665:
772:
483:
2673:
4539:
2426:
2534:
4630:
4107:
1269:
3997:
471:
2205:
1716:
1543:
562:
3613:
1452:
1339:
4154:
3254:
4045:
3557:
4344:
2995:
3098:
1154:
1028:
4300:
4198:
3932:
723:
618:
4446:
3461:
1892:
1394:
350:
2790:
4253:
3774:
2033:
3506:
4950:
3845:
2889:
2848:
3723:
1196:
1070:
765:
660:
1105:
is an empty set. Therefore, the existence of a 0-dimensional hole is equivalent to the space being empty. Hence, non-empty is equivalent to (-1)-connected. For an empty space
237:
184:
405:
2721:
2332:
1583:
5097:
3655:
3296:
3037:
2932:
3881:
3356:
3174:
1752:
4907:
4672:
2058:. Using this set, a space is 0-connected if and only if the 0th homotopy set is the one-point set. The definition of homotopy groups and this homotopy set require that
3130:
2561:
5050:
4805:
876:
3800:
3681:
2452:
1609:
1103:
4772:
4741:
1479:
957:
930:
903:
836:
805:
2733:
A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group).
4702:
4439:
2344:
42:
based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of
4546:
5391:
4050:
3937:
2730:
A connected map (0-connected map) is one that is onto path components (0th homotopy group); this corresponds to the homotopy fiber being non-empty.
2459:
5162:
1654:
5306:
5177:
3413:
Many topological proofs require lower bounds on the homotopical connectivity. There are several "recipes" for proving such lower bounds.
1208:
677:
is a cube (yellow) with a ball removed (white). The 2-dimensional sphere (blue) cannot be continuously shrunk to a single point.
410:
121:
is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point. Equivalently, it is a
5343:
2146:
5167:
24:
1491:
514:
3562:
4837:
1406:
2796:-connected map. The single point set is contractible, so all its homotopy groups vanish, and thus "isomorphism below
1287:
4112:
3189:
5298:
4006:
3518:
3508:. This is useful for computing homotopical connectivity, since the homological groups can be computed more easily.
4305:
507:. It cannot be shrunk continuously to a point in X; therefore; X has a 2-dimensional hole. Another example is the
5386:
2939:
363:
A slightly different definition of connectivity, which makes some computations simpler, is: the smallest integer
3045:
1112:
986:
5396:
4261:
4159:
3886:
3464:
684:
579:
5052:
into a more general topological space, such as the space of all continuous maps between two associated spaces
2077:
if and only if its 0th homotopy group vanishes identically, as path-connectedness implies that any two points
1482:
3428:
1855:
1344:
317:
2762:
4207:
3728:
1932:
1075:
A 0-dimensional hole is a missing 0-dimensional ball. A 0-dimensional ball is a single point; its boundary
3473:
51:
17:
3805:
4912:
2868:
2827:
3686:
1159:
1033:
771:
728:
673:
A 3-dimensional hole (a hole with a 2-dimensional boundary) is shown on the figure at the right. Here,
664:
623:
196:
143:
5269:
Frick, Florian; Soberón, Pablo (2020-05-11). "The topological
Tverberg problem beyond prime powers".
5189:
374:
2681:
2305:
1548:
482:
5121:
5055:
4358:
3618:
3259:
138:
4674:
a set of two disconnected points. There is a 1-dimensional hole between the points, so the eta is
3003:
2898:
5270:
5104:
5020:
3850:
3325:
3143:
2668:{\displaystyle \pi _{n}(X)\mathrel {\overset {\pi _{n}(f)}{\to }} \pi _{n}(Y)\to \pi _{n-1}(Ff).}
1721:
1202:
126:
31:
4886:
4635:
4534:{\displaystyle {\text{conn}}_{\pi }(K*L)\geq {\text{conn}}_{\pi }(K)+{\text{conn}}_{\pi }(L)+2.}
568:
through it. In general, a space contains a 1-dimensional-boundary hole if and only if it is not
5362:
5334:
5302:
5227:
5205:
5185:
5173:
5107:
or "h-principle". There are a number of powerful general techniques for proving h-principles.
4353:-th homology group of a simplicial complex depends only on the simplices of dimension at most
3103:
2114:
2047:
39:
23:
This article is about the concept in algebraic topology. For other connectivity concepts, see
5026:
5023:, cases when the inclusion of a geometrically-defined space, such as the space of immersions
4777:
848:
499:. An example is shown on the figure at the right. The yellow region is the topological space
5352:
5217:
5131:
3779:
3660:
3422:
2431:
2235:
1922:, respectively, which is useful in defining 0-connected and 1-connected maps, as below. The
1762:
and 0 denotes the trivial group. The two definitions are equivalent. The requirement for an
1588:
1078:
569:
4750:
4719:
1457:
935:
908:
881:
814:
783:
5116:
4410:
2421:{\displaystyle \pi _{i}(f)\colon \pi _{i}(X)\mathrel {\overset {\sim }{\to }} \pi _{i}(Y)}
2096:
1636:
508:
47:
4681:
4418:
503:; it is a pentagon with a triangle removed. The blue circle is a 1-dimensional sphere in
2302: − 1)-connected space. In terms of homotopy groups, it means that a map
5126:
4985:
4973:
2545:
2292:
2228:
2074:
1759:
976:
58:
5357:
5338:
807:- the zero-dimensional sphere. What is a zero dimensional sphere? - For every integer
5380:
5290:
4850:
1648:
257:
2129:
can be deformed continuously to a constant map. With this definition, we can define
780:
For a 1-dimensional hole (a hole with a 0-dimensional boundary) we need to consider
2122:
979:; see the figure at the right. Hence, path-connected is equivalent to 0-connected.
5245:
5222:
4814:
The general proof is based on a similar formula for the homological connectivity.
967:. If there is such a set, that cannot be continuously shrunk to a single point in
5321:
2540:
The last condition is frequently confusing; it is because the vanishing of the (
2039:
87:
491:
A 2-dimensional hole (a hole with a 1-dimensional boundary) is a circle (S) in
4713:
4406:
4374:
1481:
cannot be continuously shrunk to a single point. This can be proved using the
975:), this means that there is no path between the two points, that is, X is not
5366:
5231:
1914:
The requirements of being non-empty and path-connected can be interpreted as
3373:
This gives a more concrete explanation for the utility of the definition of
2529:{\displaystyle \pi _{n}(f)\colon \pi _{n}(X)\twoheadrightarrow \pi _{n}(Y)}
2242:
95:
1205:
has no holes of any dimension. Therefore, its connectivity is infinite:
959:
is the set of two disjoint points {0, 1}. A zero-dimensional sphere in
4625:{\displaystyle \eta _{\pi }(K*L)\geq \eta _{\pi }(K)+\eta _{\pi }(L).}
839:
565:
122:
57:
An equivalent definition of homotopical connectivity is based on the
1840:-dimensional sphere that is not homotopic to a constant map, so the
260:(- cannot be shrunk continuously to a point). Equivalently, it is a
5275:
4204:
is path-connected but not simply-connected), one should prove that
2050:; the distinguished point is the class of the trivial map, sending
4867:} (the abstract complex recording the intersection pattern of the
4361:). Therefore, the above theorem implies that a simplicial complex
770:
663:
481:
4102:{\displaystyle {\text{conn}}_{H}(X)\geq {\text{conn}}_{\pi }(X)}
3397:-sphere – has the property that any cells in dimensions between
3377:-connectedness: for example, a space where the inclusion of the
5170:: Lectures on Topological Methods in Combinatorics and Geometry
2759:-connected space if and only if the inclusion of the basepoint
2062:
be pointed (have a chosen base point), which cannot be done if
1264:{\displaystyle \eta _{\pi }(X)={\text{conn}}_{\pi }(X)=\infty }
564:. To make a 2-dimensional hole in a 3-dimensional ball, make a
4704:
is a square, which is homeomorphic to a circle, so its eta is
264:
dimensional sphere that cannot be continuously extended to a (
3992:{\displaystyle {\text{conn}}_{H}(X)={\text{conn}}_{\pi }(X).}
371:-dimensional hole. This connectivity parameter is denoted by
2740:-connectivity for spaces can in turn be defined in terms of
486:
A 2-dimensional hole (a hole with a 1-dimensional boundary).
407:, and it differs from the previous parameter by 2, that is,
82:
Homotopical connectivity is defined for maps, too. A map is
466:{\displaystyle \eta _{\pi }(X):={\text{conn}}_{\pi }(X)+2}
5206:"The intersection of a matroid and a simplicial complex"
4258:
The inequality may be strict: there are spaces in which
2200:{\displaystyle \pi _{i}(X)\simeq 0,\quad 0\leq i\leq n.}
1711:{\displaystyle \pi _{d}(X)\cong 0,\quad -1\leq d\leq n,}
572:. Hence, simply-connected is equivalent to 1-connected.
4915:
3186:
In other words, a function which is an isomorphism on
5058:
5029:
4889:
4780:
4753:
4722:
4684:
4638:
4549:
4449:
4421:
4308:
4264:
4210:
4162:
4115:
4053:
4009:
3940:
3889:
3853:
3808:
3782:
3731:
3689:
3663:
3621:
3565:
3521:
3476:
3431:
3328:
3262:
3192:
3146:
3106:
3048:
3006:
2942:
2901:
2871:
2830:
2765:
2684:
2564:
2462:
2434:
2347:
2308:
2149:
1935:
1858:
1724:
1657:
1591:
1551:
1494:
1460:
1409:
1347:
1290:
1211:
1162:
1115:
1081:
1036:
989:
938:
911:
884:
851:
817:
786:
731:
687:
626:
582:
517:
413:
377:
320:
199:
146:
5172:(2nd ed.). Berlin-Heidelberg: Springer-Verlag.
3405:
do not affect the lower-dimensional homotopy types.
3100:
means that if there is a path connecting two points
2723:
vanishes, then the map on the left is a surjection.
1538:{\displaystyle {\text{conn}}_{\pi }(S^{d})\geq d-1}
557:{\displaystyle \mathbb {R} ^{2}\setminus \{(0,0)\}}
511:- the Euclidean plane with a single point removed,
495:, that cannot be shrunk continuously to a point in
106:All definitions below consider a topological space
5091:
5044:
4944:
4901:
4799:
4766:
4735:
4696:
4666:
4624:
4533:
4433:
4338:
4294:
4247:
4192:
4148:
4101:
4039:
3991:
3926:
3875:
3839:
3794:
3768:
3717:
3675:
3649:
3608:{\displaystyle n:={\text{conn}}_{\pi }(X)+1\geq 2}
3607:
3551:
3500:
3455:
3350:
3290:
3248:
3168:
3124:
3092:
3031:
2989:
2926:
2883:
2842:
2784:
2715:
2667:
2528:
2446:
2420:
2326:
2227:is 0-connected if and only if it is non-empty and
2199:
2113:, which is equivalent to the assertion that every
2027:
1886:
1766:-connected space consists of requirements for all
1746:
1710:
1603:
1577:
1537:
1473:
1446:
1388:
1333:
1263:
1190:
1148:
1097:
1064:
1022:
951:
924:
897:
870:
830:
799:
759:
717:
654:
612:
556:
465:
399:
344:
231:
178:
5210:Transactions of the American Mathematical Society
2220:is (−1)-connected if and only if it is non-empty.
1447:{\displaystyle {\text{conn}}_{\pi }(S^{d})<d}
4708:. The join of this square with a third copy of
4543:The identity is simpler with the eta notation:
4381:containing only simplices of dimension at most
2858:can be homotoped into homotopies in the subset
2854: − 1, homotopies in the larger space
1820:is homotopic to a constant map. Therefore, the
1334:{\displaystyle {\text{conn}}_{\pi }(S^{d})=d-1}
268:+1)-dimensional ball. It is sometimes called a
4149:{\displaystyle {\text{conn}}_{\pi }(X)\leq -1}
3725:. Hurewicz theorem says that, in this case,
3249:{\displaystyle \pi _{n-1}(A)\to \pi _{n-1}(X)}
1848:is not trivial. In short, X has a hole with a
293:if it contains no holes of boundary-dimension
5320:See example 2.38 in Hatcher's book. See also
4040:{\displaystyle {\text{conn}}_{\pi }(X)\leq 0}
3552:{\displaystyle {\text{conn}}_{\pi }(X)\geq 1}
1611:can be continuously shrunk to a single point.
8:
4743:, and its eta is 3. In general, the join of
4339:{\displaystyle {\text{conn}}_{H}(X)=\infty }
3393:) – such as the inclusion of a point in the
2234:A space is 1-connected if and only if it is
725:. The smallest dimension of a hole is 3, so
681:is simply-connected but not 2-connected, so
551:
533:
2990:{\displaystyle \pi _{0}(A)\to \pi _{0}(X),}
1030:. The lowest dimension of a hole is 1, so
620:. The lowest dimension of a hole is 2, so
280:+1 is the dimension of the "missing ball").
46:-connectedness generalizes the concepts of
3093:{\displaystyle \pi _{0}(A)\to \pi _{0}(X)}
1828:is trivial. The opposite is also true: If
1149:{\displaystyle {\text{conn}}_{\pi }(X)=-2}
1023:{\displaystyle {\text{conn}}_{\pi }(X)=-1}
971:(or continuously extended to a segment in
125:that cannot be continuously extended to a
5356:
5274:
5221:
5057:
5028:
4936:
4920:
4914:
4888:
4785:
4779:
4758:
4752:
4727:
4721:
4683:
4655:
4637:
4604:
4582:
4554:
4548:
4510:
4505:
4486:
4481:
4456:
4451:
4448:
4420:
4315:
4310:
4307:
4295:{\displaystyle {\text{conn}}_{\pi }(X)=0}
4271:
4266:
4263:
4218:
4212:
4211:
4209:
4193:{\displaystyle {\text{conn}}_{\pi }(X)=0}
4169:
4164:
4161:
4122:
4117:
4114:
4084:
4079:
4060:
4055:
4052:
4016:
4011:
4008:
3971:
3966:
3947:
3942:
3939:
3927:{\displaystyle {\tilde {H_{n}}}(X)\neq 0}
3897:
3891:
3890:
3888:
3858:
3852:
3816:
3810:
3809:
3807:
3781:
3739:
3733:
3732:
3730:
3694:
3688:
3662:
3626:
3620:
3578:
3573:
3564:
3528:
3523:
3520:
3483:
3478:
3475:
3438:
3433:
3430:
3333:
3327:
3267:
3261:
3225:
3197:
3191:
3151:
3145:
3105:
3075:
3053:
3047:
3011:
3005:
2969:
2947:
2941:
2906:
2900:
2870:
2829:
2770:
2764:
2689:
2683:
2638:
2616:
2594:
2584:
2569:
2563:
2544: − 1)-st homotopy group of the
2511:
2489:
2467:
2461:
2433:
2403:
2389:
2374:
2352:
2346:
2307:
2154:
2148:
1978:
1940:
1934:
1863:
1857:
1729:
1723:
1662:
1656:
1590:
1569:
1556:
1550:
1545:, that is, that is, every continuous map
1514:
1501:
1496:
1493:
1465:
1459:
1429:
1416:
1411:
1408:
1365:
1352:
1346:
1310:
1297:
1292:
1289:
1240:
1235:
1216:
1210:
1167:
1161:
1122:
1117:
1114:
1086:
1080:
1041:
1035:
996:
991:
988:
943:
937:
916:
910:
889:
883:
856:
850:
822:
816:
791:
785:
736:
730:
718:{\displaystyle {\text{conn}}_{\pi }(X)=1}
694:
689:
686:
631:
625:
613:{\displaystyle {\text{conn}}_{\pi }(X)=0}
589:
584:
581:
524:
520:
519:
516:
442:
437:
418:
412:
382:
376:
327:
322:
319:
217:
204:
198:
164:
151:
145:
2555:homotopy groups, in the exact sequence:
1808:contains no holes of boundary dimension
576:is 0-connected but not 1-connected, so
5143:
3456:{\displaystyle {\text{conn}}_{\pi }(X)}
1887:{\displaystyle \pi _{d}(X)\not \cong 0}
1836:-dimensional boundary, then there is a
1389:{\displaystyle \eta _{\pi }(S^{d})=d+1}
1198:, which is its smallest possible value.
530:
345:{\displaystyle {\text{conn}}_{\pi }(X)}
2785:{\displaystyle x_{0}\hookrightarrow X}
1852:-dimensional boundary, if-and-only-if
5157:
5155:
5153:
5151:
5149:
5147:
4248:{\displaystyle {\tilde {H_{0}}}(X)=0}
3769:{\displaystyle {\tilde {H_{i}}}(X)=0}
3425:relates the homotopical connectivity
2820:This is instructive for a subset: an
2028:{\displaystyle \pi _{0}(X,*):=\left.}
7:
5339:"Nerves, fibers and homotopy groups"
3501:{\displaystyle {\text{conn}}_{H}(X)}
3314:may be unrelated to the homotopy in
932:, which is the segment . Therefore,
4945:{\textstyle \bigcap _{i\in J}U_{i}}
3840:{\displaystyle {\tilde {H_{n}}}(X)}
2551:corresponds to a surjection on the
1635:0, if it is non-empty, and all its
1399:The proof requires two directions:
1276:Homotopical connectivity of spheres
5204:Aharoni, Ron; Berger, Eli (2006).
4333:
3256:only implies that any elements of
2884:{\displaystyle A\hookrightarrow X}
2865:For example, for an inclusion map
2850:is one such that, up to dimension
2843:{\displaystyle A\hookrightarrow X}
1258:
14:
5103:-connected are said to satisfy a
3718:{\displaystyle \pi _{n}(X)\neq 0}
3366:can be pushed into homotopies in
1191:{\displaystyle \eta _{\pi }(X)=0}
1065:{\displaystyle \eta _{\pi }(X)=1}
760:{\displaystyle \eta _{\pi }(X)=3}
655:{\displaystyle \eta _{\pi }(X)=2}
5392:Properties of topological spaces
2291:which is defined as a map whose
232:{\displaystyle g_{d}:B^{d}\to X}
179:{\displaystyle f_{d}:S^{d}\to X}
5344:Journal of Combinatorial Theory
4960:|+1)-connected, then for every
4369:-connected if and only if its (
2891:to be 1-connected, it must be:
2744:-connectivity of maps: a space
2178:
1686:
963:is just a set of two points in
400:{\displaystyle \eta _{\pi }(X)}
256:dimensional sphere that is not
5083:
5077:
5071:
5068:
5062:
5033:
4616:
4610:
4594:
4588:
4572:
4560:
4522:
4516:
4498:
4492:
4474:
4462:
4327:
4321:
4283:
4277:
4236:
4230:
4224:
4181:
4175:
4134:
4128:
4096:
4090:
4072:
4066:
4028:
4022:
3983:
3977:
3959:
3953:
3915:
3909:
3903:
3870:
3864:
3834:
3828:
3822:
3757:
3751:
3745:
3706:
3700:
3638:
3632:
3590:
3584:
3540:
3534:
3515:is simply-connected, that is,
3495:
3489:
3450:
3444:
3362: − 1) homotopies in
3358:) means that (up to dimension
3345:
3339:
3285:
3279:
3243:
3237:
3218:
3215:
3209:
3163:
3157:
3087:
3081:
3068:
3065:
3059:
3023:
3017:
2981:
2975:
2962:
2959:
2953:
2918:
2912:
2875:
2834:
2776:
2716:{\displaystyle \pi _{n-1}(Ff)}
2710:
2701:
2659:
2650:
2631:
2628:
2622:
2606:
2600:
2586:
2581:
2575:
2523:
2517:
2504:
2501:
2495:
2479:
2473:
2415:
2409:
2391:
2386:
2380:
2364:
2358:
2327:{\displaystyle f\colon X\to Y}
2318:
2166:
2160:
1958:
1946:
1875:
1869:
1741:
1735:
1674:
1668:
1578:{\displaystyle S^{k}\to S^{d}}
1562:
1520:
1507:
1435:
1422:
1371:
1358:
1316:
1303:
1280:In general, for every integer
1252:
1246:
1228:
1222:
1179:
1173:
1134:
1128:
1053:
1047:
1008:
1002:
748:
742:
706:
700:
643:
637:
601:
595:
548:
536:
454:
448:
430:
424:
394:
388:
339:
333:
223:
170:
1:
5358:10.1016/S0097-3165(03)00015-3
5223:10.1090/S0002-9947-06-03833-5
5168:Using the Borsuk-Ulam Theorem
5092:{\displaystyle X(M)\to X(N),}
4838:abstract simplicial complexes
3650:{\displaystyle \pi _{i}(X)=0}
3291:{\displaystyle \pi _{n-1}(A)}
3176:means that in fact a path in
79:homotopy groups are trivial.
5184:Written in cooperation with
4840:, and denote their union by
3140:connecting them, while onto
3032:{\displaystyle \pi _{1}(X).}
2927:{\displaystyle \pi _{0}(X),}
5246:"n-connected space in nLab"
4716:, which is homeomorphic to
3876:{\displaystyle \pi _{n}(X)}
3351:{\displaystyle \pi _{n}(X)}
3169:{\displaystyle \pi _{1}(X)}
2804:" corresponds to the first
2338:-connected if and only if:
1747:{\displaystyle \pi _{i}(X)}
245:d-dimensional-boundary hole
38:is a property describing a
5413:
5299:Cambridge University Press
5003:-connected if-and-only-if
4902:{\displaystyle J\subset I}
4667:{\displaystyle K=L=S^{0}=}
3180:is homotopic to a path in
2726:Low-dimensional examples:
2678:If the group on the right
2252: − 1)-connected.
22:
15:
4003:is not simply-connected (
3322:-connected (so also onto
1797:should be path-connected.
352:, is the largest integer
193:is a continuous function
135:d-dimensional sphere in X
61:of the space. A space is
3465:homological connectivity
3125:{\displaystyle a,b\in A}
2095:can be connected with a
1896:homotopical connectivity
1800:The requirement for any
983:is not 0-connected, so
308:homotopical connectivity
36:homotopical connectivity
16:Not to be confused with
5045:{\displaystyle M\to N,}
4800:{\displaystyle S^{n-1}}
4415:is commonly denoted by
1902:is the largest integer
1816:-dimensional sphere in
1616:Definition using groups
871:{\displaystyle B^{d+1}}
838:is the boundary of the
367:such that X contains a
191:d-dimensional ball in X
86:-connected if it is an
5093:
5046:
4946:
4903:
4883:If, for each nonempty
4801:
4768:
4737:
4698:
4668:
4626:
4535:
4435:
4340:
4296:
4249:
4194:
4156:this is trivial. When
4150:
4103:
4041:
3993:
3928:
3877:
3841:
3796:
3795:{\displaystyle i<n}
3770:
3719:
3677:
3676:{\displaystyle i<n}
3651:
3609:
3553:
3502:
3457:
3352:
3298:that are homotopic in
3292:
3250:
3170:
3126:
3094:
3033:
2991:
2928:
2885:
2844:
2786:
2717:
2669:
2530:
2448:
2447:{\displaystyle i<n}
2428:is an isomorphism for
2422:
2328:
2201:
2042:, not a group, unless
2029:
1888:
1844:-th homotopy group of
1824:-th homotopy group of
1748:
1712:
1605:
1604:{\displaystyle k<d}
1579:
1539:
1475:
1448:
1390:
1335:
1265:
1192:
1150:
1099:
1098:{\displaystyle S^{-1}}
1066:
1024:
953:
926:
899:
872:
832:
801:
776:
761:
719:
669:
656:
614:
558:
487:
467:
401:
346:
233:
180:
102:Definition using holes
18:Homotopic connectivity
5094:
5047:
4980:is isomorphic to the
4947:
4904:
4802:
4769:
4767:{\displaystyle S^{0}}
4738:
4736:{\displaystyle S^{2}}
4699:
4669:
4627:
4536:
4436:
4341:
4297:
4250:
4195:
4151:
4104:
4042:
3994:
3929:
3878:
3842:
3797:
3771:
3720:
3678:
3652:
3610:
3554:
3503:
3458:
3353:
3293:
3251:
3171:
3127:
3095:
3034:
2992:
2929:
2886:
2845:
2824:-connected inclusion
2787:
2718:
2670:
2531:
2449:
2423:
2329:
2202:
2054:to the base point of
2030:
1889:
1749:
1713:
1606:
1580:
1540:
1476:
1474:{\displaystyle S^{d}}
1449:
1391:
1336:
1266:
1193:
1151:
1100:
1067:
1025:
954:
952:{\displaystyle S^{0}}
927:
925:{\displaystyle B^{1}}
900:
898:{\displaystyle S^{0}}
873:
833:
831:{\displaystyle S^{d}}
802:
800:{\displaystyle S^{0}}
775:A 1-dimensional hole.
774:
762:
720:
668:A 3-dimensional hole.
667:
657:
615:
559:
485:
468:
402:
347:
234:
181:
5056:
5027:
4952:is either empty or (
4913:
4887:
4778:
4751:
4720:
4682:
4636:
4547:
4447:
4419:
4306:
4262:
4208:
4160:
4113:
4051:
4007:
3938:
3887:
3851:
3806:
3780:
3729:
3687:
3661:
3619:
3563:
3519:
3474:
3429:
3326:
3260:
3190:
3144:
3104:
3046:
3004:
2940:
2899:
2869:
2828:
2763:
2682:
2562:
2460:
2432:
2345:
2306:
2147:
2069:A topological space
1933:
1856:
1789:The requirement for
1778:The requirement for
1722:
1655:
1589:
1549:
1492:
1458:
1407:
1345:
1288:
1209:
1160:
1113:
1079:
1034:
987:
936:
909:
882:
849:
844:+1)-dimensional ball
815:
784:
729:
685:
624:
580:
515:
411:
375:
318:
274:+1)-dimensional hole
197:
144:
52:simple connectedness
5122:Connective spectrum
4909:, the intersection
4774:is homeomorphic to
4697:{\displaystyle K*L}
4632:As an example, let
4434:{\displaystyle K*L}
4359:simplicial homology
4349:By definition, the
3511:Suppose first that
3136:there is a path in
3132:by passing through
2808:homotopy groups of
1926:can be defined as:
1786:should be nonempty.
1483:Borsuk–Ulam theorem
905:is the boundary of
139:continuous function
5295:Algebraic Topology
5105:homotopy principle
5089:
5042:
5021:geometric topology
5015:Homotopy principle
4942:
4931:
4899:
4797:
4764:
4733:
4694:
4664:
4622:
4531:
4431:
4336:
4292:
4245:
4190:
4146:
4109:still holds. When
4099:
4037:
3989:
3924:
3873:
3837:
3792:
3766:
3715:
3673:
3647:
3605:
3549:
3498:
3453:
3348:
3310:– the homotopy in
3288:
3246:
3166:
3122:
3090:
3029:
2987:
2924:
2881:
2840:
2782:
2713:
2665:
2526:
2444:
2418:
2324:
2264:The corresponding
2197:
2125:of two points) to
2025:
1884:
1832:has a hole with a
1744:
1708:
1601:
1575:
1535:
1471:
1444:
1386:
1331:
1261:
1188:
1146:
1095:
1062:
1020:
949:
922:
895:
868:
828:
797:
777:
757:
715:
670:
652:
610:
554:
488:
463:
397:
342:
229:
176:
48:path-connectedness
32:algebraic topology
5308:978-0-521-79160-1
5216:(11): 4895–4917.
5190:Günter M. Ziegler
5179:978-3-540-00362-5
4916:
4508:
4484:
4454:
4313:
4269:
4227:
4167:
4120:
4082:
4058:
4014:
3969:
3945:
3906:
3847:is isomorphic to
3825:
3748:
3576:
3526:
3481:
3436:
2610:
2397:
2048:topological group
1812:. That is, every
1499:
1414:
1295:
1238:
1120:
994:
692:
587:
440:
325:
90:"up to dimension
73:-simple connected
40:topological space
5404:
5387:General topology
5371:
5370:
5360:
5331:
5325:
5318:
5312:
5311:
5287:
5281:
5280:
5278:
5266:
5260:
5259:
5257:
5256:
5242:
5236:
5235:
5225:
5201:
5195:
5193:
5159:
5132:Simply connected
5098:
5096:
5095:
5090:
5051:
5049:
5048:
5043:
4951:
4949:
4948:
4943:
4941:
4940:
4930:
4908:
4906:
4905:
4900:
4806:
4804:
4803:
4798:
4796:
4795:
4773:
4771:
4770:
4765:
4763:
4762:
4742:
4740:
4739:
4734:
4732:
4731:
4703:
4701:
4700:
4695:
4673:
4671:
4670:
4665:
4660:
4659:
4631:
4629:
4628:
4623:
4609:
4608:
4587:
4586:
4559:
4558:
4540:
4538:
4537:
4532:
4515:
4514:
4509:
4506:
4491:
4490:
4485:
4482:
4461:
4460:
4455:
4452:
4440:
4438:
4437:
4432:
4373:+1)-dimensional
4345:
4343:
4342:
4337:
4320:
4319:
4314:
4311:
4301:
4299:
4298:
4293:
4276:
4275:
4270:
4267:
4254:
4252:
4251:
4246:
4229:
4228:
4223:
4222:
4213:
4199:
4197:
4196:
4191:
4174:
4173:
4168:
4165:
4155:
4153:
4152:
4147:
4127:
4126:
4121:
4118:
4108:
4106:
4105:
4100:
4089:
4088:
4083:
4080:
4065:
4064:
4059:
4056:
4046:
4044:
4043:
4038:
4021:
4020:
4015:
4012:
3998:
3996:
3995:
3990:
3976:
3975:
3970:
3967:
3952:
3951:
3946:
3943:
3933:
3931:
3930:
3925:
3908:
3907:
3902:
3901:
3892:
3882:
3880:
3879:
3874:
3863:
3862:
3846:
3844:
3843:
3838:
3827:
3826:
3821:
3820:
3811:
3801:
3799:
3798:
3793:
3775:
3773:
3772:
3767:
3750:
3749:
3744:
3743:
3734:
3724:
3722:
3721:
3716:
3699:
3698:
3682:
3680:
3679:
3674:
3656:
3654:
3653:
3648:
3631:
3630:
3614:
3612:
3611:
3606:
3583:
3582:
3577:
3574:
3558:
3556:
3555:
3550:
3533:
3532:
3527:
3524:
3507:
3505:
3504:
3499:
3488:
3487:
3482:
3479:
3462:
3460:
3459:
3454:
3443:
3442:
3437:
3434:
3423:Hurewicz theorem
3389: >
3385:-connected (for
3357:
3355:
3354:
3349:
3338:
3337:
3297:
3295:
3294:
3289:
3278:
3277:
3255:
3253:
3252:
3247:
3236:
3235:
3208:
3207:
3175:
3173:
3172:
3167:
3156:
3155:
3131:
3129:
3128:
3123:
3099:
3097:
3096:
3091:
3080:
3079:
3058:
3057:
3038:
3036:
3035:
3030:
3016:
3015:
2996:
2994:
2993:
2988:
2974:
2973:
2952:
2951:
2933:
2931:
2930:
2925:
2911:
2910:
2890:
2888:
2887:
2882:
2849:
2847:
2846:
2841:
2791:
2789:
2788:
2783:
2775:
2774:
2722:
2720:
2719:
2714:
2700:
2699:
2674:
2672:
2671:
2666:
2649:
2648:
2621:
2620:
2611:
2609:
2599:
2598:
2585:
2574:
2573:
2536:is a surjection.
2535:
2533:
2532:
2527:
2516:
2515:
2494:
2493:
2472:
2471:
2453:
2451:
2450:
2445:
2427:
2425:
2424:
2419:
2408:
2407:
2398:
2390:
2379:
2378:
2357:
2356:
2333:
2331:
2330:
2325:
2236:simply connected
2206:
2204:
2203:
2198:
2159:
2158:
2099:which starts in
2034:
2032:
2031:
2026:
2021:
2017:
2016:
2012:
1994:
1990:
1983:
1982:
1945:
1944:
1924:0th homotopy set
1893:
1891:
1890:
1885:
1868:
1867:
1753:
1751:
1750:
1745:
1734:
1733:
1717:
1715:
1714:
1709:
1667:
1666:
1610:
1608:
1607:
1602:
1584:
1582:
1581:
1576:
1574:
1573:
1561:
1560:
1544:
1542:
1541:
1536:
1519:
1518:
1506:
1505:
1500:
1497:
1480:
1478:
1477:
1472:
1470:
1469:
1453:
1451:
1450:
1445:
1434:
1433:
1421:
1420:
1415:
1412:
1395:
1393:
1392:
1387:
1370:
1369:
1357:
1356:
1340:
1338:
1337:
1332:
1315:
1314:
1302:
1301:
1296:
1293:
1270:
1268:
1267:
1262:
1245:
1244:
1239:
1236:
1221:
1220:
1197:
1195:
1194:
1189:
1172:
1171:
1155:
1153:
1152:
1147:
1127:
1126:
1121:
1118:
1104:
1102:
1101:
1096:
1094:
1093:
1071:
1069:
1068:
1063:
1046:
1045:
1029:
1027:
1026:
1021:
1001:
1000:
995:
992:
958:
956:
955:
950:
948:
947:
931:
929:
928:
923:
921:
920:
904:
902:
901:
896:
894:
893:
877:
875:
874:
869:
867:
866:
837:
835:
834:
829:
827:
826:
806:
804:
803:
798:
796:
795:
766:
764:
763:
758:
741:
740:
724:
722:
721:
716:
699:
698:
693:
690:
661:
659:
658:
653:
636:
635:
619:
617:
616:
611:
594:
593:
588:
585:
570:simply-connected
563:
561:
560:
555:
529:
528:
523:
472:
470:
469:
464:
447:
446:
441:
438:
423:
422:
406:
404:
403:
398:
387:
386:
351:
349:
348:
343:
332:
331:
326:
323:
238:
236:
235:
230:
222:
221:
209:
208:
185:
183:
182:
177:
169:
168:
156:
155:
5412:
5411:
5407:
5406:
5405:
5403:
5402:
5401:
5397:Homotopy theory
5377:
5376:
5375:
5374:
5335:Björner, Anders
5333:
5332:
5328:
5319:
5315:
5309:
5289:
5288:
5284:
5268:
5267:
5263:
5254:
5252:
5244:
5243:
5239:
5203:
5202:
5198:
5180:
5161:
5160:
5145:
5140:
5117:Connected space
5113:
5054:
5053:
5025:
5024:
5017:
4995:In particular,
4932:
4911:
4910:
4885:
4884:
4872:
4865:
4859:
4834:
4828:
4820:
4807:and its eta is
4781:
4776:
4775:
4754:
4749:
4748:
4723:
4718:
4717:
4680:
4679:
4651:
4634:
4633:
4600:
4578:
4550:
4545:
4544:
4504:
4480:
4450:
4445:
4444:
4417:
4416:
4395:
4377:(the subset of
4309:
4304:
4303:
4265:
4260:
4259:
4214:
4206:
4205:
4163:
4158:
4157:
4116:
4111:
4110:
4078:
4054:
4049:
4048:
4010:
4005:
4004:
3965:
3941:
3936:
3935:
3934:too. Therefore:
3893:
3885:
3884:
3854:
3849:
3848:
3812:
3804:
3803:
3778:
3777:
3735:
3727:
3726:
3690:
3685:
3684:
3659:
3658:
3622:
3617:
3616:
3572:
3561:
3560:
3522:
3517:
3516:
3477:
3472:
3471:
3432:
3427:
3426:
3419:
3411:
3329:
3324:
3323:
3263:
3258:
3257:
3221:
3193:
3188:
3187:
3147:
3142:
3141:
3102:
3101:
3071:
3049:
3044:
3043:
3007:
3002:
3001:
2965:
2943:
2938:
2937:
2902:
2897:
2896:
2867:
2866:
2826:
2825:
2818:
2766:
2761:
2760:
2754:
2748:with basepoint
2685:
2680:
2679:
2634:
2612:
2590:
2589:
2565:
2560:
2559:
2507:
2485:
2463:
2458:
2457:
2430:
2429:
2399:
2370:
2348:
2343:
2342:
2304:
2303:
2262:
2213:
2150:
2145:
2144:
2140:if and only if
2112:
2105:
2097:continuous path
2090:
2083:
2038:This is only a
2002:
1998:
1974:
1973:
1969:
1968:
1964:
1936:
1931:
1930:
1906:for which X is
1859:
1854:
1853:
1804:≥ 1 means that
1782:=-1 means that
1725:
1720:
1719:
1658:
1653:
1652:
1637:homotopy groups
1618:
1587:
1586:
1565:
1552:
1547:
1546:
1510:
1495:
1490:
1489:
1461:
1456:
1455:
1425:
1410:
1405:
1404:
1361:
1348:
1343:
1342:
1306:
1291:
1286:
1285:
1278:
1234:
1212:
1207:
1206:
1163:
1158:
1157:
1116:
1111:
1110:
1082:
1077:
1076:
1037:
1032:
1031:
990:
985:
984:
939:
934:
933:
912:
907:
906:
885:
880:
879:
852:
847:
846:
818:
813:
812:
787:
782:
781:
732:
727:
726:
688:
683:
682:
627:
622:
621:
583:
578:
577:
518:
513:
512:
509:punctured plane
480:
436:
414:
409:
408:
378:
373:
372:
356:for which X is
321:
316:
315:
213:
200:
195:
194:
160:
147:
142:
141:
104:
75:) if its first
59:homotopy groups
28:
21:
12:
11:
5:
5410:
5408:
5400:
5399:
5394:
5389:
5379:
5378:
5373:
5372:
5337:(2003-04-01).
5326:
5313:
5307:
5291:Hatcher, Allen
5282:
5261:
5237:
5196:
5186:Anders Björner
5178:
5163:Matoušek, Jiří
5142:
5141:
5139:
5136:
5135:
5134:
5129:
5127:Path-connected
5124:
5119:
5112:
5109:
5088:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5041:
5038:
5035:
5032:
5016:
5013:
4986:homotopy group
4974:homotopy group
4939:
4935:
4929:
4926:
4923:
4919:
4898:
4895:
4892:
4870:
4863:
4857:
4832:
4826:
4819:
4816:
4794:
4791:
4788:
4784:
4761:
4757:
4730:
4726:
4693:
4690:
4687:
4663:
4658:
4654:
4650:
4647:
4644:
4641:
4621:
4618:
4615:
4612:
4607:
4603:
4599:
4596:
4593:
4590:
4585:
4581:
4577:
4574:
4571:
4568:
4565:
4562:
4557:
4553:
4530:
4527:
4524:
4521:
4518:
4513:
4503:
4500:
4497:
4494:
4489:
4479:
4476:
4473:
4470:
4467:
4464:
4459:
4430:
4427:
4424:
4407:cell complexes
4394:
4391:
4335:
4332:
4329:
4326:
4323:
4318:
4291:
4288:
4285:
4282:
4279:
4274:
4244:
4241:
4238:
4235:
4232:
4226:
4221:
4217:
4189:
4186:
4183:
4180:
4177:
4172:
4145:
4142:
4139:
4136:
4133:
4130:
4125:
4098:
4095:
4092:
4087:
4077:
4074:
4071:
4068:
4063:
4036:
4033:
4030:
4027:
4024:
4019:
3988:
3985:
3982:
3979:
3974:
3964:
3961:
3958:
3955:
3950:
3923:
3920:
3917:
3914:
3911:
3905:
3900:
3896:
3872:
3869:
3866:
3861:
3857:
3836:
3833:
3830:
3824:
3819:
3815:
3791:
3788:
3785:
3765:
3762:
3759:
3756:
3753:
3747:
3742:
3738:
3714:
3711:
3708:
3705:
3702:
3697:
3693:
3672:
3669:
3666:
3646:
3643:
3640:
3637:
3634:
3629:
3625:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3581:
3571:
3568:
3548:
3545:
3542:
3539:
3536:
3531:
3497:
3494:
3491:
3486:
3452:
3449:
3446:
3441:
3418:
3415:
3410:
3407:
3347:
3344:
3341:
3336:
3332:
3318:– while being
3287:
3284:
3281:
3276:
3273:
3270:
3266:
3245:
3242:
3239:
3234:
3231:
3228:
3224:
3220:
3217:
3214:
3211:
3206:
3203:
3200:
3196:
3165:
3162:
3159:
3154:
3150:
3121:
3118:
3115:
3112:
3109:
3089:
3086:
3083:
3078:
3074:
3070:
3067:
3064:
3061:
3056:
3052:
3042:One-to-one on
3040:
3039:
3028:
3025:
3022:
3019:
3014:
3010:
2998:
2986:
2983:
2980:
2977:
2972:
2968:
2964:
2961:
2958:
2955:
2950:
2946:
2936:one-to-one on
2934:
2923:
2920:
2917:
2914:
2909:
2905:
2880:
2877:
2874:
2839:
2836:
2833:
2817:
2816:Interpretation
2814:
2781:
2778:
2773:
2769:
2752:
2735:
2734:
2731:
2712:
2709:
2706:
2703:
2698:
2695:
2692:
2688:
2676:
2675:
2664:
2661:
2658:
2655:
2652:
2647:
2644:
2641:
2637:
2633:
2630:
2627:
2624:
2619:
2615:
2608:
2605:
2602:
2597:
2593:
2588:
2583:
2580:
2577:
2572:
2568:
2546:homotopy fiber
2538:
2537:
2525:
2522:
2519:
2514:
2510:
2506:
2503:
2500:
2497:
2492:
2488:
2484:
2481:
2478:
2475:
2470:
2466:
2455:
2443:
2440:
2437:
2417:
2414:
2411:
2406:
2402:
2396:
2393:
2388:
2385:
2382:
2377:
2373:
2369:
2366:
2363:
2360:
2355:
2351:
2323:
2320:
2317:
2314:
2311:
2293:homotopy fiber
2268:notion to the
2261:
2260:-connected map
2255:
2254:
2253:
2239:
2232:
2229:path-connected
2221:
2212:
2209:
2208:
2207:
2196:
2193:
2190:
2187:
2184:
2181:
2177:
2174:
2171:
2168:
2165:
2162:
2157:
2153:
2110:
2103:
2088:
2081:
2075:path-connected
2036:
2035:
2024:
2020:
2015:
2011:
2008:
2005:
2001:
1997:
1993:
1989:
1986:
1981:
1977:
1972:
1967:
1963:
1960:
1957:
1954:
1951:
1948:
1943:
1939:
1916:(−1)-connected
1912:
1911:
1883:
1880:
1877:
1874:
1871:
1866:
1862:
1798:
1793:=0 means that
1787:
1760:homotopy group
1743:
1740:
1737:
1732:
1728:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1685:
1682:
1679:
1676:
1673:
1670:
1665:
1661:
1617:
1614:
1613:
1612:
1600:
1597:
1594:
1572:
1568:
1564:
1559:
1555:
1534:
1531:
1528:
1525:
1522:
1517:
1513:
1509:
1504:
1486:
1468:
1464:
1443:
1440:
1437:
1432:
1428:
1424:
1419:
1385:
1382:
1379:
1376:
1373:
1368:
1364:
1360:
1355:
1351:
1330:
1327:
1324:
1321:
1318:
1313:
1309:
1305:
1300:
1277:
1274:
1273:
1272:
1260:
1257:
1254:
1251:
1248:
1243:
1233:
1230:
1227:
1224:
1219:
1215:
1199:
1187:
1184:
1181:
1178:
1175:
1170:
1166:
1145:
1142:
1139:
1136:
1133:
1130:
1125:
1092:
1089:
1085:
1073:
1061:
1058:
1055:
1052:
1049:
1044:
1040:
1019:
1016:
1013:
1010:
1007:
1004:
999:
977:path-connected
946:
942:
919:
915:
892:
888:
865:
862:
859:
855:
825:
821:
794:
790:
769:
768:
756:
753:
750:
747:
744:
739:
735:
714:
711:
708:
705:
702:
697:
671:
651:
648:
645:
642:
639:
634:
630:
609:
606:
603:
600:
597:
592:
553:
550:
547:
544:
541:
538:
535:
532:
527:
522:
479:
476:
475:
474:
462:
459:
456:
453:
450:
445:
435:
432:
429:
426:
421:
417:
396:
393:
390:
385:
381:
361:
341:
338:
335:
330:
304:
281:
240:
228:
225:
220:
216:
212:
207:
203:
187:
175:
172:
167:
163:
159:
154:
150:
103:
100:
13:
10:
9:
6:
4:
3:
2:
5409:
5398:
5395:
5393:
5390:
5388:
5385:
5384:
5382:
5368:
5364:
5359:
5354:
5350:
5346:
5345:
5340:
5336:
5330:
5327:
5323:
5317:
5314:
5310:
5304:
5300:
5296:
5292:
5286:
5283:
5277:
5272:
5265:
5262:
5251:
5247:
5241:
5238:
5233:
5229:
5224:
5219:
5215:
5211:
5207:
5200:
5197:
5194:, Section 4.3
5192:
5191:
5187:
5181:
5175:
5171:
5169:
5164:
5158:
5156:
5154:
5152:
5150:
5148:
5144:
5137:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5114:
5110:
5108:
5106:
5102:
5086:
5080:
5074:
5065:
5059:
5039:
5036:
5030:
5022:
5014:
5012:
5010:
5006:
5002:
4998:
4993:
4991:
4987:
4983:
4979:
4975:
4971:
4967:
4963:
4959:
4955:
4937:
4933:
4927:
4924:
4921:
4917:
4896:
4893:
4890:
4881:
4879:
4878:
4873:
4866:
4856:
4852:
4851:nerve complex
4847:
4845:
4844:
4839:
4835:
4825:
4817:
4815:
4812:
4810:
4792:
4789:
4786:
4782:
4759:
4755:
4746:
4728:
4724:
4715:
4711:
4707:
4691:
4688:
4685:
4677:
4661:
4656:
4652:
4648:
4645:
4642:
4639:
4619:
4613:
4605:
4601:
4597:
4591:
4583:
4579:
4575:
4569:
4566:
4563:
4555:
4551:
4541:
4528:
4525:
4519:
4511:
4501:
4495:
4487:
4477:
4471:
4468:
4465:
4457:
4443:
4428:
4425:
4422:
4414:
4413:
4408:
4405:be non-empty
4404:
4400:
4392:
4390:
4388:
4384:
4380:
4376:
4372:
4368:
4364:
4360:
4356:
4352:
4347:
4330:
4324:
4316:
4289:
4286:
4280:
4272:
4256:
4242:
4239:
4233:
4219:
4215:
4203:
4187:
4184:
4178:
4170:
4143:
4140:
4137:
4131:
4123:
4093:
4085:
4075:
4069:
4061:
4034:
4031:
4025:
4017:
4002:
3986:
3980:
3972:
3962:
3956:
3948:
3921:
3918:
3912:
3898:
3894:
3867:
3859:
3855:
3831:
3817:
3813:
3789:
3786:
3783:
3763:
3760:
3754:
3740:
3736:
3712:
3709:
3703:
3695:
3691:
3670:
3667:
3664:
3644:
3641:
3635:
3627:
3623:
3602:
3599:
3596:
3593:
3587:
3579:
3569:
3566:
3546:
3543:
3537:
3529:
3514:
3509:
3492:
3484:
3469:
3466:
3447:
3439:
3424:
3416:
3414:
3408:
3406:
3404:
3400:
3396:
3392:
3388:
3384:
3381:-skeleton is
3380:
3376:
3371:
3369:
3365:
3361:
3342:
3334:
3330:
3321:
3317:
3313:
3309:
3306:homotopic in
3305:
3301:
3282:
3274:
3271:
3268:
3264:
3240:
3232:
3229:
3226:
3222:
3212:
3204:
3201:
3198:
3194:
3184:
3183:
3179:
3160:
3152:
3148:
3139:
3135:
3119:
3116:
3113:
3110:
3107:
3084:
3076:
3072:
3062:
3054:
3050:
3026:
3020:
3012:
3008:
2999:
2984:
2978:
2970:
2966:
2956:
2948:
2944:
2935:
2921:
2915:
2907:
2903:
2894:
2893:
2892:
2878:
2872:
2863:
2861:
2857:
2853:
2837:
2831:
2823:
2815:
2813:
2811:
2807:
2803:
2799:
2795:
2779:
2771:
2767:
2758:
2751:
2747:
2743:
2739:
2732:
2729:
2728:
2727:
2724:
2707:
2704:
2696:
2693:
2690:
2686:
2662:
2656:
2653:
2645:
2642:
2639:
2635:
2625:
2617:
2613:
2603:
2595:
2591:
2578:
2570:
2566:
2558:
2557:
2556:
2554:
2550:
2547:
2543:
2520:
2512:
2508:
2498:
2490:
2486:
2482:
2476:
2468:
2464:
2456:
2441:
2438:
2435:
2412:
2404:
2400:
2394:
2383:
2375:
2371:
2367:
2361:
2353:
2349:
2341:
2340:
2339:
2337:
2321:
2315:
2312:
2309:
2301:
2297:
2294:
2290:
2288:
2284:
2279:
2275:
2272:notion of an
2271:
2267:
2259:
2256:
2251:
2247:
2245:
2240:
2237:
2233:
2230:
2226:
2222:
2219:
2215:
2214:
2210:
2194:
2191:
2188:
2185:
2182:
2179:
2175:
2172:
2169:
2163:
2155:
2151:
2143:
2142:
2141:
2139:
2137:
2132:
2128:
2124:
2120:
2116:
2109:
2102:
2098:
2094:
2087:
2080:
2076:
2072:
2067:
2065:
2061:
2057:
2053:
2049:
2045:
2041:
2022:
2018:
2013:
2009:
2006:
2003:
1999:
1995:
1991:
1987:
1984:
1979:
1975:
1970:
1965:
1961:
1955:
1952:
1949:
1941:
1937:
1929:
1928:
1927:
1925:
1921:
1917:
1909:
1905:
1901:
1897:
1881:
1878:
1872:
1864:
1860:
1851:
1847:
1843:
1839:
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1796:
1792:
1788:
1785:
1781:
1777:
1776:
1775:
1773:
1769:
1765:
1761:
1757:
1738:
1730:
1726:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1683:
1680:
1677:
1671:
1663:
1659:
1650:
1649:trivial group
1646:
1642:
1638:
1634:
1630:
1628:
1623:
1615:
1598:
1595:
1592:
1570:
1566:
1557:
1553:
1532:
1529:
1526:
1523:
1515:
1511:
1502:
1488:Proving that
1487:
1484:
1466:
1462:
1441:
1438:
1430:
1426:
1417:
1403:Proving that
1402:
1401:
1400:
1398:
1383:
1380:
1377:
1374:
1366:
1362:
1353:
1349:
1328:
1325:
1322:
1319:
1311:
1307:
1298:
1283:
1275:
1255:
1249:
1241:
1231:
1225:
1217:
1213:
1204:
1200:
1185:
1182:
1176:
1168:
1164:
1143:
1140:
1137:
1131:
1123:
1108:
1090:
1087:
1083:
1074:
1059:
1056:
1050:
1042:
1038:
1017:
1014:
1011:
1005:
997:
982:
978:
974:
970:
966:
962:
944:
940:
917:
913:
890:
886:
863:
860:
857:
853:
845:
843:
823:
819:
811:, the sphere
810:
792:
788:
779:
778:
773:
754:
751:
745:
737:
733:
712:
709:
703:
695:
680:
676:
672:
666:
649:
646:
640:
632:
628:
607:
604:
598:
590:
575:
571:
567:
545:
542:
539:
525:
510:
506:
502:
498:
494:
490:
489:
484:
477:
460:
457:
451:
443:
433:
427:
419:
415:
391:
383:
379:
370:
366:
362:
359:
355:
336:
328:
313:
309:
305:
303:
300:
296:
292:
290:
285:
282:
279:
275:
273:
267:
263:
259:
258:nullhomotopic
255:
251:
247:
246:
241:
226:
218:
214:
210:
205:
201:
192:
188:
173:
165:
161:
157:
152:
148:
140:
136:
132:
131:
130:
128:
124:
120:
116:
111:
109:
101:
99:
97:
93:
89:
85:
80:
78:
74:
72:
67:
65:
60:
55:
53:
49:
45:
41:
37:
33:
26:
19:
5351:(1): 88–93.
5348:
5347:. Series A.
5342:
5329:
5316:
5294:
5285:
5264:
5253:. Retrieved
5249:
5240:
5213:
5209:
5199:
5183:
5166:
5100:
5018:
5011:-connected.
5008:
5004:
5000:
4996:
4994:
4989:
4981:
4977:
4969:
4965:
4961:
4957:
4953:
4882:
4876:
4875:
4868:
4861:
4854:
4848:
4842:
4841:
4830:
4823:
4821:
4813:
4808:
4744:
4709:
4705:
4675:
4542:
4442:
4411:
4402:
4398:
4396:
4389:-connected.
4386:
4382:
4378:
4370:
4366:
4362:
4354:
4350:
4348:
4257:
4201:
4000:
3512:
3510:
3467:
3420:
3412:
3409:Lower bounds
3402:
3398:
3394:
3390:
3386:
3382:
3378:
3374:
3372:
3367:
3363:
3359:
3319:
3315:
3311:
3307:
3303:
3299:
3185:
3181:
3177:
3137:
3133:
3041:
2864:
2859:
2855:
2851:
2821:
2819:
2809:
2805:
2801:
2800:and onto at
2797:
2793:
2756:
2749:
2745:
2741:
2737:
2736:
2725:
2677:
2552:
2548:
2541:
2539:
2335:
2299:
2295:
2286:
2282:
2281:
2277:
2273:
2269:
2265:
2263:
2257:
2249:
2243:
2224:
2217:
2135:
2134:
2130:
2126:
2123:discrete set
2118:
2107:
2106:and ends in
2100:
2092:
2085:
2078:
2070:
2068:
2063:
2059:
2055:
2051:
2046:is itself a
2043:
2037:
1923:
1919:
1915:
1913:
1907:
1903:
1899:
1895:
1849:
1845:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1794:
1790:
1783:
1779:
1771:
1767:
1763:
1755:
1754:denotes the
1644:
1640:
1632:
1626:
1625:
1621:
1619:
1397:
1281:
1279:
1106:
980:
972:
968:
964:
960:
841:
808:
678:
674:
573:
504:
500:
496:
492:
368:
364:
357:
353:
311:
307:
302:
298:
294:
288:
287:
283:
277:
271:
269:
265:
261:
253:
249:
244:
243:
190:
134:
129:. Formally,
118:
114:
112:
107:
105:
91:
83:
81:
76:
70:
69:
63:
62:
56:
43:
35:
29:
25:Connectivity
5322:this answer
5250:ncatlab.org
4849:Denote the
4678:. The join
3470:denoted by
2812:vanishing.
2285:-connected
2276:-connected
2040:pointed set
1920:0-connected
1910:-connected.
1454:, that is,
360:-connected.
88:isomorphism
5381:Categories
5276:2005.05251
5255:2017-09-18
5138:References
4747:copies of
4714:octahedron
3304:abstractly
2138:-connected
2066:is empty.
1629:-connected
1624:is called
314:, denoted
291:-connected
286:is called
66:-connected
5367:0097-3165
5232:0002-9947
5072:→
5034:→
4925:∈
4918:⋂
4894:⊂
4790:−
4689:∗
4606:π
4602:η
4584:π
4580:η
4576:≥
4567:∗
4556:π
4552:η
4512:π
4488:π
4478:≥
4469:∗
4458:π
4426:∗
4334:∞
4273:π
4225:~
4171:π
4141:−
4138:≤
4124:π
4086:π
4076:≥
4032:≤
4018:π
3973:π
3919:≠
3904:~
3856:π
3823:~
3746:~
3710:≠
3692:π
3624:π
3600:≥
3580:π
3544:≥
3530:π
3440:π
3331:π
3272:−
3265:π
3230:−
3223:π
3219:→
3202:−
3195:π
3149:π
3117:∈
3073:π
3069:→
3051:π
3009:π
2967:π
2963:→
2945:π
2904:π
2876:↪
2835:↪
2777:↪
2694:−
2687:π
2643:−
2636:π
2632:→
2614:π
2592:π
2587:→
2567:π
2509:π
2505:↠
2487:π
2483::
2465:π
2401:π
2395:∼
2392:→
2372:π
2368::
2350:π
2319:→
2313::
2189:≤
2183:≤
2170:≃
2152:π
2010:∗
1988:∗
1956:∗
1938:π
1861:π
1727:π
1700:≤
1694:≤
1688:−
1678:≅
1660:π
1639:of order
1563:→
1530:−
1524:≥
1503:π
1418:π
1354:π
1350:η
1326:−
1299:π
1259:∞
1242:π
1218:π
1214:η
1169:π
1165:η
1141:−
1124:π
1088:−
1043:π
1039:η
1015:−
998:π
738:π
734:η
696:π
633:π
629:η
591:π
531:∖
444:π
420:π
416:η
384:π
380:η
329:π
224:→
171:→
5293:(2001),
5165:(2007).
5111:See also
4860:, ... ,
4409:. Their
4375:skeleton
4357:+1 (see
3776:for all
3657:for all
3417:Homology
2270:absolute
2266:relative
2223:A space
2216:A space
2211:Examples
1879:≇
1647:are the
1620:A space
478:Examples
96:homotopy
4441:. Then:
4385:+1) is
4047:), then
3463:to the
2298:is an (
2246:-sphere
2115:mapping
5365:
5305:
5230:
5176:
4968:, the
3802:, and
3683:, and
3559:. Let
2792:is an
2755:is an
2280:is an
2133:to be
1718:where
1631:, for
566:tunnel
123:sphere
5271:arXiv
4874:) by
4829:,...,
4818:Nerve
4712:is a
3883:, so
3615:; so
3000:onto
2895:onto
2454:, and
2278:space
2117:from
1894:.The
1341:(and
878:. So
252:is a
137:is a
5363:ISSN
5303:ISBN
5228:ISSN
5188:and
5174:ISBN
5099:are
4984:-th
4972:-th
4853:of {
4822:Let
4507:conn
4483:conn
4453:conn
4412:join
4401:and
4397:Let
4393:Join
4312:conn
4302:but
4268:conn
4200:(so
4166:conn
4119:conn
4081:conn
4057:conn
4013:conn
3968:conn
3944:conn
3787:<
3668:<
3575:conn
3525:conn
3480:conn
3435:conn
3421:The
3401:and
3302:are
2439:<
2248:is (
2084:and
1918:and
1758:-th
1596:<
1585:for
1498:conn
1439:<
1413:conn
1294:conn
1237:conn
1203:ball
1156:and
1119:conn
993:conn
691:conn
586:conn
439:conn
324:conn
306:The
127:ball
115:hole
68:(or
50:and
30:In
5353:doi
5349:102
5218:doi
5214:358
5019:In
5007:is
4999:is
4988:of
4976:of
4836:be
4365:is
4255:.
3999:If
2997:and
2334:is
2287:map
2241:An
2121:(a
2091:in
2073:is
1898:of
1633:n ≥
310:of
248:in
117:in
98:".
94:in
5383::
5361:.
5341:.
5301:,
5297:,
5248:.
5226:.
5212:.
5208:.
5182:.
5146:^
4992:.
4964:≤
4956:−|
4880:.
4846:.
4811:.
4529:2.
4346:.
3570::=
3370:.
3182:A.
3134:X,
2862:.
2549:Ff
2296:Ff
1962::=
1774::
1770:≤
1651::
1643:≤
1284:,
1201:A
1109:,
434::=
297:≤
262:d-
254:d-
242:A
189:A
133:A
113:A
110:.
92:n,
54:.
34:,
5369:.
5355::
5324:.
5279:.
5273::
5258:.
5234:.
5220::
5101:n
5087:,
5084:)
5081:N
5078:(
5075:X
5069:)
5066:M
5063:(
5060:X
5040:,
5037:N
5031:M
5009:k
5005:K
5001:k
4997:N
4990:K
4982:j
4978:N
4970:j
4966:k
4962:j
4958:J
4954:k
4938:i
4934:U
4928:J
4922:i
4897:I
4891:J
4877:N
4871:i
4869:K
4864:n
4862:K
4858:1
4855:K
4843:K
4833:n
4831:K
4827:1
4824:K
4809:n
4793:1
4787:n
4783:S
4760:0
4756:S
4745:n
4729:2
4725:S
4710:K
4706:2
4692:L
4686:K
4676:1
4662:=
4657:0
4653:S
4649:=
4646:L
4643:=
4640:K
4620:.
4617:)
4614:L
4611:(
4598:+
4595:)
4592:K
4589:(
4573:)
4570:L
4564:K
4561:(
4526:+
4523:)
4520:L
4517:(
4502:+
4499:)
4496:K
4493:(
4475:)
4472:L
4466:K
4463:(
4429:L
4423:K
4403:L
4399:K
4387:k
4383:k
4379:K
4371:k
4367:k
4363:K
4355:k
4351:k
4331:=
4328:)
4325:X
4322:(
4317:H
4290:0
4287:=
4284:)
4281:X
4278:(
4243:0
4240:=
4237:)
4234:X
4231:(
4220:0
4216:H
4202:X
4188:0
4185:=
4182:)
4179:X
4176:(
4144:1
4135:)
4132:X
4129:(
4097:)
4094:X
4091:(
4073:)
4070:X
4067:(
4062:H
4035:0
4029:)
4026:X
4023:(
4001:X
3987:.
3984:)
3981:X
3978:(
3963:=
3960:)
3957:X
3954:(
3949:H
3922:0
3916:)
3913:X
3910:(
3899:n
3895:H
3871:)
3868:X
3865:(
3860:n
3835:)
3832:X
3829:(
3818:n
3814:H
3790:n
3784:i
3764:0
3761:=
3758:)
3755:X
3752:(
3741:i
3737:H
3713:0
3707:)
3704:X
3701:(
3696:n
3671:n
3665:i
3645:0
3642:=
3639:)
3636:X
3633:(
3628:i
3603:2
3597:1
3594:+
3591:)
3588:X
3585:(
3567:n
3547:1
3541:)
3538:X
3535:(
3513:X
3496:)
3493:X
3490:(
3485:H
3468:,
3451:)
3448:X
3445:(
3403:n
3399:k
3395:n
3391:k
3387:n
3383:n
3379:k
3375:n
3368:A
3364:X
3360:n
3346:)
3343:X
3340:(
3335:n
3320:n
3316:X
3312:A
3308:A
3300:X
3286:)
3283:A
3280:(
3275:1
3269:n
3244:)
3241:X
3238:(
3233:1
3227:n
3216:)
3213:A
3210:(
3205:1
3199:n
3178:X
3164:)
3161:X
3158:(
3153:1
3138:A
3120:A
3114:b
3111:,
3108:a
3088:)
3085:X
3082:(
3077:0
3066:)
3063:A
3060:(
3055:0
3027:.
3024:)
3021:X
3018:(
3013:1
2985:,
2982:)
2979:X
2976:(
2971:0
2960:)
2957:A
2954:(
2949:0
2922:,
2919:)
2916:X
2913:(
2908:0
2879:X
2873:A
2860:A
2856:X
2852:n
2838:X
2832:A
2822:n
2810:X
2806:n
2802:n
2798:n
2794:n
2780:X
2772:0
2768:x
2757:n
2753:0
2750:x
2746:X
2742:n
2738:n
2711:)
2708:f
2705:F
2702:(
2697:1
2691:n
2663:.
2660:)
2657:f
2654:F
2651:(
2646:1
2640:n
2629:)
2626:Y
2623:(
2618:n
2607:)
2604:f
2601:(
2596:n
2582:)
2579:X
2576:(
2571:n
2553:n
2542:n
2524:)
2521:Y
2518:(
2513:n
2502:)
2499:X
2496:(
2491:n
2480:)
2477:f
2474:(
2469:n
2442:n
2436:i
2416:)
2413:Y
2410:(
2405:i
2387:)
2384:X
2381:(
2376:i
2365:)
2362:f
2359:(
2354:i
2336:n
2322:Y
2316:X
2310:f
2300:n
2289:,
2283:n
2274:n
2258:n
2250:n
2244:n
2238:.
2231:.
2225:X
2218:X
2195:.
2192:n
2186:i
2180:0
2176:,
2173:0
2167:)
2164:X
2161:(
2156:i
2136:n
2131:X
2127:X
2119:S
2111:2
2108:x
2104:1
2101:x
2093:X
2089:2
2086:x
2082:1
2079:x
2071:X
2064:X
2060:X
2056:X
2052:S
2044:X
2023:.
2019:]
2014:)
2007:,
2004:X
2000:(
1996:,
1992:)
1985:,
1980:0
1976:S
1971:(
1966:[
1959:)
1953:,
1950:X
1947:(
1942:0
1908:n
1904:n
1900:X
1882:0
1876:)
1873:X
1870:(
1865:d
1850:d
1846:X
1842:d
1838:d
1834:d
1830:X
1826:X
1822:d
1818:X
1814:d
1810:d
1806:X
1802:d
1795:X
1791:d
1784:X
1780:d
1772:n
1768:d
1764:n
1756:i
1742:)
1739:X
1736:(
1731:i
1706:,
1703:n
1697:d
1691:1
1684:,
1681:0
1675:)
1672:X
1669:(
1664:d
1645:n
1641:d
1627:n
1622:X
1599:d
1593:k
1571:d
1567:S
1558:k
1554:S
1533:1
1527:d
1521:)
1516:d
1512:S
1508:(
1485:.
1467:d
1463:S
1442:d
1436:)
1431:d
1427:S
1423:(
1396:)
1384:1
1381:+
1378:d
1375:=
1372:)
1367:d
1363:S
1359:(
1329:1
1323:d
1320:=
1317:)
1312:d
1308:S
1304:(
1282:d
1271:.
1256:=
1253:)
1250:X
1247:(
1232:=
1229:)
1226:X
1223:(
1186:0
1183:=
1180:)
1177:X
1174:(
1144:2
1138:=
1135:)
1132:X
1129:(
1107:X
1091:1
1084:S
1072:.
1060:1
1057:=
1054:)
1051:X
1048:(
1018:1
1012:=
1009:)
1006:X
1003:(
981:X
973:X
969:X
965:X
961:X
945:0
941:S
918:1
914:B
891:0
887:S
864:1
861:+
858:d
854:B
842:d
840:(
824:d
820:S
809:d
793:0
789:S
767:.
755:3
752:=
749:)
746:X
743:(
713:1
710:=
707:)
704:X
701:(
679:X
675:X
662:.
650:2
647:=
644:)
641:X
638:(
608:0
605:=
602:)
599:X
596:(
574:X
552:}
549:)
546:0
543:,
540:0
537:(
534:{
526:2
521:R
505:X
501:X
497:X
493:X
473:.
461:2
458:+
455:)
452:X
449:(
431:)
428:X
425:(
395:)
392:X
389:(
369:d
365:d
358:n
354:n
340:)
337:X
334:(
312:X
301:.
299:n
295:d
289:n
284:X
278:d
276:(
272:d
270:(
266:d
250:X
239:.
227:X
219:d
215:B
211::
206:d
202:g
186:.
174:X
166:d
162:S
158::
153:d
149:f
119:X
108:X
84:n
77:n
71:n
64:n
44:n
27:.
20:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
