205:
has the structure of an H-group, as equipped with the standard operations of concatenation and inversion. Furthermore a continuous basepoint preserving map of pointed topological space induces a H-homomorphism of the corresponding loop spaces; this reflects the group homomorphism on fundamental
213:
from a H-space to a pointed topological space, there is a natural H-space structure on the latter space. As such, the existence of an H-space structure on a given space is only dependent on pointed homotopy type.
439:
is not a group in this way because octonion multiplication is not associative, nor can it be given any other continuous multiplication for which it is a group.
643:
173:
is an H-space as in the above definition. Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint
624:
604:
403:
that are H-spaces. Each of these spaces forms an H-space by viewing it as the subset of norm-one elements of the
64:
716:
697:. Die Grundlehren der mathematischen Wissenschaften. Vol. 212. New York-Heidelberg: Springer-Verlag.
202:
139:
711:
721:
495:(see J. R. Hubbuck. "A Short History of H-spaces", History of topology, 1999, pages 747–755).
672:
210:
8:
660:
463:
238:
453:
419:, respectively, and using the multiplication operations from these algebras. In fact,
620:
619:(Corrected reprint of the 1966 original ed.). New York-Berlin: Springer-Verlag.
600:
594:
488:
448:
245:
194:
49:
29:
652:
222:
143:
668:
269:
226:
197:, together with the fact that it is a group, can be rephrased as saying that the
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to be an exact identity, without any consideration of homotopy. In the case of a
37:
25:
687:, Lecture Notes in Mathematics, vol. 161, Berlin-New York: Springer-Verlag
408:
376:
230:
218:
705:
590:
249:
33:
680:
634:
458:
234:
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17:
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together with a continuous multiplication for which the basepoint is an
664:
492:
412:
198:
182:
432:
127:
656:
468:
416:
400:
217:
The multiplicative structure of an H-space adds structure to its
233:
H-space with finitely generated and free cohomology groups is a
491:
in recognition of the influence exerted on the subject by
185:, all three of these definitions are in fact equivalent.
209:
It is straightforward to verify that, given a pointed
372:. It is clear how to define a homotopy from to .
644:Transactions of the American Mathematical Society
703:
28:version of a generalization of the notion of
188:
599:. Cambridge: Cambridge University Press.
130:to the identity map through maps sending
695:Algebraic topology—homotopy and homology
679:
633:
692:
614:
589:
704:
685:H-spaces from a homotopy point of view
241:on the homology groups of an H-space.
146:up to basepoint-preserving homotopy.
206:groups induced by a continuous map.
637:(1963), "Homotopy associativity of
13:
487:The H in H-space was suggested by
435:) with these multiplications. But
149:One says that a topological space
14:
733:
570:
193:The standard definition of the
561:
552:
543:
534:
525:
516:
507:
498:
481:
153:is an H-space if there exists
138:. This may be thought of as a
1:
583:
43:
256:be an H-space with identity
237:. Also, one can define the
7:
693:Switzer, Robert M. (1975).
442:
55:, together with an element
10:
738:
615:Spanier, Edwin H. (1981).
504:Spanier p.34; Switzer p.14
203:pointed topological space
140:pointed topological space
48:An H-space consists of a
32:, in which the axioms on
474:
651:(2): 275–292, 293–312,
189:Examples and properties
635:Stasheff, James Dillon
379:theorem, named after
161:such that the triple
522:Stasheff (1970), p.1
252:. To see this, let
225:. For example, the
211:homotopy equivalence
717:Algebraic topology
617:Algebraic topology
596:Algebraic topology
464:Topological monoid
377:Hopf invariant one
368:) is homotopic to
239:Pontryagin product
177:, or by requiring
26:homotopy-theoretic
641:-spaces. I, II",
489:Jean-Pierre Serre
449:Topological group
248:of an H-space is
246:fundamental group
223:cohomology groups
195:fundamental group
50:topological space
30:topological group
729:
698:
688:
675:
630:
610:
577:
574:
568:
565:
559:
558:Spanier pp.35-36
556:
550:
549:Spanier pp.37-39
547:
541:
540:Spanier pp.37-39
538:
532:
529:
523:
520:
514:
511:
505:
502:
496:
485:
340:is homotopic to
276:. Define a map
180:
176:
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156:
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144:identity element
137:
133:
125:
110:
95:
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62:
58:
54:
737:
736:
732:
731:
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728:
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712:Homotopy theory
702:
701:
681:Stasheff, James
657:10.2307/1993609
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486:
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477:
454:Čech cohomology
445:
227:cohomology ring
191:
178:
174:
162:
158:
154:
150:
135:
131:
112:
97:
82:
67:
60:
56:
52:
46:
12:
11:
5:
735:
725:
724:
719:
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631:
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612:
611:. Section 3.C
605:
591:Hatcher, Allen
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560:
551:
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515:
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461:
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451:
444:
441:
383:, states that
231:path-connected
190:
187:
65:continuous map
45:
42:
9:
6:
4:
3:
2:
734:
723:
722:Hopf algebras
720:
718:
715:
713:
710:
709:
707:
696:
691:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
645:
640:
636:
632:
628:
626:0-387-90646-0
622:
618:
613:
608:
606:0-521-79540-0
602:
598:
597:
592:
588:
587:
576:Hatcher p.287
573:
567:Hatcher p.283
564:
555:
546:
537:
531:Hatcher p.291
528:
519:
513:Hatcher p.281
510:
501:
494:
490:
484:
480:
470:
467:
465:
462:
460:
457:
455:
452:
450:
447:
446:
440:
438:
434:
430:
426:
422:
418:
414:
410:
406:
402:
399:are the only
398:
394:
390:
386:
382:
378:
373:
371:
367:
363:
359:
355:
351:
347:
343:
339:
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331:
327:
323:
319:
315:
311:
307:
303:
299:
295:
291:
287:
283:
279:
275:
271:
267:
263:
259:
255:
251:
247:
242:
240:
236:
232:
228:
224:
220:
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212:
207:
204:
200:
196:
186:
184:
170:
166:
147:
145:
141:
129:
123:
119:
115:
108:
104:
100:
96:and the maps
94:
90:
86:
79:
75:
71:
66:
51:
41:
40:are removed.
39:
35:
34:associativity
31:
27:
23:
19:
694:
684:
648:
642:
638:
616:
595:
572:
563:
554:
545:
536:
527:
518:
509:
500:
483:
459:Hopf algebra
436:
431:are groups (
428:
424:
420:
396:
392:
388:
384:
374:
369:
365:
361:
357:
353:
349:
345:
341:
337:
333:
329:
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321:
317:
313:
309:
305:
301:
297:
293:
289:
285:
281:
277:
273:
265:
261:
257:
253:
243:
235:Hopf algebra
216:
208:
192:
168:
164:
148:
121:
117:
113:
106:
102:
98:
92:
88:
84:
81:, such that
77:
73:
69:
47:
21:
15:
413:quaternions
381:Frank Adams
18:mathematics
706:Categories
584:References
493:Heinz Hopf
433:Lie groups
199:loop space
183:CW complex
44:Definition
417:octonions
409:complexes
312:). Then
128:homotopic
126:are both
68:μ :
683:(1970),
593:(2002).
469:H-object
443:See also
280:: × →
260:and let
219:homology
38:inverses
673:0158400
665:1993609
401:spheres
375:Adams'
250:abelian
22:H-space
671:
663:
623:
603:
427:, and
415:, and
344:, and
328:,1) =
320:,0) =
63:and a
661:JSTOR
475:Notes
405:reals
270:loops
229:of a
201:of a
24:is a
20:, an
621:ISBN
601:ISBN
360:) =
352:) =
296:) =
264:and
244:The
221:and
171:, μ)
157:and
116:↦ μ(
111:and
101:↦ μ(
91:) =
36:and
653:doi
649:108
356:(1,
348:(0,
284:by
272:at
268:be
134:to
59:of
16:In
708::
669:MR
667:,
659:,
647:,
423:,
411:,
407:,
395:,
391:,
387:,
362:eg
167:,
120:,
105:,
87:,
83:μ(
76:→
72:×
689:.
676:.
655::
639:H
629:.
609:.
437:S
429:S
425:S
421:S
397:S
393:S
389:S
385:S
370:g
366:b
364:(
358:b
354:F
350:b
346:F
342:f
338:e
336:)
334:a
332:(
330:f
326:a
324:(
322:F
318:a
316:(
314:F
310:b
308:(
306:g
304:)
302:a
300:(
298:f
294:b
292:,
290:a
288:(
286:F
282:X
278:F
274:e
266:g
262:f
258:e
254:X
179:e
175:e
169:e
165:X
163:(
159:μ
155:e
151:X
136:e
132:e
124:)
122:x
118:e
114:x
109:)
107:e
103:x
99:x
93:e
89:e
85:e
78:X
74:X
70:X
61:X
57:e
53:X
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