543:
22:
460:
363:
236:
274:
386:
302:
584:
32:
90:
165:
62:
69:
373:
76:
608:
380:, one can always integrate a density, regardless of orientation or orientability; there is the integration map:
513:
47:
58:
603:
577:
247:
613:
490:; in particular, one can define a relative orientation sheaf using a relative dualizing complex.
570:
131:
83:
477:
is literally trivial, then the above reduces to the usual integration of a differential form.
558:
43:
8:
455:{\displaystyle \textstyle \int _{M}:\Gamma _{c}(M,{\mathcal {V}}_{M})\to \mathbb {R} .}
112:
509:
377:
277:
501:
487:
554:
470:
597:
358:{\displaystyle {\mathcal {V}}_{M}=\Omega _{M}^{n}\otimes {\mathcal {o}}_{M}}
39:
529:
Two kinds of orientability/orientation for a differentiable manifold
21:
550:
528:
120:
542:
486:
There is also a definition in terms of dualizing complex in
241:(in the integer coefficients or some other coefficients).
231:{\displaystyle o_{X,x}=\operatorname {H} _{n}(X,X-\{x\})}
390:
389:
305:
250:
168:
454:
357:
268:
230:
500:
595:
469:is oriented; i.e., the orientation sheaf of the
578:
368:is called the sheaf of (smooth) densities on
222:
216:
48:introducing citations to additional sources
372:. The point of this is that, while one can
585:
571:
444:
38:Relevant discussion may be found on the
596:
537:
15:
13:
426:
405:
344:
324:
309:
252:
189:
14:
625:
522:
541:
31:relies largely or entirely on a
20:
269:{\displaystyle \Omega _{M}^{k}}
440:
437:
414:
225:
201:
1:
494:
374:integrate a differential form
111:In the mathematical field of
557:. You can help Knowledge by
7:
504:; Schapira, Pierre (2002),
480:
10:
630:
536:
376:only if the manifold is
146:such that the stalk of
609:Orientation (geometry)
456:
359:
270:
232:
132:locally constant sheaf
457:
360:
271:
233:
508:, Berlin: Springer,
506:Sheaves on Manifolds
387:
303:
292:is the dimension of
248:
166:
44:improve this article
337:
265:
59:"Orientation sheaf"
604:Algebraic topology
452:
451:
355:
323:
266:
251:
228:
113:algebraic topology
566:
565:
502:Kashiwara, Masaki
296:, then the sheaf
117:orientation sheaf
109:
108:
94:
621:
587:
580:
573:
551:topology-related
545:
538:
518:
461:
459:
458:
453:
447:
436:
435:
430:
429:
413:
412:
400:
399:
364:
362:
361:
356:
354:
353:
348:
347:
336:
331:
319:
318:
313:
312:
276:be the sheaf of
275:
273:
272:
267:
264:
259:
237:
235:
234:
229:
197:
196:
184:
183:
104:
101:
95:
93:
52:
24:
16:
629:
628:
624:
623:
622:
620:
619:
618:
594:
593:
592:
591:
534:
525:
516:
497:
488:Verdier duality
483:
443:
431:
425:
424:
423:
408:
404:
395:
391:
388:
385:
384:
349:
343:
342:
341:
332:
327:
314:
308:
307:
306:
304:
301:
300:
260:
255:
249:
246:
245:
192:
188:
173:
169:
167:
164:
163:
154:
141:
105:
99:
96:
53:
51:
37:
25:
12:
11:
5:
627:
617:
616:
614:Topology stubs
611:
606:
590:
589:
582:
575:
567:
564:
563:
546:
532:
531:
524:
523:External links
521:
520:
519:
514:
496:
493:
492:
491:
482:
479:
471:tangent bundle
463:
462:
450:
446:
442:
439:
434:
428:
422:
419:
416:
411:
407:
403:
398:
394:
366:
365:
352:
346:
340:
335:
330:
326:
322:
317:
311:
284:on a manifold
263:
258:
254:
239:
238:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
195:
191:
187:
182:
179:
176:
172:
150:
137:
107:
106:
42:. Please help
28:
26:
19:
9:
6:
4:
3:
2:
626:
615:
612:
610:
607:
605:
602:
601:
599:
588:
583:
581:
576:
574:
569:
568:
562:
560:
556:
553:article is a
552:
547:
544:
540:
539:
535:
530:
527:
526:
517:
511:
507:
503:
499:
498:
489:
485:
484:
478:
476:
472:
468:
448:
432:
420:
417:
409:
401:
396:
392:
383:
382:
381:
379:
375:
371:
350:
338:
333:
328:
320:
315:
299:
298:
297:
295:
291:
287:
283:
281:
278:differential
261:
256:
242:
219:
213:
210:
207:
204:
198:
193:
185:
180:
177:
174:
170:
162:
161:
160:
158:
153:
149:
145:
140:
136:
133:
129:
126:of dimension
125:
122:
118:
114:
103:
92:
89:
85:
82:
78:
75:
71:
68:
64:
61: –
60:
56:
55:Find sources:
49:
45:
41:
35:
34:
33:single source
29:This article
27:
23:
18:
17:
559:expanding it
548:
533:
505:
474:
466:
464:
369:
367:
293:
289:
285:
279:
243:
240:
156:
151:
147:
143:
138:
134:
127:
123:
116:
110:
97:
87:
80:
73:
66:
54:
30:
155:at a point
598:Categories
515:3540518614
495:References
100:April 2024
70:newspapers
441:→
406:Γ
393:∫
339:⊗
325:Ω
253:Ω
214:−
199:
40:talk page
481:See also
378:oriented
121:manifold
84:scholar
512:
282:-forms
115:, the
86:
79:
72:
65:
57:
549:This
288:. If
130:is a
119:on a
91:JSTOR
77:books
555:stub
510:ISBN
244:Let
63:news
473:of
465:If
159:is
142:on
46:by
600::
586:e
579:t
572:v
561:.
475:M
467:M
449:.
445:R
438:)
433:M
427:V
421:,
418:M
415:(
410:c
402::
397:M
370:M
351:M
345:o
334:n
329:M
321:=
316:M
310:V
294:M
290:n
286:M
280:k
262:k
257:M
226:)
223:}
220:x
217:{
211:X
208:,
205:X
202:(
194:n
190:H
186:=
181:x
178:,
175:X
171:o
157:x
152:X
148:o
144:X
139:X
135:o
128:n
124:X
102:)
98:(
88:·
81:·
74:·
67:·
50:.
36:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
