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subset of the plane, has all homotopy groups zero, but the map from the Warsaw circle to a single point is not a homotopy equivalence. The study of possible generalizations of
Whitehead's theorem to more general spaces is part of the subject of
264:
395:
511:
148:
91:
in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of
442:.) The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map
730:
95:, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping.
714:
706:
174:
628:; thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the
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657:
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434:, the first condition is automatic, and it suffices to state the second condition for a single point
84:
35:
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166:
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The
Whitehead theorem does not hold for general topological spaces or even for all subspaces of
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466:, which is not at all clear from the assumptions.) This implies the same conclusion for spaces
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When a mapping that induces isomorphisms on all homotopy groups is a homotopy equivalence
17:
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614:
672:, a weak equivalence between cofibrant-fibrant objects is a homotopy equivalence.
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401:
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Spaces with isomorphic homotopy groups may not be homotopy equivalent
573:
inducing an isomorphism on homotopy groups. For instance, take
259:{\displaystyle f_{*}\colon \pi _{n}(X,x)\to \pi _{n}(Y,f(x)),}
701:, Cambridge University Press, Cambridge, 2002. xii+544 pp.
517:
CW complexes that induces an isomorphism on all integral
390:{\displaystyle f_{*}\colon \pi _{0}(X)\to \pi _{0}(Y)}
487:
332:
177:
124:
692:, Bull. Amer. Math. Soc., 55 (1949), 453–496
685:, Bull. Amer. Math. Soc., 55 (1949), 213–245
505:
389:
258:
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561:are homotopy equivalent. One really needs a map
529:A word of caution: it is not enough to assume π
474:that are homotopy equivalent to CW complexes.
8:
481:yields a useful corollary: a continuous map
620:/2, and the same universal cover, namely
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664:Generalization to model categories
521:groups is a homotopy equivalence.
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640:are not homotopy equivalent.
506:{\displaystyle f\colon X\to Y}
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143:{\displaystyle f\colon X\to Y}
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115:. Given a continuous mapping
87:. This result was proved by
731:Theorems in homotopy theory
690:Combinatorial homotopy. II.
747:
683:Combinatorial homotopy. I.
553:in order to conclude that
29:
321:weak homotopy equivalence
477:Combining this with the
404:, and the homomorphisms
307:) just means the set of
30:Not to be confused with
18:Whitehead's theorem
454:has a homotopy inverse
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411:are bijective for all
391:
287:-th homotopy group of
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688:J. H. C. Whitehead,
681:J. H. C. Whitehead,
539:) is isomorphic to π
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330:
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103:In more detail, let
85:homotopy equivalence
36:Whitehead conjecture
647:. For example, the
161:, consider for any
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113:topological spaces
93:algebraic topology
89:J. H. C. Whitehead
55:continuous mapping
717:(see Theorem 4.5)
611:fundamental group
53:states that if a
51:Whitehead theorem
32:Whitehead problem
16:(Redirected from
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515:simply connected
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695:A. Hatcher,
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167:homomorphism
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153:and a point
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73:isomorphisms
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62:CW complexes
57:
50:
40:
549:) for each
47:mathematics
676:References
423:≥ 1. (For
632:); thus,
498:→
492::
402:bijective
370:π
366:→
348:π
344::
339:∗
315:.) A map
221:π
217:→
193:π
189::
184:∗
135:→
129::
99:Statement
725:Category
565: :
519:homology
513:between
419:and all
71:induces
60:between
668:In any
653:compact
601:. Then
295:. (For
269:where π
79:, then
75:on all
49:), the
713:
705:
299:= 0, π
319:is a
83:is a
711:ISBN
709:and
703:ISBN
651:, a
636:and
605:and
589:and
557:and
470:and
427:and
107:and
67:and
438:in
415:in
400:is
311:of
157:in
111:be
41:In
34:or
727::
660:.
624:×
597:×
595:RP
593:=
586:RP
583:×
577:=
569:→
462:→
458::
450:→
446::
645:R
638:Y
634:X
626:S
622:S
618:Z
607:Y
603:X
599:S
591:Y
580:S
575:X
571:Y
567:X
563:f
559:Y
555:X
551:n
547:Y
545:(
542:n
537:X
535:(
532:n
501:Y
495:X
489:f
472:Y
468:X
464:X
460:Y
456:g
452:Y
448:X
444:f
440:X
436:x
429:Y
425:X
421:n
417:X
413:x
409:*
406:f
385:)
382:Y
379:(
374:0
363:)
360:X
357:(
352:0
335:f
317:f
313:X
305:X
303:(
301:0
297:n
293:x
289:X
285:n
281:x
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277:X
275:(
272:n
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251:)
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239:f
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230:(
225:n
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211:x
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205:X
202:(
197:n
180:f
163:n
159:X
155:x
138:Y
132:X
126:f
109:Y
105:X
81:f
69:Y
65:X
58:f
38:.
20:)
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