5875:
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5896:
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5933:
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of a one-dimensional space, the "loop of string" (or the circle), into this space, and this embedding gives a homeomorphism between the circle and its image in the embedding space. The intuitive idea behind the notion of knot equivalence is that one can
3165:
2601:
1845:
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2415:
1757:
1340:
1435:
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2451:
2739:
1163:
1982:
1555:
3058:
2270:
A solid disk is homotopy-equivalent to a single point, since you can deform the disk along radial lines continuously to a single point. However, they are not homeomorphic, since there is no
2359:
1977:
1550:
1966:
1905:
1473:
1221:
1279:
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754:
2904:
4592:
4427:
2507:
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are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. Spaces that are homotopy-equivalent to a point are called
869:
328:
3014:
593:
3957:
Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval into the real numbers defined by
2779:
444:
394:
5093:
554:
4163:
Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence. For example, a path between two smooth embeddings is a
1531:
905:
792:
697:
664:
5043:
4484:
3063:
470:
4619:
3426:
3376:
2806:
2631:
2478:
631:
4700:
3242:
5936:
3509:
3489:
3466:
3446:
3396:
3346:
3322:
3298:
3262:
3211:
3188:
2512:
812:
5001:
4730:
is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces
4079:—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots,
4373:, which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with
5967:
2281:
and an untwisted (closed) strip are homotopy equivalent, since you can deform both strips continuously to a circle. But they are not homeomorphic.
3897:
since one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space. Thus they are not ambient-isotopic.
5539:
5484:
5292:
1774:
4195:(CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be
235:
4028:
Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic using
2918:
5570:
278:
38:
5924:
5919:
5448:
5327:
5264:
3989:
to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover,
2364:
187:
136:
4132:, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots
954:
is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of
1702:
5914:
2808:
homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since
1284:
3684:
1345:
259:
58:
2811:
2420:
5816:
5522:
5504:
4742:
on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.
2684:
1119:
2266:. Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples:
3680:
214:
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with
5008:
4128:
embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy. An
3019:
5517:
5499:
4370:
4208:
4191:
is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. No
2090:{\displaystyle {\begin{aligned}H:B^{n}\times &\longrightarrow B^{n}\\(x,t)&\longmapsto (1-t)x.\end{aligned}}}
1688:{\displaystyle {\begin{aligned}H:\times &\longrightarrow C\\(s,t)&\longmapsto (1-t)f(s)+tg(s).\end{aligned}}}
5962:
5824:
5120:
223:
2323:
4672:
We can define the action of one equivalence class on another, and so we get a group. These groups are called the
4225:
3214:
1913:
1850:
1443:
215:
1168:
5957:
5895:
5623:
4926:
1226:
507:. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from
5909:
2640:
204:
5512:
705:
5844:
5839:
5765:
5642:
5630:
5603:
5563:
5141:
4933:
4922:
4914:
3568:
3268:
2877:
70:
5686:
5613:
4937:
4918:
4192:
3823:
971:
varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as
5874:
4548:
4394:
2483:
2294:
250:: as "the surface of a doughnut" and as "the surface of a coffee mug". This is also an example of an
45:
shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.
5834:
5786:
5760:
5608:
5395:
5131:
4624:
4500:
4029:
1969:
1016:
1004:
817:
283:
3923:, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of
5885:
5681:
5136:
5100:
5050:
4910:
4495:
4176:
3839:
4183:(representing something that only goes forwards, not backwards, in time, in every local frame). A
3997:
has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from
2987:
950:
that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts;
559:
5879:
5829:
5750:
5740:
5618:
5598:
5419:
5385:
5288:
5104:
4766:
4313:
4196:
4072:
3520:
2752:
1768:
399:
349:
208:
93:"place") if one can be "continuously deformed" into the other, such a deformation being called a
5849:
5056:
4948:
into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group
521:
3160:{\displaystyle \mathbb {R} ^{k}\times (\mathbb {R} ^{n-k}-\{0\})\to (\mathbb {R} ^{n-k}-\{0\})}
1486:
5867:
5733:
5691:
5556:
5535:
5494:
5480:
5454:
5444:
5411:
5352:
5333:
5323:
5270:
5260:
4906:
4726:
4715:
4703:
4487:
4430:
4184:
3701:
3679:
An example of an algebraic invariant of topological spaces which is not homotopy-invariant is
3599:
3584:
2169:
2128:
1908:
1760:
874:
759:
700:
200:
62:
669:
636:
5647:
5593:
5403:
5146:
5013:
4444:
4204:
3572:
3557:
449:
331:
148:
100:
31:
4597:
3404:
3354:
2784:
2609:
2456:
598:
5706:
5701:
4721:
4129:
3951:
3543:
2596:{\displaystyle H(t,\cdot )=t\cdot p_{0}+(1-t)\cdot \operatorname {id} _{\mathbb {R} ^{n}}}
1534:
1480:
196:
42:
4679:
4106:
one embedding to another through a path of embeddings: a continuous function starting at
3219:
2278:
5399:
5796:
5728:
4854:
4778:
4673:
4386:
4188:
3611:
3494:
3474:
3451:
3431:
3381:
3331:
3307:
3283:
3247:
3196:
3173:
910:
The animation that is looped above right provides an example of a homotopy between two
797:
4959:
4429:
being homotopic relative to a subspace is an equivalence relation, we can look at the
5951:
5806:
5716:
5696:
5423:
4834:
4738:
are isomorphic in this category if and only if they are homotopy-equivalent. Then a
2910:
2231:
339:
5899:
5791:
5711:
5657:
3894:
3708:. These are homotopies which keep the elements of the subspace fixed. Formally: if
3398:
is null-homotopic precisely when it can be continuously extended to a map from the
2747:
2219:
518:
An alternative notation is to say that a homotopy between two continuous functions
234:
17:
5889:
5801:
4374:
4076:
3880:
3511:
to itself—which is always a homotopy equivalence—is null-homotopic.
3349:
54:
3527:, that is, they respect the relation of homotopy equivalence. For example, if
5745:
5676:
5635:
5407:
5298:
5213:
5186:
3191:
1476:
219:
5415:
5355:
5165:
5770:
5458:
5360:
5337:
5274:
5126:
4363:
4098:
4094:
4033:
3920:
3399:
2271:
911:
476:
239:
195:) between the two functions. A notable use of homotopy is the definition of
37:
5237:
5755:
5672:
5579:
5438:
4200:
4180:
4061:
2634:
50:
5238:"algebraic topology - Path homotopy and separately continuous functions"
3871:
5303:
5191:
4739:
4146:
are considered equivalent when there is an ambient isotopy which moves
2274:
between them (since one is an infinite set, while the other is finite).
1840:{\displaystyle B^{n}:=\left\{x\in \mathbb {R} ^{n}:\|x\|\leq 1\right\}}
5390:
5376:
Monroe, Hunter (2008-11-01). "Are
Causality Violations Undesirable?".
4917:
have been developed. The methods for algebraic equations include the
3890:
3729:
88:
77:
2361:. The part that needs to be checked is the existence of a homotopy
4545:
as a subspace, then the equivalence classes form a group, denoted
4160:. This is the appropriate definition in the topological category.
243:
233:
36:
2977:{\displaystyle \mathbb {R} ^{n}-\mathbb {R} ^{k}\simeq S^{n-k-1}}
5552:
3304:
if it is homotopic to a constant function. (The homotopy from
2913:
is a fiber bundle with a fiber homotopy equivalent to a point.
2410:{\displaystyle H:I\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
178:
127:
4720:
The idea of homotopy can be turned into a formal category of
83:
991:
are said to be homotopic if and only if there is a homotopy
5548:
172:
163:
157:
121:
106:
1752:{\displaystyle \operatorname {id} _{B^{n}}:B^{n}\to B^{n}}
115:
4068:
are isotopic because they can be connected by rotations.
1335:{\displaystyle H:\mathbb {R} \times \to \mathbb {R} ^{2}}
975:
varies back from 1 to 0, pauses, and repeats this cycle.
3931:, in the notation used before, such that for each fixed
3614:. (Without the path-connectedness assumption, one has π
1430:{\displaystyle H(x,t)=\left(x,(1-t)x^{3}+te^{x}\right)}
27:
Continuous deformation between two continuous functions
5259:. Cambridge: Cambridge University Press. p. 185.
4925:). The methods for differential equations include the
4362:
The homotopy lifting property is used to characterize
2867:{\displaystyle \pi :\mathbb {R} ^{n}-\{0\}\to S^{n-1}}
2446:{\displaystyle \operatorname {id} _{\mathbb {R} ^{n}}}
2234:
is a special case of a homotopy equivalence, in which
5322:. Cambridge: Cambridge University Press. p. 11.
5059:
5016:
4962:
4682:
4627:
4600:
4551:
4503:
4447:
4397:
3497:
3491:
is contractible if and only if the identity map from
3477:
3454:
3434:
3407:
3384:
3357:
3334:
3310:
3286:
3250:
3222:
3199:
3176:
3066:
3022:
2990:
2921:
2880:
2814:
2787:
2755:
2687:
2643:
2612:
2515:
2486:
2459:
2423:
2367:
2326:
2297:
1980:
1916:
1853:
1777:
1705:
1553:
1489:
1446:
1348:
1287:
1229:
1171:
1122:
877:
820:
800:
762:
708:
672:
639:
601:
562:
524:
452:
402:
352:
286:
188:
166:
137:
4845:
is pointed, then the group homomorphisms induced by
2734:{\displaystyle \mathbb {R} ^{n}-\{0\}\simeq S^{n-1}}
1158:{\displaystyle f,g:\mathbb {R} \to \mathbb {R} ^{2}}
175:
169:
124:
118:
5815:
5779:
5665:
5586:
3687:, and compactification is not homotopy-invariant).
160:
154:
151:
112:
109:
103:
30:This article is about topology. For chemistry, see
5087:
5037:
4995:
4694:
4661:
4613:
4586:
4537:
4478:
4421:
4369:Another useful property involving homotopy is the
3503:
3483:
3460:
3440:
3420:
3390:
3370:
3340:
3324:to a constant function is then sometimes called a
3316:
3292:
3256:
3236:
3205:
3182:
3159:
3052:
3008:
2976:
2898:
2866:
2800:
2773:
2733:
2670:
2625:
2595:
2501:
2472:
2445:
2409:
2353:
2312:
2089:
1960:
1899:
1839:
1751:
1687:
1525:
1467:
1429:
1334:
1273:
1215:
1157:
899:
863:
806:
786:
748:
691:
658:
625:
587:
548:
464:
438:
388:
322:
3683:(which is, roughly speaking, the homology of the
3053:{\displaystyle \mathbb {R} ^{n}-\mathbb {R} ^{k}}
1972:. Then the following is a homotopy between them:
515:as the slider moves from 0 to 1, and vice versa.
5187:"Homotopy Type Theory Discussed - Computerphile"
4932:Homotopy theory can be used as a foundation for
4005:: × → given by
3993:has changed the orientation of the interval and
3167:, then applying the homotopy equivalences above.
4064:around the origin, and so the identity map and
3838:is the identity map, this is known as a strong
3471:It follows from these definitions that a space
2291:The first example of a homotopy equivalence is
5440:Introduction to numerical continuation methods
946:is some continuous function from the torus to
5564:
5107:for singular cohomology with coefficients in
5003:of based homotopy classes of based maps from
3950:A related, but different, concept is that of
3519:Homotopy equivalence is important because in
1015:. This homotopy relation is compatible with
8:
5123:(relative version of a homotopy equivalence)
3151:
3145:
3112:
3106:
2842:
2836:
2709:
2703:
2665:
2659:
2354:{\displaystyle \mathbb {R} ^{n}\simeq \{0\}}
2348:
2342:
1823:
1817:
1007:on the set of all continuous functions from
4199:by timelike curves. A manifold such as the
1961:{\displaystyle c_{\vec {0}}(x):={\vec {0}}}
1900:{\displaystyle c_{\vec {0}}:B^{n}\to B^{n}}
1468:{\displaystyle C\subseteq \mathbb {R} ^{n}}
871:. It is not sufficient to require each map
5932:
5905:
5571:
5557:
5549:
2509:onto the origin. This can be described as
1216:{\displaystyle f(x):=\left(x,x^{3}\right)}
1003:as described above. Being homotopic is an
5389:
5064:
5058:
5015:
4961:
4681:
4650:
4626:
4605:
4599:
4575:
4556:
4550:
4526:
4502:
4470:
4446:
4396:
3496:
3476:
3453:
3433:
3412:
3406:
3383:
3362:
3356:
3333:
3309:
3285:
3249:
3226:
3221:
3198:
3175:
3130:
3126:
3125:
3091:
3087:
3086:
3073:
3069:
3068:
3065:
3044:
3040:
3039:
3029:
3025:
3024:
3021:
2989:
2956:
2943:
2939:
2938:
2928:
2924:
2923:
2920:
2887:
2883:
2882:
2879:
2852:
2827:
2823:
2822:
2813:
2792:
2786:
2754:
2719:
2694:
2690:
2689:
2686:
2650:
2646:
2645:
2642:
2617:
2611:
2585:
2581:
2580:
2578:
2547:
2514:
2493:
2489:
2488:
2485:
2464:
2458:
2435:
2431:
2430:
2428:
2422:
2401:
2397:
2396:
2386:
2382:
2381:
2366:
2333:
2329:
2328:
2325:
2304:
2300:
2299:
2296:
2030:
1995:
1981:
1979:
1947:
1946:
1922:
1921:
1915:
1891:
1878:
1859:
1858:
1852:
1808:
1804:
1803:
1782:
1776:
1743:
1730:
1715:
1710:
1704:
1554:
1552:
1537:with the same endpoints, then there is a
1488:
1459:
1455:
1454:
1445:
1416:
1400:
1347:
1326:
1322:
1321:
1295:
1294:
1286:
1274:{\displaystyle g(x)=\left(x,e^{x}\right)}
1260:
1228:
1202:
1170:
1149:
1145:
1144:
1136:
1135:
1121:
882:
876:
825:
819:
799:
761:
731:
707:
677:
671:
644:
638:
600:
567:
561:
523:
451:
401:
351:
285:
4921:method and the continuation method (see
2606:There is a homotopy equivalence between
5157:
4749:homotopy invariant: this means that if
3060:as the total space of the fiber bundle
1067:are homotopic, then their compositions
814:. The two versions coincide by setting
4905:Based on the concept of the homotopy,
4179:, certain curves are distinguished as
3610:are isomorphic, and so are the higher
3535:are homotopy equivalent spaces, then:
2671:{\displaystyle \mathbb {R} ^{2}-\{0\}}
2226:Homotopy equivalence vs. homeomorphism
5103:of Eilenberg-MacLane spaces are
749:{\displaystyle (x,t)\mapsto h_{t}(x)}
7:
4391:Since the relation of two functions
3903:When two given continuous functions
2899:{\displaystyle \mathbb {R} _{>0}}
556:is a family of continuous functions
5166:"Homotopy Definition & Meaning"
4745:For example, homology groups are a
4207:(by any type of curve), and yet be
963:(X) as a function of the parameter
503:and at time 1 we have the function
5443:. Kurt Georg. Philadelphia: SIAM.
4628:
4504:
4032:. For this reason, the map of the
25:
5045:is in natural bijection with the
4587:{\displaystyle \pi _{n}(Y,y_{0})}
499:: at time 0 we have the function
258:Formally, a homotopy between two
5931:
5904:
5894:
5884:
5873:
5863:
5862:
5656:
5530:Spanier, Edwin (December 1994).
4940:a cohomology functor on a space
4621:is in the image of the subspace
4422:{\displaystyle f,g\colon X\to Y}
4220:Lifting and extension properties
3879:
3870:
2502:{\displaystyle \mathbb {R} ^{n}}
2313:{\displaystyle \mathbb {R} ^{n}}
2250:(not only homotopic to it), and
147:
99:
5299:"History of algebraic topology"
4662:{\displaystyle \partial (^{n})}
4538:{\displaystyle \partial (^{n})}
3706:homotopy relative to a subspace
2244:is equal to the identity map id
1968:which sends every point to the
864:{\displaystyle h_{t}(x)=H(x,t)}
323:{\displaystyle H:X\times \to Y}
5968:Theory of continuous functions
5082:
5070:
5032:
5020:
4990:
4987:
4975:
4963:
4656:
4647:
4634:
4631:
4581:
4562:
4532:
4523:
4510:
4507:
4467:
4454:
4413:
4060:) is isotopic to a 180-degree
3662:is a homotopy equivalence and
3154:
3121:
3118:
3115:
3082:
2845:
2765:
2568:
2556:
2531:
2519:
2392:
2194:. If such a pair exists, then
2074:
2062:
2059:
2052:
2040:
2023:
2016:
2004:
1952:
1940:
1934:
1927:
1884:
1864:
1736:
1675:
1669:
1657:
1651:
1645:
1633:
1630:
1623:
1611:
1601:
1594:
1582:
1576:
1564:
1517:
1514:
1502:
1393:
1381:
1364:
1352:
1317:
1314:
1302:
1239:
1233:
1181:
1175:
1140:
894:
888:
858:
846:
837:
831:
781:
769:
743:
737:
724:
721:
709:
620:
608:
579:
540:
433:
427:
418:
406:
383:
377:
368:
356:
314:
311:
299:
1:
3598:are path-connected, then the
2874:is a fiber bundle with fiber
2107:Given two topological spaces
3681:compactly supported homology
3009:{\displaystyle 0\leq k<n}
588:{\displaystyle h_{t}:X\to Y}
251:
89:
78:
5518:Encyclopedia of Mathematics
5500:Encyclopedia of Mathematics
4952:, and any based CW-complex
4837:, and the homotopy between
4371:homotopy extension property
4209:timelike multiply connected
4121: = 1 giving the
3911:from the topological space
3748:if there exists a homotopy
2774:{\displaystyle \pi :E\to B}
1437:is a homotopy between them.
1019:in the following sense: if
439:{\displaystyle H(x,1)=g(x)}
389:{\displaystyle H(x,0)=f(x)}
5984:
5825:Banach fixed-point theorem
5242:Mathematics Stack Exchange
5121:Fiber-homotopy equivalence
5088:{\displaystyle H^{n}(X,G)}
4713:
4384:
4223:
4110: = 0 giving the
3744:are homotopic relative to
3704:, one needs the notion of
3271:is a homotopy equivalence.
3244:is homotopy equivalent to
3213:is contractible, then the
2210:. Intuitively, two spaces
549:{\displaystyle f,g:X\to Y}
475:If we think of the second
269:from a topological space
216:compactly generated spaces
84:
74:
29:
5858:
5654:
5408:10.1007/s10701-008-9254-9
4226:Homotopy lifting property
3973:isotopic to the identity
3915:to the topological space
3893:is not equivalent to the
3716:are continuous maps from
1526:{\displaystyle f,g:\to C}
5475:Armstrong, M.A. (1979).
5437:Allgower, E. L. (2003).
5318:Allen., Hatcher (2002).
5255:Allen., Hatcher (2002).
5214:"Homotopy | mathematics"
4927:homotopy analysis method
4765:are homotopic, then the
4702:, it is also called the
4433:of maps between a fixed
4323:), then we can lift all
2127:is a pair of continuous
900:{\displaystyle h_{t}(x)}
787:{\displaystyle X\times }
5513:"Isotopy (in topology)"
5218:Encyclopedia Britannica
5009:Eilenberg–MacLane space
4494:times, and we take its
4260:and we are given a map
3927:, which is a homotopy,
3700:In order to define the
3328:.) For example, a map
692:{\displaystyle h_{1}=g}
659:{\displaystyle h_{0}=f}
273:to a topological space
238:A homotopy between two
5880:Mathematics portal
5780:Metrics and properties
5766:Second-countable space
5378:Foundations of Physics
5297:Albin, Pierre (2019).
5089:
5039:
5038:{\displaystyle K(G,n)}
4997:
4923:numerical continuation
4915:differential equations
4696:
4663:
4615:
4588:
4539:
4480:
4479:{\displaystyle X=^{n}}
4423:
4230:If we have a homotopy
4075:—for example in
3947:) gives an embedding.
3505:
3485:
3462:
3442:
3422:
3392:
3372:
3342:
3318:
3294:
3269:deformation retraction
3258:
3238:
3207:
3184:
3161:
3054:
3010:
2978:
2900:
2868:
2802:
2775:
2735:
2672:
2627:
2597:
2503:
2474:
2447:
2411:
2355:
2320:with a point, denoted
2314:
2091:
1962:
1901:
1841:
1753:
1689:
1543:straight-line homotopy
1527:
1469:
1431:
1336:
1275:
1217:
1159:
901:
865:
808:
788:
750:
693:
660:
627:
589:
550:
489:continuous deformation
466:
465:{\displaystyle x\in X}
440:
390:
324:
255:
65:to another are called
46:
5090:
5040:
4998:
4919:homotopy continuation
4697:
4664:
4616:
4614:{\displaystyle y_{0}}
4589:
4540:
4486:, the unit interval
4481:
4424:
4193:closed timelike curve
4117:embedding, ending at
3985:. Any homotopy from
3854:is a point, the term
3506:
3486:
3463:
3443:
3423:
3421:{\displaystyle D^{2}}
3393:
3373:
3371:{\displaystyle S^{1}}
3343:
3319:
3295:
3259:
3239:
3208:
3185:
3162:
3055:
3011:
2979:
2901:
2869:
2803:
2801:{\displaystyle F_{b}}
2776:
2736:
2673:
2628:
2626:{\displaystyle S^{1}}
2598:
2504:
2475:
2473:{\displaystyle p_{0}}
2448:
2412:
2356:
2315:
2092:
1963:
1902:
1842:
1754:
1690:
1528:
1470:
1432:
1337:
1276:
1218:
1160:
983:Continuous functions
902:
866:
809:
789:
751:
694:
661:
628:
626:{\displaystyle t\in }
590:
551:
467:
441:
391:
325:
237:
40:
5835:Invariance of domain
5787:Euler characteristic
5761:Bundle (mathematics)
5132:Homotopy type theory
5099:. One says that the
5057:
5014:
4960:
4857:are also the same: π
4680:
4625:
4598:
4549:
4501:
4445:
4395:
4097:space. A knot is an
3495:
3475:
3452:
3432:
3405:
3382:
3355:
3332:
3308:
3284:
3248:
3220:
3197:
3174:
3064:
3020:
2988:
2919:
2878:
2812:
2785:
2753:
2685:
2641:
2610:
2513:
2484:
2480:, the projection of
2457:
2421:
2365:
2324:
2295:
2168:is homotopic to the
2117:homotopy equivalence
2103:Homotopy equivalence
1978:
1914:
1851:
1775:
1703:
1551:
1487:
1444:
1346:
1285:
1227:
1169:
1120:
1107:are also homotopic.
1017:function composition
1005:equivalence relation
922:, of the torus into
875:
818:
798:
760:
706:
670:
637:
599:
560:
522:
450:
400:
350:
284:
260:continuous functions
82:"same, similar" and
59:continuous functions
5845:Tychonoff's theorem
5840:Poincaré conjecture
5594:General (point-set)
5400:2008FoPh...38.1065M
5142:Poincaré conjecture
5137:Mapping class group
5105:representing spaces
5095: of the space
5051:singular cohomology
4907:computation methods
4767:group homomorphisms
4695:{\displaystyle n=1}
4431:equivalence classes
4177:Lorentzian manifold
4001:to the identity is
3840:deformation retract
3736:, then we say that
3237:{\displaystyle X/A}
2204:homotopy equivalent
1440:More generally, if
1043:are homotopic, and
756:is continuous from
279:continuous function
277:is defined to be a
5830:De Rham cohomology
5751:Polyhedral complex
5741:Simplicial complex
5532:Algebraic Topology
5353:Weisstein, Eric W.
5320:Algebraic topology
5257:Algebraic topology
5085:
5035:
4993:
4692:
4659:
4611:
4584:
4535:
4476:
4419:
4197:multiply connected
4073:geometric topology
3600:fundamental groups
3525:homotopy invariant
3523:many concepts are
3521:algebraic topology
3501:
3481:
3458:
3438:
3418:
3388:
3368:
3338:
3314:
3290:
3254:
3234:
3203:
3180:
3157:
3050:
3006:
2974:
2896:
2864:
2798:
2771:
2731:
2668:
2623:
2593:
2499:
2470:
2443:
2407:
2351:
2310:
2188:is homotopic to id
2087:
2085:
1958:
1897:
1837:
1749:
1685:
1683:
1523:
1465:
1427:
1332:
1271:
1213:
1155:
907:to be continuous.
897:
861:
804:
784:
746:
689:
656:
623:
585:
546:
462:
436:
386:
320:
256:
209:algebraic topology
47:
5963:Maps of manifolds
5945:
5944:
5734:fundamental group
5541:978-0-387-94426-5
5486:978-0-387-90839-7
5384:(11): 1065–1069.
4727:homotopy category
4716:Homotopy category
4710:Homotopy category
4704:fundamental group
4185:timelike homotopy
4171:Timelike homotopy
4030:Alexander's trick
3702:fundamental group
3696:Relative homotopy
3629:) isomorphic to π
3573:cohomology groups
3504:{\displaystyle X}
3484:{\displaystyle X}
3468:on the boundary.
3461:{\displaystyle f}
3448:that agrees with
3441:{\displaystyle X}
3391:{\displaystyle X}
3341:{\displaystyle f}
3317:{\displaystyle f}
3293:{\displaystyle f}
3257:{\displaystyle X}
3206:{\displaystyle X}
3183:{\displaystyle A}
2206:, or of the same
1955:
1930:
1909:constant function
1867:
1761:identity function
807:{\displaystyle Y}
230:Formal definition
201:cohomotopy groups
63:topological space
18:Relative homotopy
16:(Redirected from
5975:
5935:
5934:
5908:
5907:
5898:
5888:
5878:
5877:
5866:
5865:
5660:
5573:
5566:
5559:
5550:
5545:
5526:
5508:
5490:
5463:
5462:
5434:
5428:
5427:
5393:
5373:
5367:
5366:
5365:
5348:
5342:
5341:
5315:
5309:
5308:
5285:
5279:
5278:
5252:
5246:
5245:
5234:
5228:
5227:
5225:
5224:
5210:
5204:
5203:
5201:
5199:
5183:
5177:
5176:
5174:
5172:
5162:
5147:Regular homotopy
5094:
5092:
5091:
5086:
5069:
5068:
5044:
5042:
5041:
5036:
5002:
5000:
4999:
4996:{\displaystyle }
4994:
4853:on the level of
4833:are in addition
4825:. Likewise, if
4777:on the level of
4701:
4699:
4698:
4693:
4668:
4666:
4665:
4660:
4655:
4654:
4620:
4618:
4617:
4612:
4610:
4609:
4593:
4591:
4590:
4585:
4580:
4579:
4561:
4560:
4544:
4542:
4541:
4536:
4531:
4530:
4485:
4483:
4482:
4477:
4475:
4474:
4428:
4426:
4425:
4420:
4361:
4355:
4344:
4343:
4338:× →
4333:
4308:
4302:
4298:
4280:
4279:
4266:
4259:
4254:
4243:
4239:× →
4205:simply connected
3883:
3874:
3856:pointed homotopy
3817:
3810:
3800:
3761:
3685:compactification
3675:
3661:
3558:simply connected
3510:
3508:
3507:
3502:
3490:
3488:
3487:
3482:
3467:
3465:
3464:
3459:
3447:
3445:
3444:
3439:
3427:
3425:
3424:
3419:
3417:
3416:
3397:
3395:
3394:
3389:
3377:
3375:
3374:
3369:
3367:
3366:
3347:
3345:
3344:
3339:
3323:
3321:
3320:
3315:
3299:
3297:
3296:
3291:
3263:
3261:
3260:
3255:
3243:
3241:
3240:
3235:
3230:
3212:
3210:
3209:
3204:
3189:
3187:
3186:
3181:
3170:If a subcomplex
3166:
3164:
3163:
3158:
3141:
3140:
3129:
3102:
3101:
3090:
3078:
3077:
3072:
3059:
3057:
3056:
3051:
3049:
3048:
3043:
3034:
3033:
3028:
3015:
3013:
3012:
3007:
2983:
2981:
2980:
2975:
2973:
2972:
2948:
2947:
2942:
2933:
2932:
2927:
2905:
2903:
2902:
2897:
2895:
2894:
2886:
2873:
2871:
2870:
2865:
2863:
2862:
2832:
2831:
2826:
2807:
2805:
2804:
2799:
2797:
2796:
2780:
2778:
2777:
2772:
2740:
2738:
2737:
2732:
2730:
2729:
2699:
2698:
2693:
2681:More generally,
2677:
2675:
2674:
2669:
2655:
2654:
2649:
2632:
2630:
2629:
2624:
2622:
2621:
2602:
2600:
2599:
2594:
2592:
2591:
2590:
2589:
2584:
2552:
2551:
2508:
2506:
2505:
2500:
2498:
2497:
2492:
2479:
2477:
2476:
2471:
2469:
2468:
2452:
2450:
2449:
2444:
2442:
2441:
2440:
2439:
2434:
2416:
2414:
2413:
2408:
2406:
2405:
2400:
2391:
2390:
2385:
2360:
2358:
2357:
2352:
2338:
2337:
2332:
2319:
2317:
2316:
2311:
2309:
2308:
2303:
2259:
2243:
2187:
2167:
2157:
2143:
2096:
2094:
2093:
2088:
2086:
2035:
2034:
2000:
1999:
1967:
1965:
1964:
1959:
1957:
1956:
1948:
1933:
1932:
1931:
1923:
1906:
1904:
1903:
1898:
1896:
1895:
1883:
1882:
1870:
1869:
1868:
1860:
1846:
1844:
1843:
1838:
1836:
1832:
1813:
1812:
1807:
1787:
1786:
1758:
1756:
1755:
1750:
1748:
1747:
1735:
1734:
1722:
1721:
1720:
1719:
1694:
1692:
1691:
1686:
1684:
1532:
1530:
1529:
1524:
1474:
1472:
1471:
1466:
1464:
1463:
1458:
1436:
1434:
1433:
1428:
1426:
1422:
1421:
1420:
1405:
1404:
1341:
1339:
1338:
1333:
1331:
1330:
1325:
1298:
1280:
1278:
1277:
1272:
1270:
1266:
1265:
1264:
1222:
1220:
1219:
1214:
1212:
1208:
1207:
1206:
1164:
1162:
1161:
1156:
1154:
1153:
1148:
1139:
1106:
1082:
1066:
1042:
941:
927:
906:
904:
903:
898:
887:
886:
870:
868:
867:
862:
830:
829:
813:
811:
810:
805:
793:
791:
790:
785:
755:
753:
752:
747:
736:
735:
698:
696:
695:
690:
682:
681:
665:
663:
662:
657:
649:
648:
632:
630:
629:
624:
594:
592:
591:
586:
572:
571:
555:
553:
552:
547:
471:
469:
468:
463:
445:
443:
442:
437:
395:
393:
392:
387:
329:
327:
326:
321:
191:
185:
184:
181:
180:
177:
174:
171:
168:
165:
162:
159:
156:
153:
141:
134:
133:
130:
129:
126:
123:
120:
117:
114:
111:
108:
105:
92:
87:
86:
81:
76:
32:Homotopic groups
21:
5983:
5982:
5978:
5977:
5976:
5974:
5973:
5972:
5958:Homotopy theory
5948:
5947:
5946:
5941:
5872:
5854:
5850:Urysohn's lemma
5811:
5775:
5661:
5652:
5624:low-dimensional
5582:
5577:
5542:
5529:
5511:
5493:
5487:
5474:
5471:
5466:
5451:
5436:
5435:
5431:
5375:
5374:
5370:
5351:
5350:
5349:
5345:
5330:
5317:
5316:
5312:
5296:
5293:Wayback Machine
5286:
5282:
5267:
5254:
5253:
5249:
5236:
5235:
5231:
5222:
5220:
5212:
5211:
5207:
5197:
5195:
5185:
5184:
5180:
5170:
5168:
5164:
5163:
5159:
5155:
5117:
5060:
5055:
5054:
5012:
5011:
4958:
4957:
4944:by mappings of
4934:homology theory
4903:
4892:
4882:
4872:
4862:
4855:homotopy groups
4816:
4806:
4796:
4786:
4781:are the same: H
4779:homology groups
4722:category theory
4718:
4712:
4678:
4677:
4674:homotopy groups
4646:
4623:
4622:
4601:
4596:
4595:
4571:
4552:
4547:
4546:
4522:
4499:
4498:
4466:
4443:
4442:
4393:
4392:
4389:
4383:
4351:
4346:
4339:
4329:
4328:
4322:
4311:
4304:
4301:
4294:
4288:
4282:
4275:
4269:
4262:
4261:
4250:
4245:
4231:
4228:
4222:
4217:
4189:timelike curves
4173:
4159:
4152:
4145:
4138:
4130:ambient isotopy
4127:
4116:
4092:
4085:
4017:) = 2
3952:ambient isotopy
3901:
3900:
3899:
3898:
3886:
3885:
3884:
3876:
3875:
3864:
3812:
3802:
3771:
3749:
3698:
3693:
3669:
3663:
3649:
3647:
3632:
3628:
3617:
3612:homotopy groups
3567:The (singular)
3560:if and only if
3546:if and only if
3517:
3493:
3492:
3473:
3472:
3450:
3449:
3430:
3429:
3408:
3403:
3402:
3380:
3379:
3358:
3353:
3352:
3330:
3329:
3306:
3305:
3282:
3281:
3278:
3246:
3245:
3218:
3217:
3195:
3194:
3172:
3171:
3124:
3085:
3067:
3062:
3061:
3038:
3023:
3018:
3017:
2986:
2985:
2952:
2937:
2922:
2917:
2916:
2881:
2876:
2875:
2848:
2821:
2810:
2809:
2788:
2783:
2782:
2751:
2750:
2715:
2688:
2683:
2682:
2644:
2639:
2638:
2613:
2608:
2607:
2579:
2574:
2543:
2511:
2510:
2487:
2482:
2481:
2460:
2455:
2454:
2429:
2424:
2419:
2418:
2395:
2380:
2363:
2362:
2327:
2322:
2321:
2298:
2293:
2292:
2288:
2265:
2255: ∘
2251:
2249:
2239: ∘
2235:
2228:
2202:are said to be
2193:
2183: ∘
2179:
2177:
2163: ∘
2159:
2145:
2131:
2105:
2084:
2083:
2055:
2037:
2036:
2026:
2019:
1991:
1976:
1975:
1917:
1912:
1911:
1887:
1874:
1854:
1849:
1848:
1802:
1795:
1791:
1778:
1773:
1772:
1771:; i.e. the set
1739:
1726:
1711:
1706:
1701:
1700:
1682:
1681:
1626:
1608:
1607:
1597:
1549:
1548:
1539:linear homotopy
1485:
1484:
1481:Euclidean space
1453:
1442:
1441:
1412:
1396:
1374:
1370:
1344:
1343:
1320:
1283:
1282:
1281:, then the map
1256:
1249:
1245:
1225:
1224:
1198:
1191:
1187:
1167:
1166:
1143:
1118:
1117:
1113:
1097:
1091: ∘
1090:
1084:
1081:
1075: ∘
1074:
1068:
1057:
1050:
1044:
1033:
1026:
1020:
981:
962:
937:
923:
878:
873:
872:
821:
816:
815:
796:
795:
758:
757:
727:
704:
703:
673:
668:
667:
640:
635:
634:
597:
596:
563:
558:
557:
520:
519:
448:
447:
398:
397:
348:
347:
282:
281:
232:
197:homotopy groups
189:
150:
146:
139:
102:
98:
41:The two dashed
35:
28:
23:
22:
15:
12:
11:
5:
5981:
5979:
5971:
5970:
5965:
5960:
5950:
5949:
5943:
5942:
5940:
5939:
5929:
5928:
5927:
5922:
5917:
5902:
5892:
5882:
5870:
5859:
5856:
5855:
5853:
5852:
5847:
5842:
5837:
5832:
5827:
5821:
5819:
5813:
5812:
5810:
5809:
5804:
5799:
5797:Winding number
5794:
5789:
5783:
5781:
5777:
5776:
5774:
5773:
5768:
5763:
5758:
5753:
5748:
5743:
5738:
5737:
5736:
5731:
5729:homotopy group
5721:
5720:
5719:
5714:
5709:
5704:
5699:
5689:
5684:
5679:
5669:
5667:
5663:
5662:
5655:
5653:
5651:
5650:
5645:
5640:
5639:
5638:
5628:
5627:
5626:
5616:
5611:
5606:
5601:
5596:
5590:
5588:
5584:
5583:
5578:
5576:
5575:
5568:
5561:
5553:
5547:
5546:
5540:
5527:
5509:
5491:
5485:
5477:Basic Topology
5470:
5467:
5465:
5464:
5449:
5429:
5368:
5343:
5328:
5310:
5280:
5265:
5247:
5229:
5205:
5178:
5156:
5154:
5151:
5150:
5149:
5144:
5139:
5134:
5129:
5124:
5116:
5113:
5101:omega-spectrum
5084:
5081:
5078:
5075:
5072:
5067:
5063:
5034:
5031:
5028:
5025:
5022:
5019:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4902:
4899:
4888:
4878:
4868:
4858:
4835:path connected
4812:
4802:
4792:
4782:
4714:Main article:
4711:
4708:
4691:
4688:
4685:
4676:. In the case
4658:
4653:
4649:
4645:
4642:
4639:
4636:
4633:
4630:
4608:
4604:
4583:
4578:
4574:
4570:
4567:
4564:
4559:
4555:
4534:
4529:
4525:
4521:
4518:
4515:
4512:
4509:
4506:
4473:
4469:
4465:
4462:
4459:
4456:
4453:
4450:
4418:
4415:
4412:
4409:
4406:
4403:
4400:
4387:Homotopy group
4385:Main article:
4382:
4379:
4320:
4309:
4299:
4286:
4267:
4224:Main article:
4221:
4218:
4216:
4213:
4172:
4169:
4165:smooth isotopy
4157:
4150:
4143:
4136:
4125:
4114:
4090:
4083:
4056:, −
3888:
3887:
3878:
3877:
3869:
3868:
3867:
3866:
3865:
3863:
3860:
3697:
3694:
3692:
3689:
3677:
3676:
3667:
3645:
3630:
3626:
3615:
3588:
3565:
3551:
3544:path-connected
3516:
3513:
3500:
3480:
3457:
3437:
3415:
3411:
3387:
3365:
3361:
3337:
3313:
3302:null-homotopic
3300:is said to be
3289:
3277:
3274:
3273:
3272:
3265:
3253:
3233:
3229:
3225:
3215:quotient space
3202:
3179:
3168:
3156:
3153:
3150:
3147:
3144:
3139:
3136:
3133:
3128:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3100:
3097:
3094:
3089:
3084:
3081:
3076:
3071:
3047:
3042:
3037:
3032:
3027:
3005:
3002:
2999:
2996:
2993:
2971:
2968:
2965:
2962:
2959:
2955:
2951:
2946:
2941:
2936:
2931:
2926:
2914:
2907:
2893:
2890:
2885:
2861:
2858:
2855:
2851:
2847:
2844:
2841:
2838:
2835:
2830:
2825:
2820:
2817:
2795:
2791:
2770:
2767:
2764:
2761:
2758:
2744:
2743:
2742:
2728:
2725:
2722:
2718:
2714:
2711:
2708:
2705:
2702:
2697:
2692:
2667:
2664:
2661:
2658:
2653:
2648:
2620:
2616:
2604:
2588:
2583:
2577:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2550:
2546:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2496:
2491:
2467:
2463:
2438:
2433:
2427:
2404:
2399:
2394:
2389:
2384:
2379:
2376:
2373:
2370:
2350:
2347:
2344:
2341:
2336:
2331:
2307:
2302:
2287:
2284:
2283:
2282:
2275:
2261:
2260:is equal to id
2245:
2227:
2224:
2189:
2173:
2104:
2101:
2100:
2099:
2098:
2097:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2056:
2054:
2051:
2048:
2045:
2042:
2039:
2038:
2033:
2029:
2025:
2022:
2020:
2018:
2015:
2012:
2009:
2006:
2003:
1998:
1994:
1990:
1987:
1984:
1983:
1954:
1951:
1945:
1942:
1939:
1936:
1929:
1926:
1920:
1894:
1890:
1886:
1881:
1877:
1873:
1866:
1863:
1857:
1835:
1831:
1828:
1825:
1822:
1819:
1816:
1811:
1806:
1801:
1798:
1794:
1790:
1785:
1781:
1746:
1742:
1738:
1733:
1729:
1725:
1718:
1714:
1709:
1697:
1696:
1695:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1627:
1625:
1622:
1619:
1616:
1613:
1610:
1609:
1606:
1603:
1600:
1598:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1556:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1462:
1457:
1452:
1449:
1438:
1425:
1419:
1415:
1411:
1408:
1403:
1399:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1373:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1329:
1324:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1297:
1293:
1290:
1269:
1263:
1259:
1255:
1252:
1248:
1244:
1241:
1238:
1235:
1232:
1211:
1205:
1201:
1197:
1194:
1190:
1186:
1183:
1180:
1177:
1174:
1152:
1147:
1142:
1138:
1134:
1131:
1128:
1125:
1112:
1109:
1095:
1088:
1079:
1072:
1055:
1048:
1031:
1024:
980:
977:
958:
932:is the torus,
896:
893:
890:
885:
881:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
828:
824:
803:
783:
780:
777:
774:
771:
768:
765:
745:
742:
739:
734:
730:
726:
723:
720:
717:
714:
711:
688:
685:
680:
676:
655:
652:
647:
643:
622:
619:
616:
613:
610:
607:
604:
584:
581:
578:
575:
570:
566:
545:
542:
539:
536:
533:
530:
527:
461:
458:
455:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
319:
316:
313:
310:
307:
304:
301:
298:
295:
292:
289:
231:
228:
53:, a branch of
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5980:
5969:
5966:
5964:
5961:
5959:
5956:
5955:
5953:
5938:
5930:
5926:
5923:
5921:
5918:
5916:
5913:
5912:
5911:
5903:
5901:
5897:
5893:
5891:
5887:
5883:
5881:
5876:
5871:
5869:
5861:
5860:
5857:
5851:
5848:
5846:
5843:
5841:
5838:
5836:
5833:
5831:
5828:
5826:
5823:
5822:
5820:
5818:
5814:
5808:
5807:Orientability
5805:
5803:
5800:
5798:
5795:
5793:
5790:
5788:
5785:
5784:
5782:
5778:
5772:
5769:
5767:
5764:
5762:
5759:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5735:
5732:
5730:
5727:
5726:
5725:
5722:
5718:
5715:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5694:
5693:
5690:
5688:
5685:
5683:
5680:
5678:
5674:
5671:
5670:
5668:
5664:
5659:
5649:
5646:
5644:
5643:Set-theoretic
5641:
5637:
5634:
5633:
5632:
5629:
5625:
5622:
5621:
5620:
5617:
5615:
5612:
5610:
5607:
5605:
5604:Combinatorial
5602:
5600:
5597:
5595:
5592:
5591:
5589:
5585:
5581:
5574:
5569:
5567:
5562:
5560:
5555:
5554:
5551:
5543:
5537:
5533:
5528:
5524:
5520:
5519:
5514:
5510:
5506:
5502:
5501:
5496:
5492:
5488:
5482:
5478:
5473:
5472:
5468:
5460:
5456:
5452:
5450:0-89871-544-X
5446:
5442:
5441:
5433:
5430:
5425:
5421:
5417:
5413:
5409:
5405:
5401:
5397:
5392:
5391:gr-qc/0609054
5387:
5383:
5379:
5372:
5369:
5363:
5362:
5357:
5354:
5347:
5344:
5339:
5335:
5331:
5329:9780521795401
5325:
5321:
5314:
5311:
5306:
5305:
5300:
5294:
5290:
5284:
5281:
5276:
5272:
5268:
5266:9780521795401
5262:
5258:
5251:
5248:
5243:
5239:
5233:
5230:
5219:
5215:
5209:
5206:
5194:
5193:
5188:
5182:
5179:
5167:
5161:
5158:
5152:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5122:
5119:
5118:
5114:
5112:
5110:
5106:
5102:
5098:
5079:
5076:
5073:
5065:
5061:
5052:
5048:
5029:
5026:
5023:
5017:
5010:
5006:
4984:
4981:
4978:
4972:
4969:
4966:
4955:
4951:
4947:
4943:
4939:
4935:
4930:
4928:
4924:
4920:
4916:
4912:
4908:
4900:
4898:
4896:
4891:
4886:
4881:
4876:
4871:
4866:
4861:
4856:
4852:
4848:
4844:
4840:
4836:
4832:
4828:
4824:
4820:
4815:
4810:
4805:
4800:
4795:
4790:
4785:
4780:
4776:
4772:
4768:
4764:
4760:
4756:
4752:
4748:
4743:
4741:
4737:
4733:
4729:
4728:
4723:
4717:
4709:
4707:
4705:
4689:
4686:
4683:
4675:
4670:
4651:
4643:
4640:
4637:
4606:
4602:
4576:
4572:
4568:
4565:
4557:
4553:
4527:
4519:
4516:
4513:
4497:
4493:
4489:
4471:
4463:
4460:
4457:
4451:
4448:
4440:
4436:
4432:
4416:
4410:
4407:
4404:
4401:
4398:
4388:
4380:
4378:
4376:
4372:
4367:
4365:
4359:
4354:
4349:
4342:
4337:
4332:
4326:
4319:
4315:
4307:
4297:
4292:
4285:
4278:
4273:
4265:
4258:
4253:
4248:
4242:
4238:
4234:
4227:
4219:
4214:
4212:
4210:
4206:
4202:
4198:
4194:
4190:
4186:
4182:
4178:
4170:
4168:
4166:
4161:
4156:
4149:
4142:
4135:
4131:
4124:
4120:
4113:
4109:
4105:
4100:
4096:
4089:
4082:
4078:
4074:
4069:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4026:
4024:
4021: −
4020:
4016:
4012:
4008:
4004:
4000:
3996:
3992:
3988:
3984:
3980:
3976:
3972:
3968:
3964:
3960:
3955:
3953:
3948:
3946:
3942:
3938:
3934:
3930:
3926:
3922:
3918:
3914:
3910:
3906:
3896:
3892:
3882:
3873:
3861:
3859:
3857:
3853:
3849:
3845:
3841:
3837:
3833:
3829:
3825:
3821:
3815:
3809:
3805:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
3769:
3765:
3760:
3756:
3752:
3747:
3743:
3739:
3735:
3731:
3727:
3723:
3719:
3715:
3711:
3707:
3703:
3695:
3690:
3688:
3686:
3682:
3673:
3666:
3660:
3656:
3652:
3644:
3640:
3636:
3625:
3621:
3613:
3609:
3605:
3601:
3597:
3593:
3589:
3586:
3582:
3578:
3574:
3570:
3566:
3563:
3559:
3555:
3552:
3549:
3545:
3541:
3538:
3537:
3536:
3534:
3530:
3526:
3522:
3514:
3512:
3498:
3478:
3469:
3455:
3435:
3413:
3409:
3401:
3385:
3378:to any space
3363:
3359:
3351:
3335:
3327:
3326:null-homotopy
3311:
3303:
3287:
3276:Null-homotopy
3275:
3270:
3266:
3251:
3231:
3227:
3223:
3216:
3200:
3193:
3177:
3169:
3148:
3142:
3137:
3134:
3131:
3109:
3103:
3098:
3095:
3092:
3079:
3074:
3045:
3035:
3030:
3016:, by writing
3003:
3000:
2997:
2994:
2991:
2969:
2966:
2963:
2960:
2957:
2953:
2949:
2944:
2934:
2929:
2915:
2912:
2911:vector bundle
2908:
2891:
2888:
2859:
2856:
2853:
2849:
2839:
2833:
2828:
2818:
2815:
2793:
2789:
2768:
2762:
2759:
2756:
2749:
2745:
2726:
2723:
2720:
2716:
2712:
2706:
2700:
2695:
2680:
2679:
2662:
2656:
2651:
2636:
2618:
2614:
2605:
2586:
2575:
2571:
2565:
2562:
2559:
2553:
2548:
2544:
2540:
2537:
2534:
2528:
2525:
2522:
2516:
2494:
2465:
2461:
2436:
2425:
2402:
2387:
2377:
2374:
2371:
2368:
2345:
2339:
2334:
2305:
2290:
2289:
2285:
2280:
2276:
2273:
2269:
2268:
2267:
2264:
2258:
2254:
2248:
2242:
2238:
2233:
2232:homeomorphism
2225:
2223:
2221:
2217:
2213:
2209:
2208:homotopy type
2205:
2201:
2197:
2192:
2186:
2182:
2176:
2171:
2166:
2162:
2156:
2152:
2148:
2142:
2138:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2102:
2080:
2077:
2071:
2068:
2065:
2057:
2049:
2046:
2043:
2031:
2027:
2021:
2013:
2010:
2007:
2001:
1996:
1992:
1988:
1985:
1974:
1973:
1971:
1949:
1943:
1937:
1924:
1918:
1910:
1892:
1888:
1879:
1875:
1871:
1861:
1855:
1833:
1829:
1826:
1820:
1814:
1809:
1799:
1796:
1792:
1788:
1783:
1779:
1770:
1766:
1762:
1744:
1740:
1731:
1727:
1723:
1716:
1712:
1707:
1698:
1678:
1672:
1666:
1663:
1660:
1654:
1648:
1642:
1639:
1636:
1628:
1620:
1617:
1614:
1604:
1599:
1591:
1588:
1585:
1579:
1573:
1570:
1567:
1561:
1558:
1547:
1546:
1544:
1540:
1536:
1520:
1511:
1508:
1505:
1499:
1496:
1493:
1490:
1482:
1478:
1460:
1450:
1447:
1439:
1423:
1417:
1413:
1409:
1406:
1401:
1397:
1390:
1387:
1384:
1378:
1375:
1371:
1367:
1361:
1358:
1355:
1349:
1327:
1311:
1308:
1305:
1299:
1291:
1288:
1267:
1261:
1257:
1253:
1250:
1246:
1242:
1236:
1230:
1209:
1203:
1199:
1195:
1192:
1188:
1184:
1178:
1172:
1165:are given by
1150:
1132:
1129:
1126:
1123:
1115:
1114:
1110:
1108:
1105:
1101:
1094:
1087:
1078:
1071:
1065:
1061:
1054:
1047:
1041:
1037:
1030:
1023:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
978:
976:
974:
970:
966:
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340:unit interval
337:
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71:Ancient Greek
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5937:Publications
5802:Chern number
5792:Betti number
5723:
5675: /
5666:Key concepts
5614:Differential
5534:. Springer.
5531:
5516:
5498:
5479:. Springer.
5476:
5439:
5432:
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5319:
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5289:Ghostarchive
5287:Archived at
5283:
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5232:
5221:. Retrieved
5217:
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5196:. Retrieved
5190:
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5169:. Retrieved
5160:
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5007:to the
5004:
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4490:with itself
4441:. If we fix
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4375:cofibrations
4368:
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4317:
4312:is called a
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2781:with fibers
2748:fiber bundle
2279:Möbius strip
2262:
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2236:
2229:
2220:contractible
2215:
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2170:identity map
2164:
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2158:, such that
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1763:on the unit
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487:describes a
484:
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270:
266:
262:
257:
247:
220:CW complexes
213:
203:, important
192:-moh-toh-pee
94:
66:
48:
5900:Wikiversity
5817:Key results
4769:induced by
4095:dimensional
4093:, in three-
4077:knot theory
4040:defined by
3965:) = −
3350:unit circle
3280:A function
1545:) given by
55:mathematics
5952:Categories
5746:CW complex
5687:Continuity
5677:Closed set
5636:cohomology
5495:"Homotopy"
5223:2019-08-17
5153:References
4956:, the set
4936:: one can
4877:) : π
4821:) for all
4801:) : H
4747:functorial
4364:fibrations
4345:such that
4281:such that
4215:Properties
3921:embeddings
3824:retraction
3770:such that
3757:× →
3585:isomorphic
3515:Invariance
3192:CW complex
1479:subset of
979:Properties
912:embeddings
699:, and the
633:such that
346:such that
240:embeddings
205:invariants
5925:geometric
5920:algebraic
5771:Cobordism
5707:Hausdorff
5702:connected
5619:Geometric
5609:Continuum
5599:Algebraic
5523:EMS Press
5505:EMS Press
5424:119707350
5416:0015-9018
5361:MathWorld
5356:"Isotopy"
5127:Homeotopy
4938:represent
4911:algebraic
4629:∂
4554:π
4505:∂
4414:→
4408::
4327:to a map
4099:embedding
4034:unit disc
3858:is used.
3818:Also, if
3648:)) where
3400:unit disk
3348:from the
3143:−
3135:−
3119:→
3104:−
3096:−
3080:×
3036:−
2995:≤
2967:−
2961:−
2950:≃
2935:−
2857:−
2846:→
2834:−
2816:π
2766:→
2757:π
2724:−
2713:≃
2701:−
2657:−
2572:⋅
2563:−
2541:⋅
2529:⋅
2393:→
2378:×
2340:≃
2272:bijection
2069:−
2060:⟼
2024:⟶
2002:×
1953:→
1928:→
1885:→
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1827:≤
1824:‖
1818:‖
1800:∈
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1602:⟶
1580:×
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1451:⊆
1388:−
1342:given by
1318:→
1300:×
1141:→
767:×
725:↦
606:∈
580:→
541:→
477:parameter
457:∈
338:with the
330:from the
315:→
297:×
67:homotopic
61:from one
5890:Wikibook
5868:Category
5756:Manifold
5724:Homotopy
5682:Interior
5673:Open set
5631:Homology
5580:Topology
5459:52377653
5338:45420394
5291:and the
5275:45420394
5198:22 April
5171:22 April
5115:See also
4594:, where
4496:boundary
4334: :
4274:→
4270: :
4255:→
4249: :
4235: :
4201:3-sphere
4181:timelike
4062:rotation
4048:,
4013:,
3943:,
3801:for all
3779:,
3762:between
3753: :
3691:Variants
3670:∈
3653: :
3622:,
3569:homology
2984:for any
2635:1-sphere
2417:between
2286:Examples
2149: :
2135: :
2119:between
1111:Examples
1098: :
1058: :
1034: :
967:, where
446:for all
95:homotopy
51:topology
5915:general
5717:uniform
5697:compact
5648:Digital
5525:, 2001
5507:, 2001
5469:Sources
5396:Bibcode
5304:YouTube
5192:YouTube
4740:functor
4488:crossed
4203:can be
3925:isotopy
3862:Isotopy
3850:. When
1907:be the
1759:be the
995:taking
332:product
252:isotopy
242:of the
224:spectra
142:-tə-pee
5910:Topics
5712:metric
5587:Fields
5538:
5483:
5457:
5447:
5422:
5414:
5336:
5326:
5273:
5263:
5053:group
4724:. The
4381:Groups
4104:deform
3891:unknot
3730:subset
2909:Every
2637:) and
1970:origin
1847:. Let
1477:convex
69:(from
57:, two
5692:Space
5420:S2CID
5386:arXiv
4887:) → π
4867:) = π
4811:) → H
4791:) = H
4757:from
4175:On a
3826:from
3822:is a
3728:is a
3190:of a
2633:(the
1535:paths
1475:is a
495:into
246:into
244:torus
222:, or
90:tópos
85:τόπος
79:homós
43:paths
5536:ISBN
5481:ISBN
5455:OCLC
5445:ISBN
5412:ISSN
5334:OCLC
5324:ISBN
5271:OCLC
5261:ISBN
5200:2022
5173:2022
5049:-th
4913:and
4909:for
4849:and
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3606:and
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2746:Any
2453:and
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2111:and
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