:
3542:
3434:
3405:
3355:
3303:
3210:
3185:conservative force
3183:(see the proof at
3169:
3140:
3098:
3019:
2959:
2930:
2891:
2838:
2727:
2611:
2553:
2525:
2497:
2473:maps each element
2463:
2425:
2394:
2360:
2338:maps each element
2328:
2281:
2178:
2140:
2110:
2035:{\displaystyle 1,}
2032:
2006:
1851:
1727:
1663:
1635:
1573:
1479:
1448:
1428:
1371:
1369:
1138:
958:
867:
802:
664:
492:abelian categories
446:
414:
383:
313:
273:
70:
54:
7785:
7784:
7574:fundamental group
7245:relative homology
7239:. It comes up in
7104:extension problem
6754:conjugate closure
6675:, in which coker(
6175:
6173:
6167:
6154:
6152:
6146:
5887:
5839:
5800:
5738:
5713:
5670:
5633:
5604:
5477:
5453:
5367:
5343:
5319:
5227:
5226:
5143:
4987:
4913:
4748:
4660:
4546:
4524:
4466:
4386:
4301:
4276:
4251:
4226:
4158:
4123:
4064:
4035:
3955:
3940:
3915:
3890:
3858:
3790:
3722:
3650:
3609:
3574:
3539:
3514:
3489:
3464:
3431:
3402:
3381:
3352:
3291:
3259:
3166:
3137:
2813:Maxwell equations
2805:
2804:
2797:
2767:quality standards
2758:This section may
1817:
1545:
1462:are exact at the
1451:{\displaystyle i}
1291:
1123:
1121:
1108:
1095:
1093:
1080:
1060:
1058:
1045:
1025:
1023:
1010:
856:
843:
654:
641:
260:
258:
245:
230:
228:
215:
193:
191:
178:
156:
154:
141:
76:is a sequence of
16:(Redirected from
7810:
7775:
7774:
7748:
7747:
7738:
7728:
7718:
7717:
7706:
7705:
7500:
7413:
7406:
7399:
7390:
7385:
7369:
7355:
7339:
7316:
7315:
7308:
7302:
7301:
7299:
7298:
7284:
7253:derived functors
7243:in the study of
7059:
7057:
7056:
7051:
7043:
7042:
7024:
7023:
7011:
7010:
6988:
6986:
6985:
6980:
6978:
6977:
6959:
6958:
6946:
6945:
6923:
6921:
6920:
6915:
6913:
6912:
6894:
6893:
6877:
6875:
6874:
6869:
6867:
6866:
6844:
6842:
6841:
6836:
6828:
6827:
6815:
6814:
6798:
6796:
6795:
6790:
6788:
6787:
6768:
6747:
6745:
6744:
6739:
6737:
6736:
6731:
6730:
6726:
6709:
6667:
6665:
6664:
6659:
6654:
6653:
6635:
6634:
6607:
6606:
6583:
6581:
6580:
6575:
6570:
6569:
6557:
6556:
6526:
6525:
6507:
6506:
6485:
6484:
6455:
6453:
6452:
6447:
6445:
6444:
6432:
6431:
6419:
6418:
6406:
6405:
6393:
6392:
6380:
6379:
6355:short five lemma
6315:
6304:
6289:
6283:
6266:
6260:
6250:
6233:
6227:
6217:
6198:
6196:
6195:
6190:
6171:
6165:
6160:
6150:
6144:
6139:
6106:
6104:
6103:
6098:
6096:
6095:
6090:
6073:
6071:
6070:
6065:
6045:
6044:
6032:
6031:
6026:
6006:
6004:
6003:
5998:
5990:
5989:
5970:
5968:
5967:
5962:
5960:
5959:
5954:
5941:
5939:
5938:
5933:
5931:
5930:
5908:
5906:
5905:
5900:
5898:
5894:
5890:
5889:
5888:
5880:
5866:
5865:
5850:
5846:
5842:
5841:
5840:
5832:
5802:
5801:
5793:
5790:
5789:
5760:
5758:
5757:
5752:
5744:
5740:
5739:
5731:
5715:
5714:
5706:
5687:
5685:
5684:
5679:
5677:
5673:
5672:
5671:
5663:
5640:
5636:
5635:
5634:
5626:
5606:
5605:
5597:
5594:
5593:
5574:
5572:
5571:
5566:
5564:
5563:
5551:
5550:
5545:
5536:
5535:
5509:
5507:
5506:
5501:
5478:
5468:
5466:
5465:
5460:
5454:
5444:
5442:
5441:
5400:simply connected
5394:
5392:
5391:
5386:
5378:
5377:
5368:
5358:
5356:
5355:
5350:
5344:
5334:
5332:
5331:
5326:
5320:
5310:
5308:
5307:
5282:
5280:
5279:
5274:
5272:
5271:
5255:
5253:
5252:
5247:
5245:
5244:
5217:
5215:
5214:
5209:
5207:
5206:
5183:
5181:
5180:
5175:
5173:
5172:
5145:
5144:
5139:
5138:
5129:
5126:
5125:
5107:
5106:
5097:
5096:
5074:
5072:
5071:
5066:
5058:
5057:
5048:
5047:
5035:
5034:
5025:
5024:
5012:
5011:
5002:
5001:
4989:
4988:
4980:
4961:
4959:
4958:
4953:
4951:
4947:
4946:
4915:
4914:
4909:
4908:
4899:
4896:
4895:
4877:
4873:
4872:
4871:
4862:
4861:
4849:
4848:
4839:
4838:
4814:
4813:
4791:
4790:
4778:
4777:
4750:
4749:
4744:
4743:
4734:
4731:
4730:
4712:
4711:
4699:
4698:
4672:
4670:
4669:
4664:
4662:
4661:
4656:
4655:
4646:
4636:
4634:
4633:
4628:
4626:
4625:
4605:
4603:
4602:
4597:
4558:
4556:
4555:
4550:
4548:
4547:
4542:
4541:
4532:
4526:
4525:
4520:
4519:
4510:
4497:
4495:
4494:
4489:
4487:
4468:
4467:
4462:
4461:
4452:
4440:
4439:
4417:
4416:
4388:
4387:
4382:
4381:
4372:
4360:
4359:
4340:
4339:
4313:
4311:
4310:
4305:
4303:
4302:
4294:
4291:
4290:
4278:
4277:
4269:
4266:
4265:
4253:
4252:
4244:
4241:
4240:
4228:
4227:
4219:
4216:
4212:
4211:
4210:
4201:
4200:
4188:
4187:
4178:
4177:
4160:
4159:
4151:
4148:
4147:
4138:
4137:
4125:
4124:
4116:
4113:
4112:
4103:
4102:
4076:
4074:
4073:
4068:
4066:
4065:
4057:
4047:
4045:
4044:
4039:
4037:
4036:
4028:
4018:
4016:
4015:
4010:
3971:
3969:
3968:
3963:
3961:
3957:
3956:
3948:
3942:
3941:
3933:
3930:
3929:
3917:
3916:
3908:
3905:
3904:
3892:
3891:
3883:
3880:
3879:
3864:
3860:
3859:
3851:
3848:
3844:
3843:
3842:
3833:
3832:
3820:
3819:
3810:
3809:
3792:
3791:
3783:
3780:
3776:
3775:
3774:
3765:
3764:
3752:
3751:
3742:
3741:
3724:
3723:
3715:
3712:
3708:
3707:
3706:
3697:
3696:
3684:
3683:
3674:
3673:
3652:
3651:
3643:
3621:
3619:
3618:
3613:
3611:
3610:
3602:
3592:
3590:
3589:
3584:
3576:
3575:
3567:
3551:
3549:
3548:
3543:
3541:
3540:
3532:
3529:
3528:
3516:
3515:
3507:
3504:
3503:
3491:
3490:
3482:
3479:
3478:
3466:
3465:
3457:
3443:
3441:
3440:
3435:
3433:
3432:
3424:
3414:
3412:
3411:
3406:
3404:
3403:
3395:
3383:
3382:
3374:
3364:
3362:
3361:
3356:
3354:
3353:
3345:
3327:
3312:
3310:
3309:
3304:
3293:
3292:
3284:
3266:
3262:
3261:
3260:
3252:
3219:
3217:
3216:
3211:
3209:
3208:
3203:
3178:
3176:
3175:
3170:
3168:
3167:
3159:
3149:
3147:
3146:
3141:
3139:
3138:
3130:
3116:
3107:
3105:
3104:
3099:
3036:First, note the
3028:
3026:
3025:
3020:
3018:
3014:
3013:
3012:
3007:
2998:
2997:
2992:
2968:
2966:
2965:
2960:
2958:
2957:
2952:
2939:
2937:
2936:
2931:
2929:
2928:
2923:
2900:
2898:
2897:
2892:
2890:
2886:
2885:
2877:
2876:
2871:
2847:
2845:
2844:
2839:
2837:
2836:
2800:
2793:
2789:
2786:
2780:
2753:
2752:
2745:
2736:
2734:
2733:
2728:
2705:
2691:
2677:
2651:
2626:quotient modules
2620:
2618:
2617:
2612:
2564:
2562:
2560:
2559:
2554:
2534:
2532:
2531:
2526:
2508:
2506:
2504:
2503:
2498:
2472:
2470:
2469:
2464:
2434:
2432:
2431:
2426:
2405:
2403:
2401:
2400:
2395:
2369:
2367:
2366:
2361:
2343:
2337:
2335:
2334:
2329:
2299:
2290:
2288:
2287:
2282:
2225:
2215:
2209:
2187:
2185:
2184:
2179:
2177:
2176:
2149:
2147:
2146:
2141:
2139:
2138:
2119:
2117:
2116:
2111:
2103:
2102:
2090:
2089:
2074:
2073:
2041:
2039:
2038:
2033:
2015:
2013:
2012:
2007:
1996:
1860:
1858:
1857:
1852:
1844:
1836:
1831:
1823:
1818:
1816:
1805:
1803:
1760:as the image of
1752:. The image of 2
1736:
1734:
1733:
1728:
1726:
1718:
1713:
1705:
1700:
1694:
1672:
1670:
1669:
1664:
1644:
1642:
1641:
1636:
1624:
1594:to the element 2
1582:
1580:
1579:
1574:
1572:
1564:
1559:
1551:
1546:
1544:
1536:
1530:
1528:
1493:). Furthermore,
1488:
1486:
1485:
1480:
1478:
1477:
1457:
1455:
1454:
1449:
1437:
1435:
1434:
1429:
1424:
1423:
1402:
1401:
1380:
1378:
1377:
1372:
1370:
1357:
1356:
1344:
1343:
1326:
1321:
1320:
1289:
1287:
1274:
1273:
1261:
1260:
1249:
1244:
1243:
1215:
1214:
1202:
1201:
1190:
1185:
1184:
1147:
1145:
1144:
1139:
1134:
1133:
1119:
1118:
1117:
1106:
1101:
1091:
1090:
1089:
1078:
1073:
1071:
1070:
1056:
1055:
1054:
1043:
1038:
1036:
1035:
1021:
1020:
1019:
1008:
1003:
1001:
1000:
967:
965:
964:
959:
876:
874:
873:
868:
847:
834:
811:
809:
808:
803:
786:
760:
673:
671:
670:
665:
645:
632:
598:exact categories
455:
453:
452:
447:
423:
421:
420:
415:
413:
412:
392:
390:
389:
384:
379:
378:
348:
347:
322:
320:
319:
314:
312:
311:
282:
280:
279:
274:
272:
271:
261:
256:
255:
254:
243:
238:
231:
226:
225:
224:
213:
208:
205:
204:
194:
189:
188:
187:
176:
171:
168:
167:
157:
152:
151:
150:
139:
134:
131:
130:
96:) such that the
94:abelian category
63:
61:
60:
55:
53:
52:
21:
7818:
7817:
7813:
7812:
7811:
7809:
7808:
7807:
7788:
7787:
7786:
7781:
7712:
7694:
7690:Urysohn's lemma
7651:
7615:
7501:
7492:
7464:low-dimensional
7422:
7417:
7382:
7360:Eisenbud, David
7358:
7352:
7328:
7320:
7319:
7310:
7309:
7305:
7296:
7294:
7286:
7285:
7281:
7271:
7210:
7200:
7191:
7178:
7168:
7159:
7150:
7093:
7070:normal subgroup
7028:
7015:
7002:
6991:
6990:
6963:
6950:
6931:
6926:
6925:
6898:
6885:
6880:
6879:
6852:
6847:
6846:
6819:
6806:
6801:
6800:
6779:
6774:
6773:
6716:
6712:
6710:
6697:
6696:
6639:
6620:
6598:
6593:
6592:
6561:
6542:
6511:
6498:
6476:
6471:
6470:
6464:
6436:
6423:
6410:
6397:
6384:
6371:
6366:
6365:
6306:
6295:
6285:
6271:
6262:
6252:
6238:
6229:
6219:
6205:
6124:
6123:
6117:splitting lemma
6113:
6085:
6080:
6079:
6036:
6021:
6016:
6015:
6009:splitting lemma
5978:
5973:
5972:
5949:
5944:
5943:
5919:
5914:
5913:
5896:
5895:
5871:
5867:
5857:
5848:
5847:
5823:
5819:
5803:
5781:
5766:
5765:
5725:
5693:
5692:
5654:
5650:
5617:
5613:
5585:
5580:
5579:
5555:
5540:
5524:
5519:
5518:
5455:
5433:
5422:
5421:
5369:
5345:
5321:
5299:
5288:
5287:
5263:
5258:
5257:
5236:
5231:
5230:
5198:
5193:
5192:
5164:
5130:
5117:
5098:
5088:
5080:
5079:
5049:
5039:
5026:
5016:
5003:
4993:
4967:
4966:
4965:By assumption,
4949:
4948:
4938:
4931:
4900:
4887:
4863:
4853:
4840:
4830:
4829:
4825:
4816:
4815:
4805:
4798:
4782:
4769:
4735:
4722:
4703:
4690:
4678:
4677:
4647:
4639:
4638:
4617:
4612:
4611:
4573:
4572:
4533:
4511:
4503:
4502:
4485:
4484:
4453:
4431:
4418:
4408:
4405:
4404:
4373:
4351:
4341:
4331:
4322:
4321:
4282:
4257:
4232:
4202:
4192:
4179:
4169:
4168:
4164:
4139:
4129:
4104:
4094:
4086:
4085:
4050:
4049:
4021:
4020:
3977:
3976:
3959:
3958:
3921:
3896:
3871:
3862:
3861:
3834:
3824:
3811:
3801:
3800:
3796:
3766:
3756:
3743:
3733:
3732:
3728:
3698:
3688:
3675:
3665:
3664:
3660:
3653:
3626:
3625:
3595:
3594:
3554:
3553:
3520:
3495:
3470:
3450:
3449:
3417:
3416:
3367:
3366:
3332:
3331:
3243:
3239:
3228:
3227:
3198:
3193:
3192:
3181:Stokes' theorem
3152:
3151:
3123:
3122:
3112:
3045:
3044:
3031:Stokes' theorem
3002:
2987:
2980:
2976:
2971:
2970:
2947:
2942:
2941:
2918:
2907:
2906:
2866:
2859:
2855:
2850:
2849:
2828:
2823:
2822:
2801:
2790:
2784:
2781:
2774:
2754:
2750:
2743:
2633:
2632:
2573:
2572:
2539:
2538:
2536:
2511:
2510:
2477:
2476:
2474:
2437:
2436:
2411:
2410:
2374:
2373:
2371:
2370:to the element
2346:
2345:
2339:
2302:
2301:
2295:
2231:
2230:
2221:
2211:
2205:
2202:
2165:
2160:
2159:
2130:
2125:
2124:
2094:
2078:
2065:
2054:
2053:
2021:
2020:
1966:
1965:
1809:
1788:
1787:
1682:
1681:
1655:
1654:
1627:
1626:
1615:
1537:
1531:
1519:
1518:
1512:
1507:
1469:
1464:
1463:
1440:
1439:
1415:
1393:
1388:
1387:
1368:
1367:
1348:
1329:
1327:
1306:
1297:
1296:
1285:
1284:
1265:
1252:
1250:
1235:
1226:
1225:
1206:
1193:
1191:
1176:
1161:
1160:
1125:
1109:
1081:
1062:
1046:
1027:
1011:
992:
987:
986:
974:
935:
934:
820:
819:
742:
741:
618:
617:
611:
500:
426:
425:
404:
399:
398:
364:
339:
325:
324:
303:
298:
297:
263:
246:
216:
196:
179:
159:
142:
122:
117:
116:
110:
44:
39:
38:
28:
23:
22:
15:
12:
11:
5:
7816:
7814:
7806:
7805:
7800:
7790:
7789:
7783:
7782:
7780:
7779:
7769:
7768:
7767:
7762:
7757:
7742:
7732:
7722:
7710:
7699:
7696:
7695:
7693:
7692:
7687:
7682:
7677:
7672:
7667:
7661:
7659:
7653:
7652:
7650:
7649:
7644:
7639:
7637:Winding number
7634:
7629:
7623:
7621:
7617:
7616:
7614:
7613:
7608:
7603:
7598:
7593:
7588:
7583:
7578:
7577:
7576:
7571:
7569:homotopy group
7561:
7560:
7559:
7554:
7549:
7544:
7539:
7529:
7524:
7519:
7509:
7507:
7503:
7502:
7495:
7493:
7491:
7490:
7485:
7480:
7479:
7478:
7468:
7467:
7466:
7456:
7451:
7446:
7441:
7436:
7430:
7428:
7424:
7423:
7418:
7416:
7415:
7408:
7401:
7393:
7387:
7386:
7380:
7356:
7350:
7325:
7324:
7318:
7317:
7303:
7278:
7277:
7276:
7275:
7270:
7267:
7259:Exact functors
7225:
7224:
7205:
7196:
7187:
7173:
7164:
7155:
7145:
7092:
7089:
7049:
7046:
7041:
7038:
7035:
7031:
7027:
7022:
7018:
7014:
7009:
7005:
7001:
6998:
6976:
6973:
6970:
6966:
6962:
6957:
6953:
6949:
6944:
6941:
6938:
6934:
6911:
6908:
6905:
6901:
6897:
6892:
6888:
6865:
6862:
6859:
6855:
6834:
6831:
6826:
6822:
6818:
6813:
6809:
6786:
6782:
6770:
6769:
6735:
6729:
6725:
6722:
6719:
6715:
6708:
6704:
6669:
6668:
6657:
6652:
6649:
6646:
6642:
6638:
6633:
6630:
6627:
6623:
6619:
6616:
6613:
6610:
6605:
6601:
6586:
6585:
6573:
6568:
6564:
6560:
6555:
6552:
6549:
6545:
6541:
6538:
6535:
6532:
6529:
6524:
6521:
6518:
6514:
6510:
6505:
6501:
6497:
6494:
6491:
6488:
6483:
6479:
6462:
6457:
6456:
6443:
6439:
6435:
6430:
6426:
6422:
6417:
6413:
6409:
6404:
6400:
6396:
6391:
6387:
6383:
6378:
6374:
6318:
6317:
6268:
6235:
6201:
6200:
6188:
6185:
6182:
6179:
6170:
6163:
6158:
6149:
6142:
6137:
6134:
6131:
6112:
6109:
6094:
6089:
6076:
6075:
6063:
6060:
6057:
6054:
6051:
6048:
6043:
6039:
6035:
6030:
6025:
5996:
5993:
5988:
5985:
5981:
5958:
5953:
5929:
5926:
5922:
5910:
5909:
5893:
5886:
5883:
5877:
5874:
5870:
5864:
5860:
5856:
5853:
5851:
5849:
5845:
5838:
5835:
5829:
5826:
5822:
5818:
5815:
5812:
5809:
5806:
5804:
5799:
5796:
5788:
5784:
5780:
5777:
5774:
5773:
5750:
5747:
5743:
5737:
5734:
5728:
5724:
5721:
5718:
5712:
5709:
5703:
5700:
5689:
5688:
5676:
5669:
5666:
5660:
5657:
5653:
5649:
5646:
5643:
5639:
5632:
5629:
5623:
5620:
5616:
5612:
5609:
5603:
5600:
5592:
5588:
5562:
5558:
5554:
5549:
5544:
5539:
5534:
5531:
5527:
5511:
5510:
5499:
5496:
5493:
5490:
5487:
5484:
5481:
5475:
5471:
5464:
5459:
5451:
5447:
5440:
5436:
5432:
5429:
5396:
5395:
5384:
5381:
5376:
5372:
5365:
5361:
5354:
5349:
5341:
5337:
5330:
5325:
5317:
5313:
5306:
5302:
5298:
5295:
5270:
5266:
5243:
5239:
5225:
5224:
5205:
5201:
5185:
5184:
5171:
5167:
5163:
5160:
5157:
5154:
5151:
5148:
5142:
5137:
5133:
5124:
5120:
5116:
5113:
5110:
5105:
5101:
5095:
5091:
5087:
5064:
5061:
5056:
5052:
5046:
5042:
5038:
5033:
5029:
5023:
5019:
5015:
5010:
5006:
5000:
4996:
4992:
4986:
4983:
4977:
4974:
4963:
4962:
4945:
4941:
4937:
4934:
4932:
4930:
4927:
4924:
4921:
4918:
4912:
4907:
4903:
4894:
4890:
4886:
4883:
4880:
4876:
4870:
4866:
4860:
4856:
4852:
4847:
4843:
4837:
4833:
4828:
4824:
4821:
4818:
4817:
4812:
4808:
4804:
4801:
4799:
4797:
4794:
4789:
4785:
4781:
4776:
4772:
4768:
4765:
4762:
4759:
4756:
4753:
4747:
4742:
4738:
4729:
4725:
4721:
4718:
4715:
4710:
4706:
4702:
4697:
4693:
4689:
4686:
4685:
4659:
4654:
4650:
4624:
4620:
4595:
4592:
4589:
4586:
4583:
4580:
4545:
4540:
4536:
4529:
4523:
4518:
4514:
4499:
4498:
4483:
4480:
4477:
4474:
4471:
4465:
4460:
4456:
4449:
4446:
4443:
4438:
4434:
4430:
4427:
4424:
4421:
4419:
4415:
4411:
4407:
4406:
4403:
4400:
4397:
4394:
4391:
4385:
4380:
4376:
4369:
4366:
4363:
4358:
4354:
4350:
4347:
4344:
4342:
4338:
4334:
4330:
4329:
4315:
4314:
4300:
4297:
4289:
4285:
4281:
4275:
4272:
4264:
4260:
4256:
4250:
4247:
4239:
4235:
4231:
4225:
4222:
4215:
4209:
4205:
4199:
4195:
4191:
4186:
4182:
4176:
4172:
4167:
4163:
4157:
4154:
4146:
4142:
4136:
4132:
4128:
4122:
4119:
4111:
4107:
4101:
4097:
4093:
4063:
4060:
4034:
4031:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3973:
3972:
3954:
3951:
3945:
3939:
3936:
3928:
3924:
3920:
3914:
3911:
3903:
3899:
3895:
3889:
3886:
3878:
3874:
3870:
3867:
3865:
3863:
3857:
3854:
3847:
3841:
3837:
3831:
3827:
3823:
3818:
3814:
3808:
3804:
3799:
3795:
3789:
3786:
3779:
3773:
3769:
3763:
3759:
3755:
3750:
3746:
3740:
3736:
3731:
3727:
3721:
3718:
3711:
3705:
3701:
3695:
3691:
3687:
3682:
3678:
3672:
3668:
3663:
3659:
3656:
3654:
3649:
3646:
3640:
3637:
3634:
3633:
3608:
3605:
3582:
3579:
3573:
3570:
3564:
3561:
3538:
3535:
3527:
3523:
3519:
3513:
3510:
3502:
3498:
3494:
3488:
3485:
3477:
3473:
3469:
3463:
3460:
3445:
3444:
3430:
3427:
3401:
3398:
3392:
3389:
3386:
3380:
3377:
3351:
3348:
3342:
3339:
3322:Poincaré lemma
3314:
3313:
3302:
3299:
3296:
3290:
3287:
3281:
3278:
3275:
3272:
3269:
3265:
3258:
3255:
3249:
3246:
3242:
3238:
3235:
3207:
3202:
3165:
3162:
3136:
3133:
3109:
3108:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3017:
3011:
3006:
3001:
2996:
2991:
2986:
2983:
2979:
2956:
2951:
2927:
2922:
2917:
2914:
2905:of a function
2889:
2884:
2880:
2875:
2870:
2865:
2862:
2858:
2835:
2831:
2803:
2802:
2757:
2755:
2748:
2742:
2739:
2738:
2737:
2726:
2723:
2720:
2717:
2714:
2711:
2708:
2704:
2700:
2697:
2694:
2690:
2686:
2683:
2680:
2676:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2650:
2646:
2643:
2640:
2622:
2621:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2552:
2549:
2546:
2524:
2521:
2518:
2496:
2493:
2490:
2487:
2484:
2462:
2459:
2456:
2453:
2450:
2447:
2444:
2424:
2421:
2418:
2393:
2390:
2387:
2384:
2381:
2359:
2356:
2353:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2292:
2291:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2201:
2198:
2190:dihedral group
2175:
2172:
2168:
2137:
2133:
2121:
2120:
2109:
2106:
2101:
2097:
2093:
2088:
2085:
2081:
2077:
2072:
2068:
2064:
2061:
2031:
2028:
2017:
2016:
2005:
2002:
1999:
1995:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1947:
1946:
1919:
1904:
1862:
1861:
1850:
1847:
1843:
1839:
1835:
1830:
1826:
1822:
1815:
1812:
1808:
1802:
1798:
1795:
1738:
1737:
1725:
1721:
1717:
1712:
1708:
1704:
1699:
1693:
1689:
1662:
1634:
1610:to an element
1584:
1583:
1571:
1567:
1563:
1558:
1554:
1550:
1543:
1540:
1535:
1527:
1511:
1508:
1506:
1503:
1476:
1472:
1447:
1427:
1422:
1418:
1414:
1411:
1408:
1405:
1400:
1396:
1366:
1363:
1360:
1355:
1351:
1347:
1342:
1339:
1336:
1332:
1328:
1324:
1319:
1316:
1313:
1309:
1305:
1302:
1299:
1298:
1295:
1288:
1286:
1283:
1280:
1277:
1272:
1268:
1264:
1259:
1255:
1251:
1247:
1242:
1238:
1234:
1231:
1228:
1227:
1224:
1221:
1218:
1213:
1209:
1205:
1200:
1196:
1192:
1188:
1183:
1179:
1175:
1172:
1169:
1168:
1137:
1132:
1128:
1116:
1112:
1104:
1099:
1088:
1084:
1076:
1069:
1065:
1053:
1049:
1041:
1034:
1030:
1018:
1014:
1006:
999:
995:
973:
970:
969:
968:
957:
954:
951:
948:
945:
942:
912:abelian groups
878:
877:
865:
862:
859:
854:
850:
846:
841:
837:
833:
830:
827:
813:
812:
801:
798:
795:
792:
789:
785:
781:
778:
775:
772:
769:
766:
763:
759:
755:
752:
749:
675:
674:
663:
660:
657:
652:
648:
644:
639:
635:
631:
628:
625:
610:
607:
606:
605:
562:
531:
499:
496:
445:
442:
439:
436:
433:
411:
407:
382:
377:
374:
371:
367:
363:
360:
357:
354:
351:
346:
342:
338:
335:
332:
310:
306:
290:is said to be
286:of groups and
284:
283:
270:
266:
253:
249:
241:
235:
223:
219:
211:
203:
199:
186:
182:
174:
166:
162:
149:
145:
137:
129:
125:
109:
106:
74:exact sequence
66:Euler diagrams
51:
47:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7815:
7804:
7801:
7799:
7796:
7795:
7793:
7778:
7770:
7766:
7763:
7761:
7758:
7756:
7753:
7752:
7751:
7743:
7741:
7737:
7733:
7731:
7727:
7723:
7721:
7716:
7711:
7709:
7701:
7700:
7697:
7691:
7688:
7686:
7683:
7681:
7678:
7676:
7673:
7671:
7668:
7666:
7663:
7662:
7660:
7658:
7654:
7648:
7647:Orientability
7645:
7643:
7640:
7638:
7635:
7633:
7630:
7628:
7625:
7624:
7622:
7618:
7612:
7609:
7607:
7604:
7602:
7599:
7597:
7594:
7592:
7589:
7587:
7584:
7582:
7579:
7575:
7572:
7570:
7567:
7566:
7565:
7562:
7558:
7555:
7553:
7550:
7548:
7545:
7543:
7540:
7538:
7535:
7534:
7533:
7530:
7528:
7525:
7523:
7520:
7518:
7514:
7511:
7510:
7508:
7504:
7499:
7489:
7486:
7484:
7483:Set-theoretic
7481:
7477:
7474:
7473:
7472:
7469:
7465:
7462:
7461:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7444:Combinatorial
7442:
7440:
7437:
7435:
7432:
7431:
7429:
7425:
7421:
7414:
7409:
7407:
7402:
7400:
7395:
7394:
7391:
7383:
7381:0-387-94269-6
7377:
7373:
7368:
7367:
7361:
7357:
7353:
7351:0-387-94426-5
7347:
7343:
7338:
7337:
7331:
7327:
7326:
7322:
7321:
7313:
7307:
7304:
7293:
7289:
7283:
7280:
7273:
7272:
7268:
7266:
7264:
7260:
7256:
7254:
7250:
7246:
7242:
7238:
7237:zig-zag lemma
7234:
7228:
7222:
7218:
7217:
7216:
7214:
7208:
7204:
7199:
7195:
7190:
7186:
7182:
7181:chain complex
7176:
7172:
7167:
7163:
7158:
7154:
7148:
7144:
7139:
7137:
7133:
7129:
7125:
7121:
7117:
7113:
7109:
7105:
7100:
7098:
7090:
7088:
7085:
7083:
7079:
7075:
7071:
7067:
7063:
7047:
7039:
7036:
7033:
7029:
7020:
7016:
7007:
7003:
6996:
6974:
6971:
6968:
6964:
6955:
6951:
6942:
6939:
6936:
6932:
6909:
6906:
6903:
6899:
6890:
6886:
6878:and morphism
6863:
6860:
6857:
6853:
6832:
6824:
6820:
6811:
6807:
6784:
6780:
6767:
6763:
6762:
6761:
6759:
6755:
6751:
6733:
6727:
6723:
6720:
6717:
6713:
6706:
6702:
6694:
6690:
6686:
6682:
6678:
6674:
6650:
6647:
6644:
6640:
6631:
6628:
6625:
6621:
6614:
6611:
6608:
6603:
6599:
6591:
6590:
6589:
6566:
6562:
6553:
6550:
6547:
6543:
6536:
6533:
6530:
6522:
6519:
6516:
6512:
6503:
6499:
6492:
6489:
6486:
6481:
6477:
6469:
6468:
6467:
6465:
6441:
6437:
6428:
6424:
6415:
6411:
6402:
6398:
6389:
6385:
6376:
6372:
6364:
6363:
6362:
6358:
6356:
6352:
6347:
6345:
6341:
6337:
6332:
6330:
6325:
6323:
6313:
6309:
6302:
6298:
6293:
6288:
6282:
6278:
6274:
6269:
6265:
6259:
6255:
6249:
6245:
6241:
6236:
6232:
6226:
6222:
6216:
6212:
6208:
6203:
6202:
6186:
6183:
6177:
6168:
6161:
6156:
6147:
6140:
6135:
6129:
6122:
6121:
6120:
6118:
6110:
6108:
6092:
6058:
6052:
6049:
6046:
6041:
6037:
6033:
6028:
6014:
6013:
6012:
6010:
5991:
5986:
5983:
5956:
5927:
5924:
5891:
5881:
5875:
5868:
5862:
5854:
5852:
5843:
5833:
5827:
5820:
5813:
5807:
5805:
5794:
5786:
5778:
5764:
5763:
5762:
5748:
5745:
5741:
5732:
5726:
5722:
5719:
5716:
5707:
5701:
5674:
5664:
5658:
5651:
5647:
5641:
5637:
5627:
5621:
5614:
5607:
5598:
5590:
5578:
5577:
5576:
5560:
5556:
5547:
5537:
5532:
5529:
5516:
5497:
5488:
5482:
5479:
5473:
5469:
5462:
5449:
5445:
5438:
5434:
5427:
5420:
5419:
5418:
5416:
5411:
5409:
5405:
5401:
5382:
5374:
5370:
5363:
5359:
5352:
5339:
5335:
5328:
5315:
5311:
5304:
5300:
5293:
5286:
5285:
5284:
5268:
5264:
5241:
5237:
5223:
5221:
5203:
5199:
5190:
5169:
5165:
5161:
5155:
5152:
5149:
5135:
5131:
5122:
5114:
5111:
5108:
5103:
5099:
5093:
5085:
5078:
5077:
5076:
5062:
5059:
5054:
5050:
5044:
5036:
5031:
5027:
5021:
5013:
5008:
5004:
4998:
4990:
4981:
4975:
4972:
4943:
4939:
4935:
4933:
4925:
4922:
4919:
4905:
4901:
4892:
4884:
4881:
4878:
4874:
4868:
4864:
4858:
4850:
4845:
4841:
4835:
4826:
4822:
4819:
4810:
4806:
4802:
4800:
4795:
4792:
4787:
4783:
4779:
4774:
4766:
4760:
4757:
4754:
4740:
4736:
4727:
4719:
4716:
4713:
4708:
4704:
4700:
4695:
4687:
4676:
4675:
4674:
4652:
4648:
4622:
4618:
4609:
4590:
4587:
4584:
4578:
4570:
4566:
4562:
4538:
4534:
4527:
4516:
4512:
4478:
4475:
4472:
4458:
4454:
4447:
4444:
4441:
4436:
4432:
4428:
4425:
4422:
4420:
4413:
4409:
4398:
4395:
4392:
4378:
4374:
4367:
4364:
4361:
4356:
4352:
4348:
4345:
4343:
4336:
4332:
4320:
4319:
4318:
4295:
4287:
4283:
4279:
4270:
4262:
4258:
4254:
4245:
4237:
4233:
4229:
4220:
4213:
4207:
4203:
4197:
4189:
4184:
4180:
4174:
4165:
4161:
4152:
4144:
4140:
4134:
4126:
4117:
4109:
4105:
4099:
4091:
4084:
4083:
4082:
4080:
4058:
4029:
4006:
4003:
3997:
3994:
3991:
3985:
3982:
3949:
3943:
3934:
3926:
3922:
3918:
3909:
3901:
3897:
3893:
3884:
3876:
3872:
3868:
3866:
3852:
3845:
3839:
3835:
3829:
3821:
3816:
3812:
3806:
3797:
3793:
3784:
3777:
3771:
3767:
3761:
3753:
3748:
3744:
3738:
3729:
3725:
3716:
3709:
3703:
3699:
3693:
3685:
3680:
3676:
3670:
3661:
3657:
3655:
3644:
3638:
3624:
3623:
3603:
3580:
3577:
3568:
3562:
3533:
3525:
3521:
3517:
3508:
3500:
3496:
3492:
3483:
3475:
3471:
3467:
3458:
3447:
3446:
3425:
3396:
3390:
3387:
3384:
3375:
3346:
3340:
3337:
3329:
3328:
3325:
3323:
3319:
3300:
3297:
3294:
3285:
3279:
3273:
3267:
3263:
3253:
3247:
3244:
3240:
3236:
3233:
3226:
3225:
3224:
3221:
3205:
3190:
3186:
3182:
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3131:
3120:
3115:
3111:for all such
3095:
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3015:
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2912:
2904:
2901:. Taking the
2887:
2873:
2863:
2860:
2856:
2833:
2829:
2821:
2820:Hilbert space
2818:Consider the
2816:
2814:
2810:
2799:
2796:
2788:
2785:December 2019
2778:
2773:
2768:
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560:
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548:
544:
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532:
529:
525:
521:
517:
513:
509:
508:
507:
505:
504:trivial group
497:
495:
493:
489:
485:
481:
477:
473:
469:
468:vector spaces
465:
460:
457:
443:
440:
437:
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409:
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396:
375:
372:
369:
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308:
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221:
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197:
184:
180:
172:
164:
160:
147:
143:
135:
127:
123:
115:
114:
113:
107:
105:
104:of the next.
103:
99:
95:
91:
87:
83:
79:
75:
67:
49:
45:
37:
32:
19:
7777:Publications
7642:Chern number
7632:Betti number
7515: /
7506:Key concepts
7454:Differential
7365:
7335:
7306:
7295:. Retrieved
7291:
7282:
7257:
7232:
7229:
7226:
7206:
7202:
7197:
7193:
7188:
7184:
7174:
7170:
7165:
7161:
7156:
7152:
7146:
7142:
7140:
7127:
7123:
7119:
7115:
7111:
7107:
7101:
7094:
7086:
7081:
7077:
7073:
7065:
7061:
6771:
6757:
6749:
6692:
6688:
6684:
6680:
6676:
6670:
6587:
6460:
6458:
6359:
6348:
6338:shows how a
6333:
6328:
6326:
6319:
6311:
6307:
6300:
6296:
6286:
6280:
6276:
6272:
6263:
6257:
6253:
6247:
6243:
6239:
6230:
6224:
6220:
6214:
6210:
6206:
6114:
6077:
5911:
5690:
5512:
5412:
5397:
5228:
5219:
5186:
4964:
4607:
4568:
4564:
4560:
4500:
4316:
4078:
3974:
3365:= 0 implies
3315:
3222:
3113:
3110:
3035:
2817:
2806:
2791:
2782:
2775:Please help
2770:
2759:
2623:
2567:
2340:
2296:
2293:
2222:
2212:
2206:
2203:
2193:
2155:
2152:cyclic group
2122:
2048:
2018:
1955:finite group
1950:
1948:
1942:
1938:
1934:
1930:
1926:
1922:
1915:
1911:
1907:
1900:
1896:
1884:
1880:
1878:factor group
1873:
1869:
1865:
1863:
1782:
1777:
1773:
1769:
1765:
1761:
1757:
1753:
1749:
1745:
1741:
1739:
1674:
1650:
1646:
1620:
1616:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1585:
1513:
1498:
1494:
1490:
1459:
1385:
1381:
1159:
1154:
1152:
1148:
985:
982:
977:
975:
927:
923:
915:
907:
903:
899:
895:
891:
887:
881:
879:
814:
734:
730:
726:
718:
714:
710:
706:
702:
694:
690:
686:
682:
678:
676:
612:
601:
593:
589:
577:
573:
569:
565:
554:
553:) is all of
550:
546:
542:
538:
534:
528:monomorphism
523:
519:
515:
511:
501:
498:Simple cases
461:
458:
394:
294:
291:
285:
111:
73:
71:
7740:Wikiversity
7657:Key results
7292:ncatlab.org
7134:. See also
6336:snake lemma
4567:and not on
3330:Proof that
3150:is 0, then
2779:if you can.
2624:Passing to
586:isomorphism
559:epimorphism
472:linear maps
7792:Categories
7586:CW complex
7527:Continuity
7517:Closed set
7476:cohomology
7297:2021-09-05
7269:References
7097:subobjects
6924:such that
6351:five lemma
6344:nine lemma
6292:direct sum
6284:such that
6251:such that
6218:such that
6111:Properties
5187:Since the
3622:such that
3552:such that
3318:divergence
2408:direct sum
2220:of a ring
2192:of order 2
1438:for every
920:direct sum
880:is called
709:embedding
582:bimorphism
108:Definition
7765:geometric
7760:algebraic
7611:Cobordism
7547:Hausdorff
7542:connected
7459:Geometric
7449:Continuum
7439:Algebraic
7274:Citations
7211:, so the
7045:→
7026:→
7013:→
7000:→
6961:→
6948:→
6940:−
6896:→
6830:→
6817:→
6721:
6679:) :
6648:−
6637:→
6629:−
6615:
6609:≅
6559:→
6551:−
6537:
6531:≅
6509:→
6493:
6487:≅
6434:→
6421:→
6408:→
6395:→
6382:→
6181:→
6133:→
6053:
6047:⊕
6034:≅
5995:∇
5992:∘
5984:−
5980:∇
5925:−
5921:∇
5885:→
5876:⋅
5873:∇
5859:∇
5837:→
5828:⋅
5825:∇
5817:∇
5814:⋅
5811:∇
5798:→
5783:∇
5779:⋅
5776:∇
5736:→
5723:
5711:→
5702:×
5699:∇
5668:→
5659:×
5656:∇
5648:×
5645:∇
5631:→
5622:⋅
5619:∇
5611:∇
5602:→
5587:∇
5553:→
5530:−
5526:∇
5515:Laplacian
5495:→
5483:
5431:→
5380:→
5297:→
5141:→
5119:∂
5090:∂
5086:∫
5075:, and so
5041:∂
5018:∂
4995:∂
4985:→
4976:
4911:→
4889:∂
4855:∂
4832:∂
4823:∫
4820:−
4780:∫
4771:∂
4767:−
4746:→
4724:∂
4701:∫
4692:∂
4688:−
4658:→
4544:→
4522:→
4464:→
4429:∫
4426:−
4384:→
4349:∫
4299:^
4274:^
4249:^
4224:^
4194:∂
4190:−
4171:∂
4156:^
4131:∂
4121:^
4096:∂
4092:−
4062:→
4033:→
3995:
3986:
3953:→
3938:^
3913:^
3888:^
3856:^
3826:∂
3822:−
3803:∂
3788:^
3758:∂
3754:−
3735:∂
3720:^
3690:∂
3686:−
3667:∂
3648:→
3639:×
3636:∇
3607:→
3572:→
3563:⋅
3560:∇
3537:^
3512:^
3487:^
3462:→
3429:→
3415:for some
3400:→
3391:
3379:→
3350:→
3341:
3289:→
3280:×
3277:∇
3274:⋅
3271:∇
3268:≡
3257:→
3248:
3237:
3164:→
3135:→
3084:∇
3078:×
3075:∇
3072:≡
3063:
3054:
3000:→
2916:∈
2879:→
2722:→
2696:→
2682:⊕
2668:→
2659:∩
2642:→
2606:→
2600:→
2594:⊕
2588:→
2582:→
2548:−
2520:⊕
2452:→
2446:⊕
2420:⊕
2355:∩
2323:⊕
2317:→
2311:∩
2276:→
2264:→
2258:⊕
2252:→
2246:∩
2240:→
2154:of order
2105:→
2092:→
2076:→
2063:→
2001:→
1987:→
1981:→
1975:→
1846:→
1825:⟶
1814:×
1807:⟶
1797:→
1707:↠
1698:↪
1661:↠
1633:↪
1553:↠
1542:×
1534:↪
1410:
1359:→
1346:→
1338:−
1323:→
1315:−
1304:→
1294:⋮
1276:→
1263:→
1246:→
1233:→
1217:→
1204:→
1187:→
1174:→
1098:⋯
950:⊕
944:≅
861:→
829:→
791:
765:
751:≅
717:, and of
699:subobject
659:→
627:→
488:cokernels
435:≤
359:
334:
234:⋯
78:morphisms
7730:Wikibook
7708:Category
7596:Manifold
7564:Homotopy
7522:Interior
7513:Open set
7471:Homology
7420:Topology
7362:(1995).
7332:(1995).
7263:functors
7213:homology
7169:to 0 in
6728:⟩
6714:⟨
6209: :
6162:→
6141:→
5470:→
5446:→
5360:→
5336:→
5312:→
3189:gradient
3119:gradient
2903:gradient
2760:require
1764:through
1505:Examples
1103:→
1075:→
1040:→
1005:→
890: :
849:→
836:→
723:quotient
647:→
634:→
480:category
424:for all
240:→
210:→
173:→
136:→
7755:general
7557:uniform
7537:compact
7488:Digital
7323:Sources
7221:acyclic
6752:by the
6687:is not
6290:is the
2762:cleanup
2563:
2537:
2507:
2475:
2406:of the
2404:
2372:
2226:. Then
2216:be two
2188:is the
2150:is the
2044:abelian
1876:to the
733:, with
484:kernels
90:modules
7750:Topics
7552:metric
7427:Fields
7378:
7348:
7247:; the
6756:of im(
6695:) but
6329:splits
6172:
6166:
6151:
6145:
2218:ideals
2123:where
1889:modulo
1386:where
1290:
1120:
1107:
1092:
1079:
1057:
1044:
1022:
1009:
257:
244:
227:
214:
190:
177:
153:
140:
102:kernel
82:groups
64:using
36:groups
7532:Space
7160:maps
7122:have
6612:coker
1623:mod 2
1153:with
883:split
713:into
705:with
697:as a
600:like
588:from
482:with
395:exact
292:exact
98:image
86:rings
7376:ISBN
7346:ISBN
7261:are
7110:and
7102:The
7080:/im(
6691:/im(
6349:The
6334:The
6305:and
6115:The
6059:curl
5720:curl
5489:curl
5474:curl
5450:grad
5340:curl
5316:grad
4563:and
3992:grad
3983:curl
3388:curl
3245:curl
3060:grad
3051:curl
3038:curl
2210:and
2204:Let
2158:and
1961:is
926:and
486:and
470:and
441:<
7372:785
7342:179
6490:ker
6324:".
6294:of
5364:div
4973:div
3338:div
3324:):
3234:div
3220:.
2535:to
2509:of
2344:of
2046:).
1868:to
1768:↦ 2
1744:↦ 2
1649:to
1606:in
1598:in
1499:(2)
1495:(1)
1491:(1)
1460:(2)
1382:(2)
1155:n ≥
1149:(1)
922:of
788:ker
725:),
701:of
602:Set
592:to
576:to
549:to
522:to
356:ker
323:if
72:An
7794::
7374:.
7344:.
7290:.
7255:.
7209:+1
7177:+2
7151:∘
7149:+1
7138:.
6718:im
6683:→
6534:im
6331:.
6279:→
6275::
6256:∘
6246:→
6242::
6223:∘
6213:→
6050:im
6011:,
5498:0.
5480:im
2815:.
2565:.
1941:/2
1933:/2
1925:/2
1883:/2
1619:=
1407:im
930::
914:,
902:∘
894:→
762:im
662:0.
604:).
568:→
537:→
514:→
331:im
295:at
88:,
84:,
7412:e
7405:t
7398:v
7384:.
7354:.
7300:.
7223:.
7207:i
7203:f
7198:i
7194:A
7189:i
7185:f
7175:i
7171:A
7166:i
7162:A
7157:i
7153:f
7147:i
7143:f
7128:C
7124:A
7120:B
7116:B
7112:C
7108:A
7082:f
7078:H
7074:f
7066:H
7062:f
7048:0
7040:1
7037:+
7034:k
7030:C
7021:k
7017:A
7008:k
7004:C
6997:0
6975:1
6972:+
6969:k
6965:A
6956:k
6952:A
6943:1
6937:k
6933:A
6910:1
6907:+
6904:k
6900:A
6891:k
6887:A
6864:1
6861:+
6858:k
6854:A
6833:0
6825:7
6821:C
6812:6
6808:A
6785:7
6781:C
6758:f
6750:H
6734:H
6724:f
6707:/
6703:H
6693:f
6689:H
6685:H
6681:G
6677:f
6656:)
6651:1
6645:k
6641:A
6632:2
6626:k
6622:A
6618:(
6604:k
6600:C
6584:.
6572:)
6567:k
6563:A
6554:1
6548:k
6544:A
6540:(
6528:)
6523:1
6520:+
6517:k
6513:A
6504:k
6500:A
6496:(
6482:k
6478:C
6463:k
6461:C
6442:6
6438:A
6429:5
6425:A
6416:4
6412:A
6403:3
6399:A
6390:2
6386:A
6377:1
6373:A
6316:.
6314:)
6312:C
6310:(
6308:u
6303:)
6301:A
6299:(
6297:f
6287:B
6281:B
6277:C
6273:u
6267:.
6264:C
6258:u
6254:g
6248:B
6244:C
6240:u
6234:.
6231:A
6225:f
6221:t
6215:A
6211:B
6207:t
6187:,
6184:0
6178:C
6169:g
6157:B
6148:f
6136:A
6130:0
6093:3
6088:R
6074:,
6062:)
6056:(
6042:2
6038:L
6029:3
6024:H
5987:1
5957:3
5952:H
5928:1
5892:)
5882:A
5869:(
5863:2
5855:=
5844:)
5834:A
5821:(
5808:=
5795:A
5787:2
5749:0
5746:=
5742:)
5733:A
5727:(
5717:=
5708:A
5675:)
5665:A
5652:(
5642:+
5638:)
5628:A
5615:(
5608:=
5599:A
5591:2
5561:2
5557:L
5548:3
5543:H
5538::
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5492:)
5486:(
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5220:z
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5170:z
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5162:=
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5147:(
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5060:=
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4991:=
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4936:=
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4917:(
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4851:+
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4827:(
4811:z
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4803:=
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4793:d
4788:y
4784:F
4775:y
4764:)
4761:y
4758:,
4755:x
4752:(
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4470:(
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4423:=
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4346:=
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4280:+
4271:j
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4255:+
4246:i
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4230:=
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4166:(
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4004:=
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3998:f
3989:(
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3944:=
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3927:z
3923:F
3919:+
3910:j
3902:y
3898:F
3894:+
3885:i
3877:x
3873:F
3869:=
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3836:A
3830:y
3817:y
3813:A
3807:x
3798:(
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3785:j
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3762:x
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3662:(
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3578:=
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3518:+
3509:j
3501:y
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3493:+
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3468:=
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3385:=
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3347:F
3301:,
3298:0
3295:=
3286:v
3264:)
3254:v
3241:(
3206:3
3201:H
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3096:0
3093:=
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3057:(
3016:}
3010:3
3005:R
2995:3
2990:R
2985::
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2978:{
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2703:/
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2689:/
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2071:n
2067:C
2060:1
2030:,
2027:1
2004:1
1998:N
1994:/
1990:G
1984:G
1978:N
1972:1
1951:Z
1945:.
1943:Z
1939:Z
1935:Z
1931:Z
1927:Z
1923:Z
1918:.
1916:Z
1912:Z
1908:Z
1903:.
1901:Z
1897:Z
1885:Z
1881:Z
1874:Z
1870:Z
1866:Z
1849:0
1842:Z
1838:2
1834:/
1829:Z
1821:Z
1811:2
1801:Z
1794:0
1778:Z
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1766:n
1762:Z
1758:Z
1754:Z
1750:Z
1746:n
1742:n
1724:Z
1720:2
1716:/
1711:Z
1703:Z
1692:Z
1688:2
1675:Z
1651:Z
1647:Z
1621:i
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1612:j
1608:Z
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1600:Z
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1592:Z
1588:i
1570:Z
1566:2
1562:/
1557:Z
1549:Z
1539:2
1526:Z
1475:i
1471:K
1446:i
1426:)
1421:i
1417:f
1413:(
1404:=
1399:i
1395:K
1365:,
1362:0
1354:n
1350:K
1341:1
1335:n
1331:A
1318:1
1312:n
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1301:0
1282:,
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1271:3
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1254:A
1241:2
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1230:0
1223:,
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1212:2
1208:K
1199:1
1195:A
1182:1
1178:K
1171:0
1136:,
1131:n
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1115:n
1111:f
1087:3
1083:f
1068:2
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1052:2
1048:f
1033:1
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1017:1
1013:f
998:0
994:A
956:.
953:C
947:A
941:B
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924:A
916:B
908:C
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892:C
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864:0
858:C
853:g
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840:f
832:A
826:0
800:)
797:g
794:(
784:/
780:B
777:=
774:)
771:f
768:(
758:/
754:B
748:C
735:g
731:A
729:/
727:B
719:C
715:B
711:A
707:f
703:B
695:A
691:g
687:f
683:g
679:f
656:C
651:g
643:B
638:f
630:A
624:0
594:Y
590:X
578:Y
574:X
570:Y
566:X
555:C
551:C
547:B
543:C
539:C
535:B
524:B
520:A
516:B
512:A
444:n
438:i
432:1
410:i
406:G
381:)
376:1
373:+
370:i
366:f
362:(
353:=
350:)
345:i
341:f
337:(
309:i
305:G
269:n
265:G
252:n
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222:3
218:f
202:2
198:G
185:2
181:f
165:1
161:G
148:1
144:f
128:0
124:G
68:.
50:i
46:G
20:)
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