5864:
5647:
5885:
5853:
3870:
5922:
5895:
5875:
3861:
2084:
1682:
4090:
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
3154:
2590:
1834:
2971:
2404:
1746:
1329:
1424:
2861:
2440:
2728:
1152:
1971:
1544:
3047:
2259:
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
2348:
1966:
1539:
1955:
1894:
1462:
1210:
224:
1268:
2665:
743:
2893:
27:
4581:
4416:
2496:
2307:
4656:
4532:
2207:
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
858:
317:
3003:
582:
3946:
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
2768:
433:
383:
5082:
543:
4152:
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
1520:
894:
781:
686:
653:
5032:
4473:
3052:
459:
4608:
3415:
3365:
2795:
2620:
2467:
620:
4689:
3231:
3498:
3478:
3455:
3435:
3385:
3335:
3311:
3287:
3251:
3200:
3177:
2501:
801:
4990:
5925:
4719:
is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces
4068:—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots,
4362:, 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
2270:
and an untwisted (closed) strip are homotopy equivalent, since you can deform both strips continuously to a circle. But they are not homeomorphic.
3886:
since one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space. Thus they are not ambient-isotopic.
5281:
1763:
4184:(CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be
267:
4017:
Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic using
2907:
3978:
to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover,
2353:
4121:, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots
943:
is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of
5956:
1691:
2797:
homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since
1273:
5528:
5473:
1334:
2800:
2409:
4731:
on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.
2673:
1108:
2255:. Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples:
203:
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with
5559:
4117:
embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy. An
3008:
4180:
is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. No
5913:
5908:
5437:
5316:
5253:
2079:{\displaystyle {\begin{aligned}H:B^{n}\times &\longrightarrow B^{n}\\(x,t)&\longmapsto (1-t)x.\end{aligned}}}
1677:{\displaystyle {\begin{aligned}H:\times &\longrightarrow C\\(s,t)&\longmapsto (1-t)f(s)+tg(s).\end{aligned}}}
176:
125:
2312:
5903:
4661:
We can define the action of one equivalence class on another, and so we get a group. These groups are called the
1902:
1839:
1432:
1157:
3673:
1215:
248:
47:
5805:
5511:
5493:
496:. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from
2629:
5501:
3669:
694:
2866:
5506:
5488:
4997:
4359:
4197:
960:
varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as
5951:
5813:
5109:
4537:
212:
4383:
2472:
2283:
239:: as "the surface of a doughnut" and as "the surface of a coffee mug". This is also an example of an
34:
shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.
4613:
4489:
4214:
3203:
806:
272:
204:
3912:, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of
5946:
5612:
4915:
4172:(representing something that only goes forwards, not backwards, in time, in every local frame). A
3986:
has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from
2976:
939:
that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts;
548:
5898:
5884:
5277:
2741:
388:
338:
193:
82:"place") if one can be "continuously deformed" into the other, such a deformation being called a
5045:
4937:
into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group
510:
5833:
5754:
5631:
5619:
5592:
5552:
4922:
4911:
4903:
3557:
3257:
3149:{\displaystyle \mathbb {R} ^{k}\times (\mathbb {R} ^{n-k}-\{0\})\to (\mathbb {R} ^{n-k}-\{0\})}
1475:
59:
5828:
5483:
5130:
3668:
An example of an algebraic invariant of topological spaces which is not homotopy-invariant is
863:
748:
5675:
5602:
4926:
4907:
4181:
3812:
658:
625:
5002:
4433:
438:
5823:
5775:
5749:
5597:
5384:
5120:
4586:
4018:
3393:
3343:
2773:
2598:
2445:
1958:
1005:
993:
587:
2585:{\displaystyle H(t,\cdot )=t\cdot p_{0}+(1-t)\cdot \operatorname {id} _{\mathbb {R} ^{n}}}
8:
5670:
5125:
5089:
5039:
4899:
4668:
4484:
4165:
4095:
one embedding to another through a path of embeddings: a continuous function starting at
3828:
3208:
5874:
5388:
5868:
5838:
5818:
5739:
5729:
5607:
5587:
5408:
5374:
5093:
4755:
4302:
4185:
4061:
3509:
3483:
3463:
3440:
3420:
3370:
3320:
3296:
3272:
3236:
3185:
3162:
1757:
899:
The animation that is looped above right provides an example of a homotopy between two
786:
197:
4948:
4418:
being homotopic relative to a subspace is an equivalence relation, we can look at the
5863:
5856:
5722:
5680:
5545:
5524:
5469:
5443:
5433:
5412:
5400:
5341:
5322:
5312:
5259:
5249:
4895:
4727:
are isomorphic in this category if and only if they are homotopy-equivalent. Then a
4715:
4704:
4692:
4476:
4419:
4173:
3690:
3588:
3573:
2158:
2117:
1897:
1749:
689:
189:
51:
5888:
5636:
5582:
5392:
5135:
4193:
3697:. These are homotopies which keep the elements of the subspace fixed. Formally: if
3561:
3546:
3387:
is null-homotopic precisely when it can be continuously extended to a map from the
507:
An alternative notation is to say that a homotopy between two continuous functions
320:
137:
89:
20:
5695:
5690:
4710:
4118:
3940:
3532:
1523:
1469:
185:
31:
5878:
3500:
to itself—which is always a homotopy equivalence—is null-homotopic.
2267:
5785:
5717:
4843:
4767:
4662:
4375:
4177:
3600:
3516:, that is, they respect the relation of homotopy equivalence. For example, if
5396:
5287:
5202:
5175:
5940:
5795:
5705:
5685:
5404:
5344:
5154:
4823:
2899:
2220:
328:
5447:
5326:
5263:
184:) between the two functions. A notable use of homotopy is the definition of
5780:
5700:
5646:
3883:
2736:
2208:
5226:
5790:
5427:
4363:
4065:
3338:
43:
5227:"algebraic topology - Path homotopy and separately continuous functions"
5734:
5665:
5624:
4135:
are considered equivalent when there is an ambient isotopy which moves
3180:
2263:
between them (since one is an infinite set, while the other is finite).
1829:{\displaystyle B^{n}:=\left\{x\in \mathbb {R} ^{n}:\|x\|\leq 1\right\}}
1465:
208:
5379:
5365:
Monroe, Hunter (2008-11-01). "Are
Causality Violations Undesirable?".
223:
5759:
5349:
5115:
4906:
have been developed. The methods for algebraic equations include the
4352:
4087:
4083:
4022:
3909:
3388:
2260:
900:
465:
228:
77:
66:
3869:
5744:
5661:
5568:
4189:
4169:
4050:
2623:
2350:. The part that needs to be checked is the existence of a homotopy
39:
4534:
as a subspace, then the equivalence classes form a group, denoted
4149:. This is the appropriate definition in the topological category.
5292:
5180:
4728:
2966:{\displaystyle \mathbb {R} ^{n}-\mathbb {R} ^{k}\simeq S^{n-k-1}}
26:
3879:
3718:
3293:
if it is homotopic to a constant function. (The homotopy from
3860:
2902:
is a fiber bundle with a fiber homotopy equivalent to a point.
2399:{\displaystyle H:I\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
167:
116:
4709:
The idea of homotopy can be turned into a formal category of
232:
72:
980:
are said to be homotopic if and only if there is a homotopy
5537:
161:
152:
146:
110:
95:
1741:{\displaystyle \operatorname {id} _{B^{n}}:B^{n}\to B^{n}}
104:
4057:
are isotopic because they can be connected by rotations.
2214:
1324:{\displaystyle H:\mathbb {R} \times \to \mathbb {R} ^{2}}
964:
varies back from 1 to 0, pauses, and repeats this cycle.
3920:, in the notation used before, such that for each fixed
3603:. (Without the path-connectedness assumption, one has π
1419:{\displaystyle H(x,t)=\left(x,(1-t)x^{3}+te^{x}\right)}
16:
Continuous deformation between two continuous functions
5248:. Cambridge: Cambridge University Press. p. 185.
4914:). The methods for differential equations include the
4351:
The homotopy lifting property is used to characterize
2856:{\displaystyle \pi :\mathbb {R} ^{n}-\{0\}\to S^{n-1}}
2435:{\displaystyle \operatorname {id} _{\mathbb {R} ^{n}}}
2223:
is a special case of a homotopy equivalence, in which
5311:. Cambridge: Cambridge University Press. p. 11.
5048:
5005:
4951:
4671:
4616:
4589:
4540:
4492:
4436:
4386:
3486:
3480:
is contractible if and only if the identity map from
3466:
3443:
3423:
3396:
3373:
3346:
3323:
3299:
3275:
3239:
3211:
3188:
3165:
3055:
3011:
2979:
2910:
2869:
2803:
2776:
2744:
2676:
2632:
2601:
2504:
2475:
2448:
2412:
2356:
2315:
2286:
1969:
1905:
1842:
1766:
1694:
1542:
1478:
1435:
1337:
1276:
1218:
1160:
1111:
866:
809:
789:
751:
697:
661:
628:
590:
551:
513:
441:
391:
341:
275:
177:
155:
126:
4834:
is pointed, then the group homomorphisms induced by
2723:{\displaystyle \mathbb {R} ^{n}-\{0\}\simeq S^{n-1}}
1147:{\displaystyle f,g:\mathbb {R} \to \mathbb {R} ^{2}}
164:
158:
113:
107:
4208:
3676:, and compactification is not homotopy-invariant).
149:
143:
140:
101:
98:
92:
19:This article is about topology. For chemistry, see
5076:
5026:
4984:
4683:
4650:
4602:
4575:
4526:
4467:
4410:
4358:Another useful property involving homotopy is the
3492:
3472:
3449:
3429:
3409:
3379:
3359:
3329:
3313:to a constant function is then sometimes called a
3305:
3281:
3245:
3225:
3194:
3171:
3148:
3041:
2997:
2965:
2887:
2855:
2789:
2762:
2722:
2659:
2614:
2584:
2490:
2461:
2434:
2398:
2342:
2301:
2078:
1949:
1888:
1828:
1740:
1676:
1514:
1456:
1418:
1323:
1262:
1204:
1146:
888:
852:
795:
775:
737:
680:
647:
614:
576:
537:
453:
427:
377:
311:
3672:(which is, roughly speaking, the homology of the
3042:{\displaystyle \mathbb {R} ^{n}-\mathbb {R} ^{k}}
1961:. Then the following is a homotopy between them:
504:as the slider moves from 0 to 1, and vice versa.
5938:
5339:
5176:"Homotopy Type Theory Discussed - Computerphile"
4921:Homotopy theory can be used as a foundation for
3994:: × → given by
3982:has changed the orientation of the interval and
3156:, then applying the homotopy equivalences above.
4053:around the origin, and so the identity map and
3827:is the identity map, this is known as a strong
3460:It follows from these definitions that a space
2280:The first example of a homotopy equivalence is
5429:Introduction to numerical continuation methods
935:is some continuous function from the torus to
5553:
5096:for singular cohomology with coefficients in
4992:of based homotopy classes of based maps from
3939:A related, but different, concept is that of
3508:Homotopy equivalence is important because in
1004:. This homotopy relation is compatible with
5112:(relative version of a homotopy equivalence)
3140:
3134:
3101:
3095:
2831:
2825:
2698:
2692:
2654:
2648:
2343:{\displaystyle \mathbb {R} ^{n}\simeq \{0\}}
2337:
2331:
1812:
1806:
996:on the set of all continuous functions from
4188:by timelike curves. A manifold such as the
1950:{\displaystyle c_{\vec {0}}(x):={\vec {0}}}
1889:{\displaystyle c_{\vec {0}}:B^{n}\to B^{n}}
1457:{\displaystyle C\subseteq \mathbb {R} ^{n}}
860:. It is not sufficient to require each map
5921:
5894:
5560:
5546:
2498:onto the origin. This can be described as
1205:{\displaystyle f(x):=\left(x,x^{3}\right)}
992:as described above. Being homotopic is an
5463:
5378:
3115:
3076:
3058:
3029:
3014:
2928:
2913:
2872:
2812:
2679:
2635:
2570:
2478:
2420:
2386:
2371:
2318:
2289:
1793:
1526:with the same endpoints, then there is a
1444:
1311:
1284:
1263:{\displaystyle g(x)=\left(x,e^{x}\right)}
1134:
1125:
5425:
4910:method and the continuation method (see
2595:There is a homotopy equivalence between
222:
25:
5518:
4738:homotopy invariant: this means that if
3049:as the total space of the fiber bundle
2091:
1056:are homotopic, then their compositions
803:. The two versions coincide by setting
5939:
5364:
5306:
5243:
4894:Based on the concept of the homotopy,
4168:, certain curves are distinguished as
3599:are isomorphic, and so are the higher
3524:are homotopy equivalent spaces, then:
2660:{\displaystyle \mathbb {R} ^{2}-\{0\}}
2215:Homotopy equivalence vs. homeomorphism
5541:
5340:
5285:
5092:of Eilenberg-MacLane spaces are
738:{\displaystyle (x,t)\mapsto h_{t}(x)}
4698:
4380:Since the relation of two functions
4159:
3892:When two given continuous functions
3684:
2888:{\displaystyle \mathbb {R} _{>0}}
545:is a family of continuous functions
218:
5155:"Homotopy Definition & Meaning"
4734:For example, homology groups are a
4196:(by any type of curve), and yet be
952:(X) as a function of the parameter
492:and at time 1 we have the function
13:
5432:. Kurt Georg. Philadelphia: SIAM.
4617:
4493:
4021:. For this reason, the map of the
14:
5968:
5034:is in natural bijection with the
4576:{\displaystyle \pi _{n}(Y,y_{0})}
488:: at time 0 we have the function
247:Formally, a homotopy between two
5920:
5893:
5883:
5873:
5862:
5852:
5851:
5645:
5519:Spanier, Edwin (December 1994).
4929:a cohomology functor on a space
4610:is in the image of the subspace
4411:{\displaystyle f,g\colon X\to Y}
4209:Lifting and extension properties
3868:
3859:
3264:
2491:{\displaystyle \mathbb {R} ^{n}}
2302:{\displaystyle \mathbb {R} ^{n}}
2239:(not only homotopic to it), and
136:
88:
5419:
5288:"History of algebraic topology"
4889:
4651:{\displaystyle \partial (^{n})}
4527:{\displaystyle \partial (^{n})}
4203:
3695:homotopy relative to a subspace
2233:is equal to the identity map id
1957:which sends every point to the
853:{\displaystyle h_{t}(x)=H(x,t)}
312:{\displaystyle H:X\times \to Y}
5957:Theory of continuous functions
5358:
5333:
5300:
5270:
5237:
5219:
5195:
5168:
5147:
5071:
5059:
5021:
5009:
4979:
4976:
4964:
4952:
4645:
4636:
4623:
4620:
4570:
4551:
4521:
4512:
4499:
4496:
4456:
4443:
4402:
4049:) is isotopic to a 180-degree
3651:is a homotopy equivalence and
3143:
3110:
3107:
3104:
3071:
2834:
2754:
2557:
2545:
2520:
2508:
2381:
2183:. If such a pair exists, then
2063:
2051:
2048:
2041:
2029:
2012:
2005:
1993:
1941:
1929:
1923:
1916:
1873:
1853:
1725:
1664:
1658:
1646:
1640:
1634:
1622:
1619:
1612:
1600:
1590:
1583:
1571:
1565:
1553:
1506:
1503:
1491:
1382:
1370:
1353:
1341:
1306:
1303:
1291:
1228:
1222:
1170:
1164:
1129:
883:
877:
847:
835:
826:
820:
770:
758:
732:
726:
713:
710:
698:
609:
597:
568:
529:
422:
416:
407:
395:
372:
366:
357:
345:
303:
300:
288:
1:
5141:
3587:are path-connected, then the
3503:
2863:is a fiber bundle with fiber
2274:
2096:Given two topological spaces
967:
5567:
3670:compactly supported homology
2998:{\displaystyle 0\leq k<n}
577:{\displaystyle h_{t}:X\to Y}
240:
78:
67:
7:
5507:Encyclopedia of Mathematics
5489:Encyclopedia of Mathematics
5103:
4941:, and any based CW-complex
4826:, and the homotopy between
4360:homotopy extension property
4198:timelike multiply connected
4110: = 1 giving the
3900:from the topological space
3737:if there exists a homotopy
3679:
2763:{\displaystyle \pi :E\to B}
1426:is a homotopy between them.
1099:
1008:in the following sense: if
428:{\displaystyle H(x,1)=g(x)}
378:{\displaystyle H(x,0)=f(x)}
10:
5973:
5814:Banach fixed-point theorem
5457:
5231:Mathematics Stack Exchange
5110:Fiber-homotopy equivalence
5077:{\displaystyle H^{n}(X,G)}
4702:
4373:
4212:
4099: = 0 giving the
3850:
3733:are homotopic relative to
3693:, one needs the notion of
3260:is a homotopy equivalence.
3233:is homotopy equivalent to
3202:is contractible, then the
2199:. Intuitively, two spaces
538:{\displaystyle f,g:X\to Y}
464:If we think of the second
258:from a topological space
205:compactly generated spaces
73:
63:
18:
5847:
5804:
5768:
5654:
5643:
5575:
5397:10.1007/s10701-008-9254-9
4369:
4215:Homotopy lifting property
3962:isotopic to the identity
3904:to the topological space
3882:is not equivalent to the
3705:are continuous maps from
1515:{\displaystyle f,g:\to C}
5464:Armstrong, M.A. (1979).
5426:Allgower, E. L. (2003).
5307:Allen., Hatcher (2002).
5244:Allen., Hatcher (2002).
5203:"Homotopy | mathematics"
4916:homotopy analysis method
4754:are homotopic, then the
4691:, it is also called the
4422:of maps between a fixed
4312:), then we can lift all
2116:is a pair of continuous
889:{\displaystyle h_{t}(x)}
776:{\displaystyle X\times }
5502:"Isotopy (in topology)"
5207:Encyclopedia Britannica
4998:Eilenberg–MacLane space
4483:times, and we take its
4249:and we are given a map
3916:, which is a homotopy,
3689:In order to define the
3317:.) For example, a map
681:{\displaystyle h_{1}=g}
648:{\displaystyle h_{0}=f}
262:to a topological space
227:A homotopy between two
5869:Mathematics portal
5769:Metrics and properties
5755:Second-countable space
5367:Foundations of Physics
5286:Albin, Pierre (2019).
5078:
5028:
5027:{\displaystyle K(G,n)}
4986:
4912:numerical continuation
4904:differential equations
4685:
4652:
4604:
4577:
4528:
4469:
4468:{\displaystyle X=^{n}}
4412:
4219:If we have a homotopy
4064:—for example in
3936:) gives an embedding.
3494:
3474:
3451:
3431:
3411:
3381:
3361:
3331:
3307:
3283:
3258:deformation retraction
3247:
3227:
3196:
3173:
3150:
3043:
2999:
2967:
2889:
2857:
2791:
2764:
2724:
2661:
2616:
2586:
2492:
2463:
2436:
2400:
2344:
2309:with a point, denoted
2303:
2080:
1951:
1890:
1830:
1742:
1678:
1532:straight-line homotopy
1516:
1458:
1420:
1325:
1264:
1206:
1148:
890:
854:
797:
777:
739:
682:
649:
616:
578:
539:
478:continuous deformation
455:
454:{\displaystyle x\in X}
429:
379:
313:
244:
54:to another are called
35:
5079:
5029:
4987:
4908:homotopy continuation
4686:
4653:
4605:
4603:{\displaystyle y_{0}}
4578:
4529:
4475:, the unit interval
4470:
4413:
4182:closed timelike curve
4106:embedding, ending at
3974:. Any homotopy from
3843:is a point, the term
3495:
3475:
3452:
3432:
3412:
3410:{\displaystyle D^{2}}
3382:
3362:
3360:{\displaystyle S^{1}}
3332:
3308:
3284:
3248:
3228:
3197:
3174:
3151:
3044:
3000:
2968:
2890:
2858:
2792:
2790:{\displaystyle F_{b}}
2765:
2725:
2662:
2617:
2615:{\displaystyle S^{1}}
2587:
2493:
2464:
2462:{\displaystyle p_{0}}
2437:
2401:
2345:
2304:
2081:
1952:
1891:
1831:
1743:
1679:
1517:
1459:
1421:
1326:
1265:
1207:
1149:
972:Continuous functions
891:
855:
798:
778:
740:
683:
650:
617:
615:{\displaystyle t\in }
579:
540:
456:
430:
380:
314:
226:
29:
5824:Invariance of domain
5776:Euler characteristic
5750:Bundle (mathematics)
5121:Homotopy type theory
5088:. One says that the
5046:
5003:
4949:
4846:are also the same: π
4669:
4614:
4587:
4538:
4490:
4434:
4384:
4086:space. A knot is an
3484:
3464:
3441:
3421:
3394:
3371:
3344:
3321:
3297:
3273:
3237:
3209:
3186:
3163:
3053:
3009:
2977:
2908:
2867:
2801:
2774:
2742:
2674:
2630:
2599:
2502:
2473:
2469:, the projection of
2446:
2410:
2354:
2313:
2284:
2157:is homotopic to the
2106:homotopy equivalence
2092:Homotopy equivalence
1967:
1903:
1840:
1764:
1692:
1540:
1476:
1433:
1335:
1274:
1216:
1158:
1109:
1096:are also homotopic.
1006:function composition
994:equivalence relation
911:, of the torus into
864:
807:
787:
749:
695:
659:
626:
588:
549:
511:
439:
389:
339:
273:
249:continuous functions
71:"same, similar" and
48:continuous functions
5834:Tychonoff's theorem
5829:Poincaré conjecture
5583:General (point-set)
5389:2008FoPh...38.1065M
5131:Poincaré conjecture
5126:Mapping class group
5094:representing spaces
5084: of the space
5040:singular cohomology
4896:computation methods
4756:group homomorphisms
4684:{\displaystyle n=1}
4420:equivalence classes
4166:Lorentzian manifold
3990:to the identity is
3829:deformation retract
3725:, then we say that
3226:{\displaystyle X/A}
2193:homotopy equivalent
1429:More generally, if
1032:are homotopic, and
745:is continuous from
268:continuous function
266:is defined to be a
5819:De Rham cohomology
5740:Polyhedral complex
5730:Simplicial complex
5521:Algebraic Topology
5342:Weisstein, Eric W.
5309:Algebraic topology
5246:Algebraic topology
5074:
5024:
4982:
4681:
4648:
4600:
4573:
4524:
4465:
4408:
4186:multiply connected
4062:geometric topology
3589:fundamental groups
3514:homotopy invariant
3512:many concepts are
3510:algebraic topology
3490:
3470:
3447:
3427:
3407:
3377:
3357:
3327:
3303:
3279:
3243:
3223:
3192:
3169:
3146:
3039:
2995:
2963:
2885:
2853:
2787:
2760:
2720:
2657:
2612:
2582:
2488:
2459:
2432:
2396:
2340:
2299:
2177:is homotopic to id
2076:
2074:
1947:
1886:
1826:
1738:
1674:
1672:
1512:
1454:
1416:
1321:
1260:
1202:
1144:
896:to be continuous.
886:
850:
793:
773:
735:
678:
645:
612:
574:
535:
451:
425:
375:
309:
245:
198:algebraic topology
36:
5952:Maps of manifolds
5934:
5933:
5723:fundamental group
5530:978-0-387-94426-5
5475:978-0-387-90839-7
5373:(11): 1065–1069.
4716:homotopy category
4705:Homotopy category
4699:Homotopy category
4693:fundamental group
4174:timelike homotopy
4160:Timelike homotopy
4019:Alexander's trick
3691:fundamental group
3685:Relative homotopy
3618:) isomorphic to π
3562:cohomology groups
3493:{\displaystyle X}
3473:{\displaystyle X}
3457:on the boundary.
3450:{\displaystyle f}
3437:that agrees with
3430:{\displaystyle X}
3380:{\displaystyle X}
3330:{\displaystyle f}
3306:{\displaystyle f}
3282:{\displaystyle f}
3246:{\displaystyle X}
3195:{\displaystyle X}
3172:{\displaystyle A}
2195:, or of the same
1944:
1919:
1898:constant function
1856:
1750:identity function
796:{\displaystyle Y}
219:Formal definition
190:cohomotopy groups
52:topological space
5964:
5924:
5923:
5897:
5896:
5887:
5877:
5867:
5866:
5855:
5854:
5649:
5562:
5555:
5548:
5539:
5538:
5534:
5515:
5497:
5479:
5452:
5451:
5423:
5417:
5416:
5382:
5362:
5356:
5355:
5354:
5337:
5331:
5330:
5304:
5298:
5297:
5274:
5268:
5267:
5241:
5235:
5234:
5223:
5217:
5216:
5214:
5213:
5199:
5193:
5192:
5190:
5188:
5172:
5166:
5165:
5163:
5161:
5151:
5136:Regular homotopy
5083:
5081:
5080:
5075:
5058:
5057:
5033:
5031:
5030:
5025:
4991:
4989:
4988:
4985:{\displaystyle }
4983:
4842:on the level of
4822:are in addition
4814:. Likewise, if
4766:on the level of
4690:
4688:
4687:
4682:
4657:
4655:
4654:
4649:
4644:
4643:
4609:
4607:
4606:
4601:
4599:
4598:
4582:
4580:
4579:
4574:
4569:
4568:
4550:
4549:
4533:
4531:
4530:
4525:
4520:
4519:
4474:
4472:
4471:
4466:
4464:
4463:
4417:
4415:
4414:
4409:
4350:
4344:
4333:
4332:
4327:× →
4322:
4297:
4291:
4287:
4269:
4268:
4255:
4248:
4243:
4232:
4228:× →
4194:simply connected
3872:
3863:
3845:pointed homotopy
3806:
3799:
3789:
3750:
3674:compactification
3664:
3650:
3547:simply connected
3499:
3497:
3496:
3491:
3479:
3477:
3476:
3471:
3456:
3454:
3453:
3448:
3436:
3434:
3433:
3428:
3416:
3414:
3413:
3408:
3406:
3405:
3386:
3384:
3383:
3378:
3366:
3364:
3363:
3358:
3356:
3355:
3336:
3334:
3333:
3328:
3312:
3310:
3309:
3304:
3288:
3286:
3285:
3280:
3252:
3250:
3249:
3244:
3232:
3230:
3229:
3224:
3219:
3201:
3199:
3198:
3193:
3178:
3176:
3175:
3170:
3159:If a subcomplex
3155:
3153:
3152:
3147:
3130:
3129:
3118:
3091:
3090:
3079:
3067:
3066:
3061:
3048:
3046:
3045:
3040:
3038:
3037:
3032:
3023:
3022:
3017:
3004:
3002:
3001:
2996:
2972:
2970:
2969:
2964:
2962:
2961:
2937:
2936:
2931:
2922:
2921:
2916:
2894:
2892:
2891:
2886:
2884:
2883:
2875:
2862:
2860:
2859:
2854:
2852:
2851:
2821:
2820:
2815:
2796:
2794:
2793:
2788:
2786:
2785:
2769:
2767:
2766:
2761:
2729:
2727:
2726:
2721:
2719:
2718:
2688:
2687:
2682:
2670:More generally,
2666:
2664:
2663:
2658:
2644:
2643:
2638:
2621:
2619:
2618:
2613:
2611:
2610:
2591:
2589:
2588:
2583:
2581:
2580:
2579:
2578:
2573:
2541:
2540:
2497:
2495:
2494:
2489:
2487:
2486:
2481:
2468:
2466:
2465:
2460:
2458:
2457:
2441:
2439:
2438:
2433:
2431:
2430:
2429:
2428:
2423:
2405:
2403:
2402:
2397:
2395:
2394:
2389:
2380:
2379:
2374:
2349:
2347:
2346:
2341:
2327:
2326:
2321:
2308:
2306:
2305:
2300:
2298:
2297:
2292:
2248:
2232:
2176:
2156:
2146:
2132:
2085:
2083:
2082:
2077:
2075:
2024:
2023:
1989:
1988:
1956:
1954:
1953:
1948:
1946:
1945:
1937:
1922:
1921:
1920:
1912:
1895:
1893:
1892:
1887:
1885:
1884:
1872:
1871:
1859:
1858:
1857:
1849:
1835:
1833:
1832:
1827:
1825:
1821:
1802:
1801:
1796:
1776:
1775:
1747:
1745:
1744:
1739:
1737:
1736:
1724:
1723:
1711:
1710:
1709:
1708:
1683:
1681:
1680:
1675:
1673:
1521:
1519:
1518:
1513:
1463:
1461:
1460:
1455:
1453:
1452:
1447:
1425:
1423:
1422:
1417:
1415:
1411:
1410:
1409:
1394:
1393:
1330:
1328:
1327:
1322:
1320:
1319:
1314:
1287:
1269:
1267:
1266:
1261:
1259:
1255:
1254:
1253:
1211:
1209:
1208:
1203:
1201:
1197:
1196:
1195:
1153:
1151:
1150:
1145:
1143:
1142:
1137:
1128:
1095:
1071:
1055:
1031:
930:
916:
895:
893:
892:
887:
876:
875:
859:
857:
856:
851:
819:
818:
802:
800:
799:
794:
782:
780:
779:
774:
744:
742:
741:
736:
725:
724:
687:
685:
684:
679:
671:
670:
654:
652:
651:
646:
638:
637:
621:
619:
618:
613:
583:
581:
580:
575:
561:
560:
544:
542:
541:
536:
460:
458:
457:
452:
434:
432:
431:
426:
384:
382:
381:
376:
318:
316:
315:
310:
180:
174:
173:
170:
169:
166:
163:
160:
157:
154:
151:
148:
145:
142:
130:
123:
122:
119:
118:
115:
112:
109:
106:
103:
100:
97:
94:
81:
76:
75:
70:
65:
21:Homotopic groups
5972:
5971:
5967:
5966:
5965:
5963:
5962:
5961:
5947:Homotopy theory
5937:
5936:
5935:
5930:
5861:
5843:
5839:Urysohn's lemma
5800:
5764:
5650:
5641:
5613:low-dimensional
5571:
5566:
5531:
5500:
5482:
5476:
5460:
5455:
5440:
5424:
5420:
5363:
5359:
5338:
5334:
5319:
5305:
5301:
5282:Wayback Machine
5275:
5271:
5256:
5242:
5238:
5225:
5224:
5220:
5211:
5209:
5201:
5200:
5196:
5186:
5184:
5174:
5173:
5169:
5159:
5157:
5153:
5152:
5148:
5144:
5106:
5053:
5049:
5047:
5044:
5043:
5004:
5001:
5000:
4950:
4947:
4946:
4933:by mappings of
4923:homology theory
4892:
4881:
4871:
4861:
4851:
4844:homotopy groups
4805:
4795:
4785:
4775:
4770:are the same: H
4768:homology groups
4711:category theory
4707:
4701:
4670:
4667:
4666:
4663:homotopy groups
4639:
4635:
4615:
4612:
4611:
4594:
4590:
4588:
4585:
4584:
4564:
4560:
4545:
4541:
4539:
4536:
4535:
4515:
4511:
4491:
4488:
4487:
4459:
4455:
4435:
4432:
4431:
4385:
4382:
4381:
4378:
4372:
4340:
4335:
4328:
4318:
4317:
4311:
4300:
4293:
4290:
4283:
4277:
4271:
4264:
4258:
4251:
4250:
4239:
4234:
4220:
4217:
4211:
4206:
4178:timelike curves
4162:
4148:
4141:
4134:
4127:
4119:ambient isotopy
4116:
4105:
4081:
4074:
4006:) = 2
3941:ambient isotopy
3890:
3889:
3888:
3887:
3875:
3874:
3873:
3865:
3864:
3853:
3801:
3791:
3760:
3738:
3687:
3682:
3658:
3652:
3638:
3636:
3621:
3617:
3606:
3601:homotopy groups
3556:The (singular)
3549:if and only if
3535:if and only if
3506:
3485:
3482:
3481:
3465:
3462:
3461:
3442:
3439:
3438:
3422:
3419:
3418:
3401:
3397:
3395:
3392:
3391:
3372:
3369:
3368:
3351:
3347:
3345:
3342:
3341:
3322:
3319:
3318:
3298:
3295:
3294:
3274:
3271:
3270:
3267:
3238:
3235:
3234:
3215:
3210:
3207:
3206:
3187:
3184:
3183:
3164:
3161:
3160:
3119:
3114:
3113:
3080:
3075:
3074:
3062:
3057:
3056:
3054:
3051:
3050:
3033:
3028:
3027:
3018:
3013:
3012:
3010:
3007:
3006:
2978:
2975:
2974:
2945:
2941:
2932:
2927:
2926:
2917:
2912:
2911:
2909:
2906:
2905:
2876:
2871:
2870:
2868:
2865:
2864:
2841:
2837:
2816:
2811:
2810:
2802:
2799:
2798:
2781:
2777:
2775:
2772:
2771:
2743:
2740:
2739:
2708:
2704:
2683:
2678:
2677:
2675:
2672:
2671:
2639:
2634:
2633:
2631:
2628:
2627:
2606:
2602:
2600:
2597:
2596:
2574:
2569:
2568:
2567:
2563:
2536:
2532:
2503:
2500:
2499:
2482:
2477:
2476:
2474:
2471:
2470:
2453:
2449:
2447:
2444:
2443:
2424:
2419:
2418:
2417:
2413:
2411:
2408:
2407:
2390:
2385:
2384:
2375:
2370:
2369:
2355:
2352:
2351:
2322:
2317:
2316:
2314:
2311:
2310:
2293:
2288:
2287:
2285:
2282:
2281:
2277:
2254:
2244: ∘
2240:
2238:
2228: ∘
2224:
2217:
2191:are said to be
2182:
2172: ∘
2168:
2166:
2152: ∘
2148:
2134:
2120:
2094:
2073:
2072:
2044:
2026:
2025:
2019:
2015:
2008:
1984:
1980:
1970:
1968:
1965:
1964:
1936:
1935:
1911:
1910:
1906:
1904:
1901:
1900:
1880:
1876:
1867:
1863:
1848:
1847:
1843:
1841:
1838:
1837:
1797:
1792:
1791:
1784:
1780:
1771:
1767:
1765:
1762:
1761:
1760:; i.e. the set
1732:
1728:
1719:
1715:
1704:
1700:
1699:
1695:
1693:
1690:
1689:
1671:
1670:
1615:
1597:
1596:
1586:
1543:
1541:
1538:
1537:
1528:linear homotopy
1477:
1474:
1473:
1470:Euclidean space
1448:
1443:
1442:
1434:
1431:
1430:
1405:
1401:
1389:
1385:
1363:
1359:
1336:
1333:
1332:
1315:
1310:
1309:
1283:
1275:
1272:
1271:
1270:, then the map
1249:
1245:
1238:
1234:
1217:
1214:
1213:
1191:
1187:
1180:
1176:
1159:
1156:
1155:
1138:
1133:
1132:
1124:
1110:
1107:
1106:
1102:
1086:
1080: ∘
1079:
1073:
1070:
1064: ∘
1063:
1057:
1046:
1039:
1033:
1022:
1015:
1009:
970:
951:
926:
912:
871:
867:
865:
862:
861:
814:
810:
808:
805:
804:
788:
785:
784:
750:
747:
746:
720:
716:
696:
693:
692:
666:
662:
660:
657:
656:
633:
629:
627:
624:
623:
589:
586:
585:
556:
552:
550:
547:
546:
512:
509:
508:
440:
437:
436:
390:
387:
386:
340:
337:
336:
274:
271:
270:
221:
186:homotopy groups
178:
139:
135:
128:
91:
87:
30:The two dashed
24:
17:
12:
11:
5:
5970:
5960:
5959:
5954:
5949:
5932:
5931:
5929:
5928:
5918:
5917:
5916:
5911:
5906:
5891:
5881:
5871:
5859:
5848:
5845:
5844:
5842:
5841:
5836:
5831:
5826:
5821:
5816:
5810:
5808:
5802:
5801:
5799:
5798:
5793:
5788:
5786:Winding number
5783:
5778:
5772:
5770:
5766:
5765:
5763:
5762:
5757:
5752:
5747:
5742:
5737:
5732:
5727:
5726:
5725:
5720:
5718:homotopy group
5710:
5709:
5708:
5703:
5698:
5693:
5688:
5678:
5673:
5668:
5658:
5656:
5652:
5651:
5644:
5642:
5640:
5639:
5634:
5629:
5628:
5627:
5617:
5616:
5615:
5605:
5600:
5595:
5590:
5585:
5579:
5577:
5573:
5572:
5565:
5564:
5557:
5550:
5542:
5536:
5535:
5529:
5516:
5498:
5480:
5474:
5466:Basic Topology
5459:
5456:
5454:
5453:
5438:
5418:
5357:
5332:
5317:
5299:
5269:
5254:
5236:
5218:
5194:
5167:
5145:
5143:
5140:
5139:
5138:
5133:
5128:
5123:
5118:
5113:
5105:
5102:
5090:omega-spectrum
5073:
5070:
5067:
5064:
5061:
5056:
5052:
5023:
5020:
5017:
5014:
5011:
5008:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4954:
4891:
4888:
4877:
4867:
4857:
4847:
4824:path connected
4801:
4791:
4781:
4771:
4703:Main article:
4700:
4697:
4680:
4677:
4674:
4665:. In the case
4647:
4642:
4638:
4634:
4631:
4628:
4625:
4622:
4619:
4597:
4593:
4572:
4567:
4563:
4559:
4556:
4553:
4548:
4544:
4523:
4518:
4514:
4510:
4507:
4504:
4501:
4498:
4495:
4462:
4458:
4454:
4451:
4448:
4445:
4442:
4439:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4376:Homotopy group
4374:Main article:
4371:
4368:
4309:
4298:
4288:
4275:
4256:
4213:Main article:
4210:
4207:
4205:
4202:
4161:
4158:
4154:smooth isotopy
4146:
4139:
4132:
4125:
4114:
4103:
4079:
4072:
4045:, −
3877:
3876:
3867:
3866:
3858:
3857:
3856:
3855:
3854:
3852:
3849:
3686:
3683:
3681:
3678:
3666:
3665:
3656:
3634:
3619:
3615:
3604:
3577:
3554:
3540:
3533:path-connected
3505:
3502:
3489:
3469:
3446:
3426:
3404:
3400:
3376:
3354:
3350:
3326:
3302:
3291:null-homotopic
3289:is said to be
3278:
3266:
3263:
3262:
3261:
3254:
3242:
3222:
3218:
3214:
3204:quotient space
3191:
3168:
3157:
3145:
3142:
3139:
3136:
3133:
3128:
3125:
3122:
3117:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3089:
3086:
3083:
3078:
3073:
3070:
3065:
3060:
3036:
3031:
3026:
3021:
3016:
2994:
2991:
2988:
2985:
2982:
2960:
2957:
2954:
2951:
2948:
2944:
2940:
2935:
2930:
2925:
2920:
2915:
2903:
2896:
2882:
2879:
2874:
2850:
2847:
2844:
2840:
2836:
2833:
2830:
2827:
2824:
2819:
2814:
2809:
2806:
2784:
2780:
2759:
2756:
2753:
2750:
2747:
2733:
2732:
2731:
2717:
2714:
2711:
2707:
2703:
2700:
2697:
2694:
2691:
2686:
2681:
2656:
2653:
2650:
2647:
2642:
2637:
2609:
2605:
2593:
2577:
2572:
2566:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2539:
2535:
2531:
2528:
2525:
2522:
2519:
2516:
2513:
2510:
2507:
2485:
2480:
2456:
2452:
2427:
2422:
2416:
2393:
2388:
2383:
2378:
2373:
2368:
2365:
2362:
2359:
2339:
2336:
2333:
2330:
2325:
2320:
2296:
2291:
2276:
2273:
2272:
2271:
2264:
2250:
2249:is equal to id
2234:
2216:
2213:
2178:
2162:
2093:
2090:
2089:
2088:
2087:
2086:
2071:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2045:
2043:
2040:
2037:
2034:
2031:
2028:
2027:
2022:
2018:
2014:
2011:
2009:
2007:
2004:
2001:
1998:
1995:
1992:
1987:
1983:
1979:
1976:
1973:
1972:
1943:
1940:
1934:
1931:
1928:
1925:
1918:
1915:
1909:
1883:
1879:
1875:
1870:
1866:
1862:
1855:
1852:
1846:
1824:
1820:
1817:
1814:
1811:
1808:
1805:
1800:
1795:
1790:
1787:
1783:
1779:
1774:
1770:
1735:
1731:
1727:
1722:
1718:
1714:
1707:
1703:
1698:
1686:
1685:
1684:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1616:
1614:
1611:
1608:
1605:
1602:
1599:
1598:
1595:
1592:
1589:
1587:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1545:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1481:
1451:
1446:
1441:
1438:
1427:
1414:
1408:
1404:
1400:
1397:
1392:
1388:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1362:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1318:
1313:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1286:
1282:
1279:
1258:
1252:
1248:
1244:
1241:
1237:
1233:
1230:
1227:
1224:
1221:
1200:
1194:
1190:
1186:
1183:
1179:
1175:
1172:
1169:
1166:
1163:
1141:
1136:
1131:
1127:
1123:
1120:
1117:
1114:
1101:
1098:
1084:
1077:
1068:
1061:
1044:
1037:
1020:
1013:
969:
966:
947:
921:is the torus,
885:
882:
879:
874:
870:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
817:
813:
792:
772:
769:
766:
763:
760:
757:
754:
734:
731:
728:
723:
719:
715:
712:
709:
706:
703:
700:
677:
674:
669:
665:
644:
641:
636:
632:
611:
608:
605:
602:
599:
596:
593:
573:
570:
567:
564:
559:
555:
534:
531:
528:
525:
522:
519:
516:
450:
447:
444:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
344:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
220:
217:
42:, a branch of
15:
9:
6:
4:
3:
2:
5969:
5958:
5955:
5953:
5950:
5948:
5945:
5944:
5942:
5927:
5919:
5915:
5912:
5910:
5907:
5905:
5902:
5901:
5900:
5892:
5890:
5886:
5882:
5880:
5876:
5872:
5870:
5865:
5860:
5858:
5850:
5849:
5846:
5840:
5837:
5835:
5832:
5830:
5827:
5825:
5822:
5820:
5817:
5815:
5812:
5811:
5809:
5807:
5803:
5797:
5796:Orientability
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5774:
5773:
5771:
5767:
5761:
5758:
5756:
5753:
5751:
5748:
5746:
5743:
5741:
5738:
5736:
5733:
5731:
5728:
5724:
5721:
5719:
5716:
5715:
5714:
5711:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5687:
5684:
5683:
5682:
5679:
5677:
5674:
5672:
5669:
5667:
5663:
5660:
5659:
5657:
5653:
5648:
5638:
5635:
5633:
5632:Set-theoretic
5630:
5626:
5623:
5622:
5621:
5618:
5614:
5611:
5610:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5593:Combinatorial
5591:
5589:
5586:
5584:
5581:
5580:
5578:
5574:
5570:
5563:
5558:
5556:
5551:
5549:
5544:
5543:
5540:
5532:
5526:
5522:
5517:
5513:
5509:
5508:
5503:
5499:
5495:
5491:
5490:
5485:
5481:
5477:
5471:
5467:
5462:
5461:
5449:
5445:
5441:
5439:0-89871-544-X
5435:
5431:
5430:
5422:
5414:
5410:
5406:
5402:
5398:
5394:
5390:
5386:
5381:
5380:gr-qc/0609054
5376:
5372:
5368:
5361:
5352:
5351:
5346:
5343:
5336:
5328:
5324:
5320:
5318:9780521795401
5314:
5310:
5303:
5295:
5294:
5289:
5283:
5279:
5273:
5265:
5261:
5257:
5255:9780521795401
5251:
5247:
5240:
5232:
5228:
5222:
5208:
5204:
5198:
5183:
5182:
5177:
5171:
5156:
5150:
5146:
5137:
5134:
5132:
5129:
5127:
5124:
5122:
5119:
5117:
5114:
5111:
5108:
5107:
5101:
5099:
5095:
5091:
5087:
5068:
5065:
5062:
5054:
5050:
5041:
5037:
5018:
5015:
5012:
5006:
4999:
4995:
4973:
4970:
4967:
4961:
4958:
4955:
4944:
4940:
4936:
4932:
4928:
4924:
4919:
4917:
4913:
4909:
4905:
4901:
4897:
4887:
4885:
4880:
4875:
4870:
4865:
4860:
4855:
4850:
4845:
4841:
4837:
4833:
4829:
4825:
4821:
4817:
4813:
4809:
4804:
4799:
4794:
4789:
4784:
4779:
4774:
4769:
4765:
4761:
4757:
4753:
4749:
4745:
4741:
4737:
4732:
4730:
4726:
4722:
4718:
4717:
4712:
4706:
4696:
4694:
4678:
4675:
4672:
4664:
4659:
4640:
4632:
4629:
4626:
4595:
4591:
4565:
4561:
4557:
4554:
4546:
4542:
4516:
4508:
4505:
4502:
4486:
4482:
4478:
4460:
4452:
4449:
4446:
4440:
4437:
4429:
4425:
4421:
4405:
4399:
4396:
4393:
4390:
4387:
4377:
4367:
4365:
4361:
4356:
4354:
4348:
4343:
4338:
4331:
4326:
4321:
4315:
4308:
4304:
4296:
4286:
4281:
4274:
4267:
4262:
4254:
4247:
4242:
4237:
4231:
4227:
4223:
4216:
4201:
4199:
4195:
4191:
4187:
4183:
4179:
4175:
4171:
4167:
4157:
4155:
4150:
4145:
4138:
4131:
4124:
4120:
4113:
4109:
4102:
4098:
4094:
4089:
4085:
4078:
4071:
4067:
4063:
4058:
4056:
4052:
4048:
4044:
4040:
4036:
4032:
4028:
4024:
4020:
4015:
4013:
4010: −
4009:
4005:
4001:
3997:
3993:
3989:
3985:
3981:
3977:
3973:
3969:
3965:
3961:
3957:
3953:
3949:
3944:
3942:
3937:
3935:
3931:
3927:
3923:
3919:
3915:
3911:
3907:
3903:
3899:
3895:
3885:
3881:
3871:
3862:
3848:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3818:
3814:
3810:
3804:
3798:
3794:
3787:
3783:
3779:
3775:
3771:
3767:
3763:
3758:
3754:
3749:
3745:
3741:
3736:
3732:
3728:
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3677:
3675:
3671:
3662:
3655:
3649:
3645:
3641:
3633:
3629:
3625:
3614:
3610:
3602:
3598:
3594:
3590:
3586:
3582:
3578:
3575:
3571:
3567:
3563:
3559:
3555:
3552:
3548:
3544:
3541:
3538:
3534:
3530:
3527:
3526:
3525:
3523:
3519:
3515:
3511:
3501:
3487:
3467:
3458:
3444:
3424:
3402:
3398:
3390:
3374:
3367:to any space
3352:
3348:
3340:
3324:
3316:
3315:null-homotopy
3300:
3292:
3276:
3265:Null-homotopy
3259:
3255:
3240:
3220:
3216:
3212:
3205:
3189:
3182:
3166:
3158:
3137:
3131:
3126:
3123:
3120:
3098:
3092:
3087:
3084:
3081:
3068:
3063:
3034:
3024:
3019:
3005:, by writing
2992:
2989:
2986:
2983:
2980:
2958:
2955:
2952:
2949:
2946:
2942:
2938:
2933:
2923:
2918:
2904:
2901:
2900:vector bundle
2897:
2880:
2877:
2848:
2845:
2842:
2838:
2828:
2822:
2817:
2807:
2804:
2782:
2778:
2757:
2751:
2748:
2745:
2738:
2734:
2715:
2712:
2709:
2705:
2701:
2695:
2689:
2684:
2669:
2668:
2651:
2645:
2640:
2625:
2607:
2603:
2594:
2575:
2564:
2560:
2554:
2551:
2548:
2542:
2537:
2533:
2529:
2526:
2523:
2517:
2514:
2511:
2505:
2483:
2454:
2450:
2425:
2414:
2391:
2376:
2366:
2363:
2360:
2357:
2334:
2328:
2323:
2294:
2279:
2278:
2269:
2265:
2262:
2258:
2257:
2256:
2253:
2247:
2243:
2237:
2231:
2227:
2222:
2221:homeomorphism
2212:
2210:
2206:
2202:
2198:
2197:homotopy type
2194:
2190:
2186:
2181:
2175:
2171:
2165:
2160:
2155:
2151:
2145:
2141:
2137:
2131:
2127:
2123:
2119:
2115:
2111:
2107:
2103:
2099:
2069:
2066:
2060:
2057:
2054:
2046:
2038:
2035:
2032:
2020:
2016:
2010:
2002:
1999:
1996:
1990:
1985:
1981:
1977:
1974:
1963:
1962:
1960:
1938:
1932:
1926:
1913:
1907:
1899:
1881:
1877:
1868:
1864:
1860:
1850:
1844:
1822:
1818:
1815:
1809:
1803:
1798:
1788:
1785:
1781:
1777:
1772:
1768:
1759:
1755:
1751:
1733:
1729:
1720:
1716:
1712:
1705:
1701:
1696:
1687:
1667:
1661:
1655:
1652:
1649:
1643:
1637:
1631:
1628:
1625:
1617:
1609:
1606:
1603:
1593:
1588:
1580:
1577:
1574:
1568:
1562:
1559:
1556:
1550:
1547:
1536:
1535:
1533:
1529:
1525:
1509:
1500:
1497:
1494:
1488:
1485:
1482:
1479:
1471:
1467:
1449:
1439:
1436:
1428:
1412:
1406:
1402:
1398:
1395:
1390:
1386:
1379:
1376:
1373:
1367:
1364:
1360:
1356:
1350:
1347:
1344:
1338:
1316:
1300:
1297:
1294:
1288:
1280:
1277:
1256:
1250:
1246:
1242:
1239:
1235:
1231:
1225:
1219:
1198:
1192:
1188:
1184:
1181:
1177:
1173:
1167:
1161:
1154:are given by
1139:
1121:
1118:
1115:
1112:
1104:
1103:
1097:
1094:
1090:
1083:
1076:
1067:
1060:
1054:
1050:
1043:
1036:
1030:
1026:
1019:
1012:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
965:
963:
959:
955:
950:
946:
942:
938:
934:
929:
924:
920:
915:
910:
906:
902:
897:
880:
872:
868:
844:
841:
838:
832:
829:
823:
815:
811:
790:
767:
764:
761:
755:
752:
729:
721:
717:
707:
704:
701:
691:
675:
672:
667:
663:
642:
639:
634:
630:
606:
603:
600:
594:
591:
571:
565:
562:
557:
553:
532:
526:
523:
520:
517:
514:
505:
503:
499:
495:
491:
487:
483:
479:
475:
472:as time then
471:
467:
462:
448:
445:
442:
419:
413:
410:
404:
401:
398:
392:
369:
363:
360:
354:
351:
348:
342:
334:
330:
329:unit interval
326:
323:of the space
322:
306:
297:
294:
291:
285:
282:
279:
276:
269:
265:
261:
257:
253:
250:
242:
238:
234:
230:
225:
216:
214:
210:
206:
201:
199:
195:
191:
187:
183:
182:
172:
133:
132:
121:
85:
80:
69:
61:
60:Ancient Greek
57:
53:
49:
45:
41:
33:
28:
22:
5926:Publications
5791:Chern number
5781:Betti number
5712:
5664: /
5655:Key concepts
5603:Differential
5523:. Springer.
5520:
5505:
5487:
5468:. Springer.
5465:
5428:
5421:
5370:
5366:
5360:
5348:
5335:
5308:
5302:
5291:
5278:Ghostarchive
5276:Archived at
5272:
5245:
5239:
5230:
5221:
5210:. Retrieved
5206:
5197:
5185:. Retrieved
5179:
5170:
5158:. Retrieved
5149:
5097:
5085:
5035:
4996:to the
4993:
4942:
4938:
4934:
4930:
4920:
4893:
4890:Applications
4883:
4878:
4873:
4868:
4863:
4858:
4853:
4848:
4839:
4835:
4831:
4827:
4819:
4815:
4811:
4807:
4802:
4797:
4792:
4787:
4782:
4777:
4772:
4763:
4759:
4751:
4747:
4743:
4739:
4735:
4733:
4724:
4720:
4714:
4708:
4660:
4480:
4479:with itself
4430:. If we fix
4427:
4423:
4379:
4364:cofibrations
4357:
4346:
4341:
4336:
4329:
4324:
4319:
4313:
4306:
4301:is called a
4294:
4284:
4279:
4272:
4265:
4260:
4252:
4245:
4240:
4235:
4233:and a cover
4229:
4225:
4221:
4218:
4176:between two
4163:
4153:
4151:
4143:
4136:
4129:
4122:
4111:
4107:
4100:
4096:
4092:
4076:
4069:
4059:
4054:
4046:
4042:
4041:) = (−
4038:
4034:
4030:
4026:
4016:
4011:
4007:
4003:
3999:
3995:
3991:
3987:
3983:
3979:
3975:
3971:
3967:
3963:
3959:
3955:
3951:
3947:
3945:
3938:
3933:
3929:
3925:
3921:
3917:
3913:
3905:
3901:
3897:
3893:
3891:
3884:trefoil knot
3844:
3840:
3836:
3832:
3824:
3820:
3816:
3808:
3802:
3796:
3792:
3785:
3781:
3777:
3773:
3769:
3765:
3761:
3756:
3752:
3747:
3743:
3739:
3734:
3730:
3726:
3722:
3714:
3710:
3706:
3702:
3698:
3694:
3688:
3667:
3660:
3653:
3647:
3643:
3639:
3631:
3627:
3623:
3612:
3608:
3596:
3592:
3584:
3580:
3569:
3565:
3550:
3542:
3536:
3528:
3521:
3517:
3513:
3507:
3459:
3314:
3290:
3268:
2770:with fibers
2737:fiber bundle
2268:Möbius strip
2251:
2245:
2241:
2235:
2229:
2225:
2218:
2209:contractible
2204:
2200:
2196:
2192:
2188:
2184:
2179:
2173:
2169:
2163:
2159:identity map
2153:
2149:
2147:, such that
2143:
2139:
2135:
2129:
2125:
2121:
2113:
2109:
2105:
2101:
2097:
2095:
1753:
1752:on the unit
1531:
1527:
1092:
1088:
1081:
1074:
1065:
1058:
1052:
1048:
1041:
1034:
1028:
1024:
1017:
1010:
1001:
997:
989:
985:
981:
977:
973:
971:
961:
957:
953:
948:
944:
940:
936:
932:
927:
922:
918:
913:
908:
904:
898:
506:
501:
497:
493:
489:
485:
481:
477:
476:describes a
473:
469:
463:
332:
324:
263:
259:
255:
251:
246:
236:
209:CW complexes
202:
192:, important
181:-moh-toh-pee
83:
55:
37:
5889:Wikiversity
5806:Key results
4758:induced by
4084:dimensional
4082:, in three-
4066:knot theory
4029:defined by
3954:) = −
3339:unit circle
3269:A function
1534:) given by
44:mathematics
5941:Categories
5735:CW complex
5676:Continuity
5666:Closed set
5625:cohomology
5484:"Homotopy"
5212:2019-08-17
5142:References
4945:, the set
4925:: one can
4866:) : π
4810:) for all
4790:) : H
4736:functorial
4353:fibrations
4334:such that
4270:such that
4204:Properties
3910:embeddings
3813:retraction
3759:such that
3746:× →
3574:isomorphic
3504:Invariance
3181:CW complex
1468:subset of
968:Properties
901:embeddings
688:, and the
622:such that
335:such that
229:embeddings
194:invariants
5914:geometric
5909:algebraic
5760:Cobordism
5696:Hausdorff
5691:connected
5608:Geometric
5598:Continuum
5588:Algebraic
5512:EMS Press
5494:EMS Press
5413:119707350
5405:0015-9018
5350:MathWorld
5345:"Isotopy"
5116:Homeotopy
4927:represent
4900:algebraic
4618:∂
4543:π
4494:∂
4403:→
4397::
4316:to a map
4088:embedding
4023:unit disc
3847:is used.
3807:Also, if
3637:)) where
3389:unit disk
3337:from the
3132:−
3124:−
3108:→
3093:−
3085:−
3069:×
3025:−
2984:≤
2956:−
2950:−
2939:≃
2924:−
2846:−
2835:→
2823:−
2805:π
2755:→
2746:π
2713:−
2702:≃
2690:−
2646:−
2561:⋅
2552:−
2530:⋅
2518:⋅
2382:→
2367:×
2329:≃
2261:bijection
2058:−
2049:⟼
2013:⟶
1991:×
1942:→
1917:→
1874:→
1854:→
1816:≤
1813:‖
1807:‖
1789:∈
1726:→
1629:−
1620:⟼
1591:⟶
1569:×
1507:→
1440:⊆
1377:−
1331:given by
1307:→
1289:×
1130:→
756:×
714:↦
595:∈
569:→
530:→
466:parameter
446:∈
327:with the
319:from the
304:→
286:×
56:homotopic
50:from one
5879:Wikibook
5857:Category
5745:Manifold
5713:Homotopy
5671:Interior
5662:Open set
5620:Homology
5569:Topology
5448:52377653
5327:45420394
5280:and the
5264:45420394
5187:22 April
5160:22 April
5104:See also
4583:, where
4485:boundary
4323: :
4263:→
4259: :
4244:→
4238: :
4224: :
4190:3-sphere
4170:timelike
4051:rotation
4037:,
4002:,
3932:,
3790:for all
3768:,
3751:between
3742: :
3680:Variants
3659:∈
3642: :
3611:,
3558:homology
2973:for any
2624:1-sphere
2406:between
2275:Examples
2138: :
2124: :
2108:between
1100:Examples
1087: :
1047: :
1023: :
956:, where
435:for all
84:homotopy
40:topology
5904:general
5706:uniform
5686:compact
5637:Digital
5514:, 2001
5496:, 2001
5458:Sources
5385:Bibcode
5293:YouTube
5181:YouTube
4729:functor
4477:crossed
4192:can be
3914:isotopy
3851:Isotopy
3839:. When
1896:be the
1748:be the
984:taking
321:product
241:isotopy
231:of the
213:spectra
131:-tə-pee
5899:Topics
5701:metric
5576:Fields
5527:
5472:
5446:
5436:
5411:
5403:
5325:
5315:
5262:
5252:
5042:group
4713:. The
4370:Groups
4093:deform
3880:unknot
3719:subset
2898:Every
2626:) and
1959:origin
1836:. Let
1466:convex
58:(from
46:, two
5681:Space
5409:S2CID
5375:arXiv
4876:) → π
4856:) = π
4800:) → H
4780:) = H
4746:from
4164:On a
3815:from
3811:is a
3717:is a
3179:of a
2622:(the
1524:paths
1464:is a
484:into
235:into
233:torus
211:, or
79:tópos
74:τόπος
68:homós
32:paths
5525:ISBN
5470:ISBN
5444:OCLC
5434:ISBN
5401:ISSN
5323:OCLC
5313:ISBN
5260:OCLC
5250:ISBN
5189:2022
5162:2022
5038:-th
4902:and
4898:for
4838:and
4830:and
4818:and
4762:and
4742:and
4723:and
4426:and
4303:lift
4128:and
4075:and
3970:) =
3908:are
3896:and
3878:The
3823:and
3800:and
3780:) =
3772:) =
3755:and
3729:and
3713:and
3701:and
3595:and
3583:and
3572:are
3568:and
3560:and
3520:and
2990:<
2878:>
2735:Any
2442:and
2266:The
2203:and
2187:and
2167:and
2133:and
2118:maps
2112:and
2104:, a
2100:and
1758:disk
1688:Let
1530:(or
1522:are
1472:and
1212:and
1072:and
976:and
907:and
655:and
584:for
385:and
254:and
188:and
64:ὁμός
5393:doi
5100:.
4886:).
4750:to
4305:of
4142:to
4060:In
4025:in
3960:not
3958:is
3835:to
3831:of
3819:to
3805:∈ .
3721:of
3709:to
3591:of
3579:If
3564:of
3553:is.
3545:is
3539:is.
3531:is
3417:to
1105:If
1000:to
988:to
925:is
917:.
783:to
690:map
500:to
480:of
468:of
461:.
331:to
196:in
179:HOH
127:hə-
38:In
5943::
5510:,
5504:,
5492:,
5486:,
5442:.
5407:.
5399:.
5391:.
5383:.
5371:38
5369:.
5347:.
5321:.
5290:.
5284::
5258:.
5229:.
5205:.
5178:.
4918:.
4695:.
4658:.
4366:.
4355:.
4345:=
4339:○
4282:○
4278:=
4200:.
4156:.
4014:.
4008:yx
3943:.
3924:,
3795:∈
3663:.)
3646:→
3626:,
3256:A
2667:.
2565:id
2415:id
2219:A
2211:.
2161:id
2142:→
2128:→
1933::=
1778::=
1697:id
1174::=
1091:→
1051:→
1040:,
1027:→
1016:,
931:,
903:,
215:.
207:,
200:.
175:,
168:iː
162:oʊ
153:oʊ
147:oʊ
134:;
129:MO
124:,
117:iː
62::
5561:e
5554:t
5547:v
5533:.
5478:.
5450:.
5415:.
5395::
5387::
5377::
5353:.
5329:.
5296:.
5266:.
5233:.
5215:.
5191:.
5164:.
5098:G
5086:X
5072:)
5069:G
5066:,
5063:X
5060:(
5055:n
5051:H
5036:n
5022:)
5019:n
5016:,
5013:G
5010:(
5007:K
4994:X
4980:]
4977:)
4974:n
4971:,
4968:G
4965:(
4962:K
4959:,
4956:X
4953:[
4943:X
4939:G
4935:X
4931:X
4884:Y
4882:(
4879:n
4874:X
4872:(
4869:n
4864:g
4862:(
4859:n
4854:f
4852:(
4849:n
4840:g
4836:f
4832:g
4828:f
4820:Y
4816:X
4812:n
4808:Y
4806:(
4803:n
4798:X
4796:(
4793:n
4788:g
4786:(
4783:n
4778:f
4776:(
4773:n
4764:g
4760:f
4752:Y
4748:X
4744:g
4740:f
4725:Y
4721:X
4679:1
4676:=
4673:n
4646:)
4641:n
4637:]
4633:1
4630:,
4627:0
4624:[
4621:(
4596:0
4592:y
4571:)
4566:0
4562:y
4558:,
4555:Y
4552:(
4547:n
4522:)
4517:n
4513:]
4509:1
4506:,
4503:0
4500:[
4497:(
4481:n
4461:n
4457:]
4453:1
4450:,
4447:0
4444:[
4441:=
4438:X
4428:Y
4424:X
4406:Y
4400:X
4394:g
4391:,
4388:f
4349:.
4347:H
4342:H
4337:p
4330:Y
4325:X
4320:H
4314:H
4310:0
4307:h
4299:0
4295:h
4292:(
4289:0
4285:h
4280:p
4276:0
4273:H
4266:Y
4261:X
4257:0
4253:h
4246:Y
4241:Y
4236:p
4230:Y
4226:X
4222:H
4147:2
4144:K
4140:1
4137:K
4133:2
4130:K
4126:1
4123:K
4115:2
4112:K
4108:t
4104:1
4101:K
4097:t
4080:2
4077:K
4073:1
4070:K
4055:f
4047:y
4043:x
4039:y
4035:x
4033:(
4031:f
4027:R
4012:x
4004:y
4000:x
3998:(
3996:H
3992:H
3988:f
3984:g
3980:f
3976:f
3972:x
3968:x
3966:(
3964:g
3956:x
3952:x
3950:(
3948:f
3934:t
3930:x
3928:(
3926:H
3922:t
3918:H
3906:Y
3902:X
3898:g
3894:f
3841:K
3837:K
3833:X
3825:f
3821:K
3817:X
3809:g
3803:t
3797:K
3793:k
3788:)
3786:k
3784:(
3782:g
3778:k
3776:(
3774:f
3770:t
3766:k
3764:(
3762:H
3757:g
3753:f
3748:Y
3744:X
3740:H
3735:K
3731:g
3727:f
3723:X
3715:K
3711:Y
3707:X
3703:g
3699:f
3661:X
3657:0
3654:x
3648:Y
3644:X
3640:f
3635:0
3632:x
3630:(
3628:f
3624:Y
3622:(
3620:1
3616:0
3613:x
3609:X
3607:(
3605:1
3597:Y
3593:X
3585:Y
3581:X
3576:.
3570:Y
3566:X
3551:Y
3543:X
3537:Y
3529:X
3522:Y
3518:X
3488:X
3468:X
3445:f
3425:X
3403:2
3399:D
3375:X
3353:1
3349:S
3325:f
3301:f
3277:f
3253:.
3241:X
3221:A
3217:/
3213:X
3190:X
3167:A
3144:)
3141:}
3138:0
3135:{
3127:k
3121:n
3116:R
3111:(
3105:)
3102:}
3099:0
3096:{
3088:k
3082:n
3077:R
3072:(
3064:k
3059:R
3035:k
3030:R
3020:n
3015:R
2993:n
2987:k
2981:0
2959:1
2953:k
2947:n
2943:S
2934:k
2929:R
2919:n
2914:R
2895:.
2881:0
2873:R
2849:1
2843:n
2839:S
2832:}
2829:0
2826:{
2818:n
2813:R
2808::
2783:b
2779:F
2758:B
2752:E
2749::
2730:.
2716:1
2710:n
2706:S
2699:}
2696:0
2693:{
2685:n
2680:R
2655:}
2652:0
2649:{
2641:2
2636:R
2608:1
2604:S
2592:.
2576:n
2571:R
2558:)
2555:t
2549:1
2546:(
2543:+
2538:0
2534:p
2527:t
2524:=
2521:)
2515:,
2512:t
2509:(
2506:H
2484:n
2479:R
2455:0
2451:p
2426:n
2421:R
2392:n
2387:R
2377:n
2372:R
2364:I
2361::
2358:H
2338:}
2335:0
2332:{
2324:n
2319:R
2295:n
2290:R
2252:Y
2246:g
2242:f
2236:X
2230:f
2226:g
2205:Y
2201:X
2189:Y
2185:X
2180:Y
2174:g
2170:f
2164:X
2154:f
2150:g
2144:X
2140:Y
2136:g
2130:Y
2126:X
2122:f
2114:Y
2110:X
2102:Y
2098:X
2070:.
2067:x
2064:)
2061:t
2055:1
2052:(
2042:)
2039:t
2036:,
2033:x
2030:(
2021:n
2017:B
2006:]
2003:1
2000:,
1997:0
1994:[
1986:n
1982:B
1978::
1975:H
1939:0
1930:)
1927:x
1924:(
1914:0
1908:c
1882:n
1878:B
1869:n
1865:B
1861::
1851:0
1845:c
1823:}
1819:1
1810:x
1804::
1799:n
1794:R
1786:x
1782:{
1773:n
1769:B
1756:-
1754:n
1734:n
1730:B
1721:n
1717:B
1713::
1706:n
1702:B
1668:.
1665:)
1662:s
1659:(
1656:g
1653:t
1650:+
1647:)
1644:s
1641:(
1638:f
1635:)
1632:t
1626:1
1623:(
1613:)
1610:t
1607:,
1604:s
1601:(
1594:C
1584:]
1581:1
1578:,
1575:0
1572:[
1566:]
1563:1
1560:,
1557:0
1554:[
1551::
1548:H
1510:C
1504:]
1501:1
1498:,
1495:0
1492:[
1489::
1486:g
1483:,
1480:f
1450:n
1445:R
1437:C
1413:)
1407:x
1403:e
1399:t
1396:+
1391:3
1387:x
1383:)
1380:t
1374:1
1371:(
1368:,
1365:x
1361:(
1357:=
1354:)
1351:t
1348:,
1345:x
1342:(
1339:H
1317:2
1312:R
1304:]
1301:1
1298:,
1295:0
1292:[
1285:R
1281::
1278:H
1257:)
1251:x
1247:e
1243:,
1240:x
1236:(
1232:=
1229:)
1226:x
1223:(
1220:g
1199:)
1193:3
1189:x
1185:,
1182:x
1178:(
1171:)
1168:x
1165:(
1162:f
1140:2
1135:R
1126:R
1122::
1119:g
1116:,
1113:f
1093:Z
1089:X
1085:1
1082:g
1078:2
1075:g
1069:1
1066:f
1062:2
1059:f
1053:Z
1049:Y
1045:2
1042:g
1038:2
1035:f
1029:Y
1025:X
1021:1
1018:g
1014:1
1011:f
1002:Y
998:X
990:g
986:f
982:H
978:g
974:f
962:t
958:t
954:t
949:t
945:h
941:g
937:R
933:f
928:R
923:Y
919:X
914:R
909:g
905:f
884:)
881:x
878:(
873:t
869:h
848:)
845:t
842:,
839:x
836:(
833:H
830:=
827:)
824:x
821:(
816:t
812:h
791:Y
771:]
768:1
765:,
762:0
759:[
753:X
733:)
730:x
727:(
722:t
718:h
711:)
708:t
705:,
702:x
699:(
676:g
673:=
668:1
664:h
643:f
640:=
635:0
631:h
610:]
607:1
604:,
601:0
598:[
592:t
572:Y
566:X
563::
558:t
554:h
533:Y
527:X
524::
521:g
518:,
515:f
502:g
498:f
494:g
490:f
486:g
482:f
474:H
470:H
449:X
443:x
423:)
420:x
417:(
414:g
411:=
408:)
405:1
402:,
399:x
396:(
393:H
373:)
370:x
367:(
364:f
361:=
358:)
355:0
352:,
349:x
346:(
343:H
333:Y
325:X
307:Y
301:]
298:1
295:,
292:0
289:[
283:X
280::
277:H
264:Y
260:X
256:g
252:f
243:.
237:R
171:/
165:p
159:t
156:ˌ
150:m
144:h
141:ˈ
138:/
120:/
114:p
111:ə
108:t
105:ɒ
102:m
99:ˈ
96:ə
93:h
90:/
86:(
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
