449:
592:
327:
201:
358:
95:
488:
350:
672:
495:
710:
744:
247:
121:
775:
770:
611:
444:{\displaystyle 0\to \mathbb {Z} /p\mathbb {Z} \to \mathbb {Z} /p^{2}\mathbb {Z} \to \mathbb {Z} /p\mathbb {Z} \to 0}
702:
215:
44:
56:
112:
48:
670:
Bockstein, Meyer (1958), "Sur la formule des coefficients universels pour les groupes d'homologie",
464:
20:
211:
740:
706:
455:
335:
238:
754:
720:
685:
663:
640:
750:
736:
716:
681:
659:
647:
636:
624:
28:
696:
650:(1943), "A complete system of fields of coefficients for the ∇-homological dimension",
219:
764:
728:
692:
231:
105:
101:
227:
587:{\displaystyle \beta (a\cup b)=\beta (a)\cup b+(-1)^{\dim a}a\cup \beta (b)}
230:
condition should enter). The construction of β is by the usual argument (
598:
in other words, it is a superderivation acting on the cohomology mod
218:, abelian groups, and the homology is of the complexes formed by
458:. This Bockstein homomorphism has the following two properties:
322:{\displaystyle \beta \colon H^{i}(C,R)\to H^{i+1}(C,P).}
196:{\displaystyle \beta \colon H_{i}(C,R)\to H_{i-1}(C,P).}
498:
467:
361:
338:
250:
124:
59:
241:, this time increasing degree by one. Thus we have
673:Comptes Rendus de l'Académie des Sciences, Série I
586:
482:
443:
344:
321:
195:
104:, when they are introduced as coefficients into a
89:
627:(1942), "Universal systems of ∇-homology rings",
115:groups as a homomorphism reducing degree by one,
8:
554:
497:
466:
454:is used as one of the generators of the
431:
430:
422:
418:
417:
410:
409:
403:
394:
390:
389:
382:
381:
373:
369:
368:
360:
337:
289:
261:
249:
163:
135:
123:
58:
40:
36:
32:
352:associated to the coefficient sequence
733:Algebraic topology. Corrected reprint
90:{\displaystyle 0\to P\to Q\to R\to 0}
7:
237:A similar construction applies to
14:
652:C. R. (Doklady) Acad. Sci. URSS
629:C. R. (Doklady) Acad. Sci. URSS
581:
575:
551:
541:
529:
523:
514:
502:
483:{\displaystyle \beta \beta =0}
435:
414:
386:
365:
313:
301:
282:
279:
267:
187:
175:
156:
153:
141:
81:
75:
69:
63:
1:
612:Bockstein spectral sequence
332:The Bockstein homomorphism
111:, and which appears in the
792:
703:Cambridge University Press
210:should be a complex of
45:connecting homomorphism
588:
484:
445:
346:
345:{\displaystyle \beta }
323:
197:
91:
25:Bockstein homomorphism
589:
485:
446:
347:
324:
198:
92:
739:, pp. xvi+528,
496:
465:
359:
336:
248:
206:To be more precise,
122:
57:
49:short exact sequence
776:Homological algebra
735:, New York-Berlin:
29:Meyer Bockstein
21:homological algebra
771:Algebraic topology
698:Algebraic Topology
584:
480:
441:
342:
319:
193:
87:
47:associated with a
729:Spanier, Edwin H.
712:978-0-521-79540-1
239:cohomology groups
783:
757:
723:
688:
666:
648:Bockstein, Meyer
643:
625:Bockstein, Meyer
593:
591:
590:
585:
565:
564:
489:
487:
486:
481:
456:Steenrod algebra
450:
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393:
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372:
351:
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266:
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202:
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140:
139:
96:
94:
93:
88:
27:, introduced by
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790:
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782:
781:
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761:
760:
747:
737:Springer-Verlag
727:
713:
691:
669:
646:
623:
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608:
550:
494:
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463:
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399:
357:
356:
334:
333:
285:
257:
246:
245:
159:
131:
120:
119:
55:
54:
17:
16:Homological map
12:
11:
5:
789:
787:
779:
778:
773:
763:
762:
759:
758:
745:
725:
711:
693:Hatcher, Allen
689:
667:
654:, New Series,
644:
631:, New Series,
619:
616:
615:
614:
607:
604:
596:
595:
583:
580:
577:
574:
571:
568:
563:
560:
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553:
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543:
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534:
531:
528:
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519:
516:
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491:
479:
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440:
437:
433:
429:
425:
420:
416:
412:
406:
402:
397:
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388:
384:
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376:
371:
367:
364:
341:
330:
329:
318:
315:
312:
309:
306:
303:
298:
295:
292:
288:
284:
281:
278:
275:
272:
269:
264:
260:
256:
253:
220:tensor product
214:, or at least
204:
203:
192:
189:
186:
183:
180:
177:
172:
169:
166:
162:
158:
155:
152:
149:
146:
143:
138:
134:
130:
127:
102:abelian groups
98:
97:
86:
83:
80:
77:
74:
71:
68:
65:
62:
15:
13:
10:
9:
6:
4:
3:
2:
788:
777:
774:
772:
769:
768:
766:
756:
752:
748:
746:0-387-90646-0
742:
738:
734:
730:
726:
722:
718:
714:
708:
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572:
569:
566:
561:
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555:
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544:
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532:
526:
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517:
511:
508:
505:
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492:
477:
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378:
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362:
355:
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353:
339:
316:
310:
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296:
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160:
150:
147:
144:
136:
132:
128:
125:
118:
117:
116:
114:
110:
107:
106:chain complex
103:
84:
78:
72:
66:
60:
53:
52:
51:
50:
46:
42:
38:
34:
30:
26:
22:
732:
697:
677:
671:
655:
651:
632:
628:
602:of a space.
599:
597:
453:
331:
236:
223:
216:torsion-free
207:
205:
108:
99:
24:
18:
680:: 396–398,
658:: 187–189,
635:: 243–245,
232:snake lemma
228:flat module
765:Categories
618:References
573:β
570:∪
559:
545:−
533:∪
521:β
509:∪
500:β
472:β
469:β
436:→
415:→
387:→
366:→
340:β
283:→
255::
252:β
168:−
157:→
129::
126:β
82:→
76:→
70:→
64:→
731:(1981),
695:(2002),
606:See also
113:homology
43:), is a
755:0666554
721:1867354
686:0103918
664:0009115
641:0008701
31: (
753:
743:
719:
709:
684:
662:
639:
226:(some
23:, the
222:with
741:ISBN
707:ISBN
212:free
41:1958
37:1943
33:1942
19:In
678:247
556:dim
234:).
100:of
767::
751:MR
749:,
717:MR
715:,
705:,
701:,
682:MR
676:,
660:MR
656:38
637:MR
633:37
39:,
35:,
724:.
600:p
594:;
582:)
579:b
576:(
567:a
562:a
552:)
548:1
542:(
539:+
536:b
530:)
527:a
524:(
518:=
515:)
512:b
506:a
503:(
490:,
478:0
475:=
439:0
432:Z
428:p
424:/
419:Z
411:Z
405:2
401:p
396:/
391:Z
383:Z
379:p
375:/
370:Z
363:0
317:.
314:)
311:P
308:,
305:C
302:(
297:1
294:+
291:i
287:H
280:)
277:R
274:,
271:C
268:(
263:i
259:H
224:C
208:C
191:.
188:)
185:P
182:,
179:C
176:(
171:1
165:i
161:H
154:)
151:R
148:,
145:C
142:(
137:i
133:H
109:C
85:0
79:R
73:Q
67:P
61:0
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