86:
845:
1379:
1190:(a differential form with distributional coefficients) version of the de Rham theorem, which says the singular cohomology can be computed as the cohomology of the complex of currents. This version is weaker in the sense that the isomorphism is not a ring homomorphism (since currents cannot be multiplied and so the space of currents is not a ring).
1099:
673:
77:. Thus, for abstract reason, the de Rham cohomology is isomorphic as a group to the singular cohomology. But the de Rham theorem gives a more explicit isomorphism between the two cohomologies; thus, connecting analysis and topology more directly.
944:
1107:
This theorem has the following consequence (familiar from calculus); namely, a closed differential form is exact if and only if the integrations of it over arbitrary cycles are all zero. For a one-form, it means that a closed one-form
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1094:{\displaystyle \operatorname {H} _{\mathrm {deRham} }^{*}(M)\to \operatorname {H} _{*}^{\mathrm {sing} }(M)^{*},\,\mapsto \left(\mapsto \int _{\sigma }\omega \right)}
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There is also a version of the theorem involving singular homology instead of cohomology. It says the pairing
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is isomorphic as a ring with the Čech cohomology of it. This Čech version is essentially due to André Weil.
668:{\displaystyle :\operatorname {H} _{\textrm {deRham}}^{*}(M)\to \operatorname {H} _{\mathrm {sing} }^{*}(M)}
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498:{\displaystyle \omega \mapsto \left(\sigma \mapsto \int _{\sigma }\omega \right).}
276:{\displaystyle \int _{\widetilde {c}}\omega -\int _{{\widetilde {c}}'}\omega =0'}
41:
349:
The key part of the theorem is a construction of the de Rham homomorphism. Let
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570:
198:{\displaystyle {\widetilde {c}}-{\widetilde {c}}'=\partial {\widetilde {b}}}
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between the de Rham cohomology and the (smooth) singular homology; namely,
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There is also a variant of the theorem that says the de Rham cohomology of
1378:
1298:
1386:
37:
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429:{\displaystyle k:\Omega ^{p}(M)\to S_{{\mathcal {C}}^{\infty }}^{p}(M)}
84:
924:{\displaystyle (\omega ,\sigma )\mapsto \int _{\sigma }\omega }
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1128:
is exact (i.e., admits a potential function) if and only if
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Finally, the theorem says that the induced homomorphism
89:
The deRham homomorphism is well defined because for any
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759:{\displaystyle S_{{\mathcal {C}}^{\infty }}^{*}(M)}
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131:{\displaystyle {\widetilde {c}},{\widetilde {c}}'}
130:
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678:where these cohomologies are the cohomologies of
1278:
339:{\displaystyle \int _{\widetilde {c}}\omega =0}
1414:
8:
1289:Griffiths, Phillip; Harris, Joseph (1994).
1178:. This is exactly a statement in calculus.
51:implies that the de Rham cohomology is the
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792:is a ring homomorphism and is called the
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542:{\displaystyle k\circ d=\partial \circ k}
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1338:(1952). "Sur les théorèmes de de Rham".
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443:-forms to the space of smooth singular
1266:
1218:
1206:
1151:{\displaystyle \int _{\gamma }\omega }
825:is an isomorphism (i.e., bijective).
138:representing the same homology class
16:In mathematics, more specifically in
7:
1375:
1373:
1104:is an isomorphism of vector spaces.
353:be a manifold. Then there is a map
1393:. You can help Knowledge (XXG) by
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1445:Theorems in differential geometry
1340:Commentarii Mathematici Helvetici
766:, respectively. As it turns out,
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1291:Principles of Algebraic Geometry
843:
439:from the space of differential
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707:{\displaystyle \Omega ^{*}(M)}
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70:{\displaystyle \mathbb {R} }
1279:Griffiths & Harris 1994
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1372:
1238:(2nd ed.). Springer.
1316:Warner, Frank W. (1983).
1244:10.1007/978-0-8176-4767-4
1232:Conlon, Lawrence (2001).
1158:is independent of a path
876:Singular-homology version
1235:Differentiable Manifolds
55:with the constant sheaf
1171:{\displaystyle \gamma }
1121:{\displaystyle \omega }
296:{\displaystyle \omega }
1320:. New York: Springer.
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303:is an exact form then
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18:differential geometry
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794:de Rham homomorphism
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573:and so it induces:
447:-cochains given by
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34:singular cohomology
1352:10.1007/BF02564296
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948:
921:
855:. You can help by
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30:de Rham cohomology
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1308:978-0-471-05059-9
1253:978-0-8176-4766-7
873:
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562:{\displaystyle k}
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26:ring homomorphism
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1387:geometry-related
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1186:There is also a
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818:{\displaystyle }
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785:{\displaystyle }
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53:sheaf cohomology
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1230:Appendix D. to
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1184:
1182:Current version
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936:perfect pairing
908:
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853:needs expansion
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768:
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509:Stokes' formula
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22:de Rham theorem
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5:
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1440:Geometry stubs
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1221:, 5.36., 5.45.
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49:Poincaré lemma
24:says that the
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9:
6:
4:
3:
2:
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1389:article is a
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1327:0-387-90894-3
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867:
858:
854:
851:This section
849:
846:
842:
841:
836:Idea of proof
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27:
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19:
1395:expanding it
1384:
1369:
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1317:
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1274:
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1234:
1226:
1214:
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857:adding to it
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444:
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438:
350:
348:
46:
21:
15:
1346:: 119–145.
1336:Weil, André
1267:Warner 1983
1219:Warner 1983
1207:Warner 1983
42:isomorphism
38:integration
1434:Categories
1194:References
934:induces a
1360:124799328
1166:γ
1146:ω
1141:γ
1137:∫
1116:ω
1084:ω
1079:σ
1075:∫
1071:↦
1065:σ
1054:↦
1048:ω
1036:∗
1022:
1001:∗
993:→
981:
976:∗
919:ω
914:σ
910:∫
906:↦
900:σ
894:ω
864:July 2024
743:∗
736:∞
691:∗
687:Ω
654:
649:∗
625:→
613:
608:∗
571:chain map
534:∘
531:∂
522:∘
511:implies:
485:ω
480:σ
476:∫
472:↦
469:σ
461:↦
458:ω
406:∞
389:→
371:Ω
328:ω
321:~
312:∫
291:ω
260:ω
248:~
237:∫
233:−
230:ω
223:~
214:∫
190:~
181:∂
168:~
158:−
152:~
119:~
103:~
81:Statement
36:given by
28:from the
549:; i.e.,
270:′
254:′
174:′
125:′
1269:, 4.17.
1209:, 5.35.
1188:current
283:and if
32:to the
1358:
1324:
1305:
1250:
602:deRham
40:is an
20:, the
1385:This
1356:S2CID
569:is a
205:, so
1391:stub
1322:ISBN
1303:ISBN
1248:ISBN
714:and
47:The
1348:doi
1295:doi
1240:doi
859:.
1436::
1354:.
1344:26
1342:.
1301:.
1293:.
1246:.
796:.
44:.
1422:e
1415:t
1408:v
1397:.
1362:.
1350::
1330:.
1311:.
1297::
1256:.
1242::
1088:)
1068:]
1062:[
1058:(
1051:]
1045:[
1041:,
1032:)
1028:M
1025:(
1016:g
1013:n
1010:i
1007:s
997:H
990:)
987:M
984:(
970:m
967:a
964:h
961:R
958:e
955:d
950:H
903:)
897:,
891:(
866:)
862:(
830:M
813:]
810:k
807:[
780:]
777:k
774:[
754:)
751:M
748:(
730:C
723:S
702:)
699:M
696:(
663:)
660:M
657:(
643:g
640:n
637:i
634:s
629:H
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619:M
616:(
597:H
593::
590:]
587:k
584:[
557:k
537:k
528:=
525:d
519:k
493:.
489:)
465:(
445:p
441:p
424:)
421:M
418:(
413:p
400:C
393:S
386:)
383:M
380:(
375:p
367::
364:k
351:M
334:0
331:=
318:c
267:0
263:=
245:c
220:c
187:b
178:=
165:c
149:c
116:c
109:,
100:c
64:R
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