25:
90:
2872:
454:
2071:
662:); in the rose, a neighborhood of the intersection point might not fully contain any of the circles. Additionally, the rose is not compact: the complement of the distinguished point is an infinite union of open intervals; to those add a small open neighborhood of the distinguished point to get an
646:
with infinitely many petals, but these two spaces are not homeomorphic. The difference between their topologies is seen in the fact that, in the
Hawaiian earring, every open neighborhood of the point of intersection of the circles contains all but finitely many of the circles (an
2640:
293:
1948:
2545:
2469:
1822:
2867:{\displaystyle \mathbb {H} _{k}=\bigcup _{n\in \mathbb {N} }\left\{(x_{0},x_{1},\ldots ,x_{k})\in \mathbb {R} ^{k+1}:\left(x_{0}-{\frac {1}{n}}\right)^{2}+x_{1}^{2}+\cdots +x_{k}^{2}={\frac {1}{n^{2}}}\right\}.}
3514:
2342:
3379:
2269:
2119:
823:
2223:
209:
971:
3330:
3429:
3139:
1013:
arise from loops whose image is not contained in finitely many of the
Hawaiian earring's circles; in fact, some of them are surjective. For example, the path that on the interval
2380:
1941:
1528:
3104:
1862:
1226:
3069:
3040:
3011:
2978:
2904:
164:
282:
238:
2949:
2601:
2575:
2402:
2173:
1408:
1386:
754:
629:
571:
549:
527:
478:
124:
449:{\displaystyle \mathbb {H} =\bigcup _{n=1}^{\infty }\left\{(x,y)\in \mathbb {R} ^{2}\mid \left(x-{\frac {1}{n}}\right)^{2}+y^{2}=\left({\frac {1}{n}}\right)^{2}\right\}.}
1153:
1100:
917:
728:
701:
3458:
3285:
3168:
1317:
1126:
3257:
3221:
2066:{\displaystyle \left(\prod _{i=1}^{\infty }\mathbb {Z} \right)\oplus \left(\prod _{i=1}^{\infty }\mathbb {Z} {\Big /}\bigoplus _{i=1}^{\infty }\mathbb {Z} \right).}
1344:
1284:
1257:
607:
2474:
1900:
3538:
3188:
2632:
2293:
2151:
1428:
1364:
1180:
1011:
991:
887:
867:
847:
1069:
3697:
3579:
3816:
54:
2407:
1533:
3733:
3689:
3466:
2298:
3764:
3338:
2606:
The
Hawaiian earring can be generalized to higher dimensions. Such a generalization was used by Michael Barratt and
2228:
2078:
763:
76:
2181:
169:
47:
925:
3762:
Fabel, Paul (2005), "The topological
Hawaiian earring group does not embed in the inverse limit of free groups",
3290:
3107:
1410:
only finitely many times. An example of an element of the inverse limit that does not correspond an element of
3386:
3625:
3112:
574:
3623:; Conner, Gregory R. (2000), "The big fundamental group, big Hawaiian earrings, and the big free groups",
3854:
3811:
2981:
2347:
1908:
1159:
496:
3074:
1433:
2547:
is proven abstractly using infinite abelian group theory and does not have a geometric interpretation.
2176:
1827:
1191:
485:
3045:
3016:
2987:
2954:
2880:
140:
243:
214:
37:
2925:
2122:
41:
33:
2614:
spaces with nontrivial singular homology groups in dimensions larger than that of the space. The
2584:
2558:
2385:
2156:
1391:
1369:
737:
612:
554:
532:
510:
461:
107:
3226:
2133:). This factor represents the singular homology classes of loops that do not have winding number
1903:
674:
The
Hawaiian earring is neither simply connected nor semilocally simply connected since, for all
504:
3333:
2506:
2130:
2024:
1131:
1078:
895:
706:
489:
58:
2382:
consists of homology classes represented by loops whose winding number around every circle of
677:
3434:
3264:
3147:
2611:
2540:{\textstyle \prod _{i=1}^{\infty }\mathbb {Z} {\Big /}\bigoplus _{i=1}^{\infty }\mathbb {Z} }
2126:
1289:
1105:
3837:
3803:
3783:
3754:
3720:
3677:
3648:
3602:
3230:
3194:
1322:
1262:
1235:
580:
8:
3656:
Conner, Gregory; Spencer, K. (2005), "Anomalous behavior of the
Hawaiian earring group",
3787:
1882:
3773:
3549:
3523:
3173:
2617:
2278:
2136:
1413:
1349:
1165:
996:
976:
872:
852:
832:
131:
3639:
3593:
3574:
1016:
3461:
3191:
2915:
757:
285:
127:
3825:
3791:
3742:
3706:
3665:
3634:
3588:
2578:
3711:
3833:
3799:
3750:
3716:
3673:
3644:
3620:
3598:
3517:
643:
135:
3690:"The fundamental groups of one-dimensional wild spaces and the Hawaiian earring"
635:
and is usually given as the simplest example of a space with this complication.
3829:
1876:
973:
on a countably infinite number of generators, which forms a proper subgroup of
632:
3746:
3731:; Kawamura, Kazuhiro (2000), "The singular homology of the Hawaiian earring",
3669:
3848:
2907:
1186:
500:
481:
3795:
2344:
is represented by an infinite product of commutators. The second summand of
2919:
3728:
3685:
3570:
2607:
1872:
98:
1075:
th circle. More generally, one may form infinite products of the loops
920:
869:
is locally free in the sense that every finitely generated subgroup of
663:
2464:{\textstyle H_{1}(\mathbb {H} )\to \prod _{i=1}^{\infty }\mathbb {Z} }
3778:
1817:{\displaystyle \left(1,^{-1}^{-1},^{-1}^{-1}^{-1}^{-1},\dots \right)}
1182:
639:
1155:
and its inverse appear within the product only finitely many times.
89:
2295:, since every element in the kernel of the natural homomorphism
3814:; Morrison, Ian (1986), "A van Kampen theorem for weak joins",
1102:
indexed over any countable linear order provided that for each
3400:
1366:
is a proper subgroup of the inverse limit since each loop in
93:
The
Hawaiian earring. Only the ten largest circles are shown.
2914:-spheres which have one single point in common, and the
3341:
2477:
2410:
2301:
2231:
2081:
1436:
219:
179:
3526:
3469:
3437:
3389:
3293:
3267:
3233:
3197:
3176:
3150:
3115:
3077:
3048:
3019:
2990:
2957:
2928:
2883:
2643:
2620:
2587:
2561:
2404:
is zero, i.e. the kernel of the natural homomorphism
2388:
2350:
2281:
2184:
2159:
2139:
1951:
1911:
1885:
1830:
1536:
1416:
1394:
1372:
1352:
1325:
1292:
1265:
1238:
1194:
1168:
1134:
1108:
1081:
1019:
999:
979:
928:
898:
875:
855:
849:
is uncountable, and it is not a free group. However,
835:
766:
740:
709:
680:
615:
583:
557:
535:
513:
464:
296:
246:
217:
172:
143:
110:
3509:{\displaystyle H_{q}(\mathbb {H} _{k};\mathbb {Q} )}
2922:
in which the sphere's diameters converge to zero as
2337:{\textstyle G\to \prod _{i=1}^{\infty }\mathbb {Z} }
734:
th circle is not homotopic to a trivial loop. Thus,
642:of countably infinitely many circles; that is, the
3532:
3508:
3452:
3423:
3373:
3324:
3279:
3251:
3215:
3182:
3162:
3133:
3098:
3063:
3034:
3005:
2972:
2943:
2898:
2866:
2626:
2595:
2581:, i.e. all higher homotopy and homology groups of
2569:
2539:
2463:
2396:
2374:
2336:
2287:
2263:
2217:
2167:
2145:
2113:
2065:
1935:
1894:
1856:
1816:
1522:
1422:
1402:
1380:
1358:
1338:
1311:
1278:
1251:
1220:
1174:
1147:
1120:
1094:
1063:
1005:
985:
965:
911:
881:
861:
841:
817:
748:
722:
695:
623:
601:
565:
543:
521:
472:
448:
276:
232:
203:
158:
118:
16:Topological space defined by the union of circles
3846:
3698:Proceedings of the American Mathematical Society
3580:Proceedings of the American Mathematical Society
3374:{\textstyle \prod _{i=1}^{\infty }\mathbb {Z} .}
2264:{\textstyle \prod _{i=1}^{\infty }\mathbb {Z} ,}
2114:{\textstyle \prod _{i=1}^{\infty }\mathbb {Z} ,}
655:contains every circle whose radius is less than
46:but its sources remain unclear because it lacks
818:{\displaystyle G=\pi _{1}(\mathbb {H} ,(0,0)),}
638:The Hawaiian earring looks very similar to the
488:of the union of a countable family of disjoint
3817:Proceedings of the London Mathematical Society
3810:
2218:{\displaystyle {\check {H}}_{1}(\mathbb {H} )}
204:{\displaystyle \left({\tfrac {1}{n}},0\right)}
3655:
3568:
892:The homotopy classes of the individual loops
3619:
3190:-dimensional Hawaiian earring is a compact,
2634:-dimensional Hawaiian earring is defined as
966:{\displaystyle \langle \mid n\geq 1\rangle }
960:
929:
3727:
3575:"An example of anomalous singular homology"
3734:Journal of the London Mathematical Society
3325:{\displaystyle \pi _{k}(\mathbb {H} _{k})}
1867:
3777:
3710:
3638:
3592:
3499:
3485:
3364:
3309:
3121:
3080:
3051:
3022:
2993:
2960:
2886:
2735:
2671:
2646:
2589:
2563:
2533:
2500:
2457:
2425:
2390:
2365:
2330:
2254:
2208:
2161:
2104:
2051:
2018:
1979:
1926:
1530:, which appears formally as the sequence
1396:
1374:
993:. The uncountably many other elements of
787:
742:
617:
559:
537:
515:
466:
351:
298:
146:
112:
77:Learn how and when to remove this message
3424:{\displaystyle q\equiv 1{\bmod {(}}k-1)}
2471:. The existence of the isomorphism with
88:
3134:{\displaystyle \Sigma \mathbb {H} _{k}}
3847:
3071:is the original Hawaiian earring, and
1875:and Kazuhiro Kawamura proved that the
1430:is an infinite product of commutators
3761:
2550:
669:
18:
3684:
3460:Barratt and Milnor showed that the
3042:consists of a convergent sequence,
2375:{\displaystyle H_{1}(\mathbb {H} )}
1936:{\displaystyle H_{1}(\mathbb {H} )}
1523:{\textstyle \prod _{n=2}^{\infty }}
1319:simply kills the last generator of
13:
3765:Algebraic & Geometric Topology
3613:
3358:
3116:
3099:{\displaystyle \mathbb {H} _{k+1}}
2935:
2527:
2494:
2451:
2324:
2248:
2098:
2045:
2012:
1973:
1453:
321:
14:
3866:
3594:10.1090/s0002-9939-1962-0137110-9
2984:of a countable union of disjoint
2125:of infinitely many copies of the
1857:{\displaystyle \varprojlim F_{n}}
1221:{\displaystyle \varprojlim F_{n}}
631:does not have a simply connected
3064:{\displaystyle \mathbb {H} _{1}}
3035:{\displaystyle \mathbb {H} _{0}}
3006:{\displaystyle \mathbb {R} ^{k}}
2973:{\displaystyle \mathbb {H} _{k}}
2899:{\displaystyle \mathbb {H} _{k}}
2610:to provide examples of compact,
159:{\displaystyle \mathbb {R} ^{2}}
23:
277:{\displaystyle n=1,2,3,\ldots }
233:{\displaystyle {\tfrac {1}{n}}}
3562:
3503:
3480:
3418:
3404:
3319:
3304:
3246:
3234:
3210:
3198:
2932:
2727:
2682:
2432:
2429:
2421:
2369:
2361:
2305:
2212:
2204:
2192:
1930:
1922:
1791:
1777:
1765:
1751:
1748:
1735:
1732:
1719:
1707:
1693:
1681:
1667:
1664:
1651:
1648:
1635:
1620:
1606:
1594:
1580:
1577:
1564:
1561:
1548:
1517:
1458:
1058:
1020:
945:
932:
809:
806:
794:
783:
596:
584:
343:
331:
1:
3712:10.1090/S0002-9939-01-06431-0
3640:10.1016/S0166-8641(99)00104-2
3626:Topology and Its Applications
3555:
3013:s. Recursively, one has that
2944:{\displaystyle n\to \infty .}
1259:, where the bonding map from
829:. The Hawaiian earring group
825:sometimes referred to as the
575:semi-locally simply connected
2596:{\displaystyle \mathbb {H} }
2570:{\displaystyle \mathbb {H} }
2397:{\displaystyle \mathbb {H} }
2177:Cech Singular homology group
2168:{\displaystyle \mathbb {H} }
1403:{\displaystyle \mathbb {H} }
1388:may traverse each circle of
1381:{\displaystyle \mathbb {H} }
749:{\displaystyle \mathbb {H} }
624:{\displaystyle \mathbb {H} }
566:{\displaystyle \mathbb {H} }
544:{\displaystyle \mathbb {R} }
522:{\displaystyle \mathbb {H} }
473:{\displaystyle \mathbb {H} }
119:{\displaystyle \mathbb {H} }
7:
3543:
2982:Alexandrov compactification
2175:and is precisely the first
1943:is isomorphic to the group
529:is locally homeomorphic to
507:metrizable space. Although
10:
3871:
2980:may be constructed as the
551:at all non-origin points,
495:The Hawaiian earring is a
486:one-point compactification
3747:10.1112/S0024610700001071
3670:10.1515/jgth.2005.8.2.223
2271:may be considered as the
1148:{\displaystyle \ell _{n}}
1095:{\displaystyle \ell _{n}}
912:{\displaystyle \ell _{n}}
723:{\displaystyle \ell _{n}}
666:with no finite subcover.
3830:10.1112/plms/s3-53.3.562
1902:and therefore the first
1228:of the free groups with
696:{\displaystyle n\geq 1,}
32:This article includes a
3796:10.2140/agt.2005.5.1585
3658:Journal of Group Theory
3462:singular homology group
3453:{\displaystyle q>1,}
3280:{\displaystyle k\geq 2}
3163:{\displaystyle k\geq 1}
3106:is homeomorphic to the
2273:infinite abelianization
2153:around every circle of
1904:singular homology group
1868:First singular homology
1312:{\displaystyle F_{n-1}}
1121:{\displaystyle n\geq 1}
61:more precise citations.
3534:
3510:
3454:
3425:
3375:
3362:
3326:
3281:
3253:
3217:
3184:
3164:
3135:
3100:
3065:
3036:
3007:
2974:
2945:
2900:
2868:
2628:
2597:
2571:
2541:
2531:
2498:
2465:
2455:
2398:
2376:
2338:
2328:
2289:
2265:
2252:
2219:
2169:
2147:
2115:
2102:
2067:
2049:
2016:
1977:
1937:
1896:
1858:
1818:
1524:
1457:
1424:
1404:
1382:
1360:
1340:
1313:
1280:
1253:
1222:
1176:
1162:and Ian Morrison that
1149:
1122:
1096:
1065:
1007:
987:
967:
913:
883:
863:
843:
827:Hawaiian earring group
819:
750:
724:
697:
625:
603:
567:
545:
523:
505:locally path-connected
474:
450:
325:
278:
234:
205:
160:
120:
94:
3535:
3511:
3455:
3426:
3376:
3342:
3332:is isomorphic to the
3327:
3282:
3254:
3252:{\displaystyle (k-1)}
3218:
3216:{\displaystyle (k-1)}
3185:
3165:
3136:
3101:
3066:
3037:
3008:
2975:
2946:
2901:
2869:
2629:
2598:
2572:
2542:
2511:
2478:
2466:
2435:
2399:
2377:
2339:
2308:
2290:
2266:
2232:
2220:
2170:
2148:
2127:infinite cyclic group
2116:
2082:
2068:
2029:
1996:
1957:
1938:
1897:
1859:
1824:in the inverse limit
1819:
1525:
1437:
1425:
1405:
1383:
1361:
1341:
1339:{\displaystyle F_{n}}
1314:
1281:
1279:{\displaystyle F_{n}}
1254:
1252:{\displaystyle F_{n}}
1223:
1177:
1150:
1123:
1097:
1066:
1008:
988:
968:
914:
884:
864:
844:
820:
751:
725:
698:
626:
604:
602:{\displaystyle (0,0)}
568:
546:
524:
475:
451:
305:
279:
235:
206:
161:
121:
92:
3524:
3520:group for each such
3467:
3435:
3387:
3339:
3291:
3265:
3231:
3195:
3174:
3148:
3113:
3075:
3046:
3017:
2988:
2955:
2926:
2881:
2641:
2618:
2585:
2559:
2475:
2408:
2386:
2348:
2299:
2279:
2229:
2182:
2157:
2137:
2079:
1949:
1909:
1883:
1828:
1534:
1434:
1414:
1392:
1370:
1350:
1323:
1290:
1263:
1236:
1192:
1166:
1132:
1106:
1079:
1071:circumnavigates the
1017:
997:
977:
926:
896:
873:
853:
833:
764:
738:
707:
678:
613:
581:
555:
533:
511:
462:
294:
244:
215:
170:
141:
108:
3788:2005math......1482F
3287:, it is known that
2835:
2811:
1516:
1498:
730:parameterizing the
3855:Topological spaces
3569:Barratt, Michael;
3550:List of topologies
3530:
3506:
3450:
3421:
3371:
3334:Baer–Specker group
3322:
3277:
3249:
3213:
3180:
3160:
3131:
3108:reduced suspension
3096:
3061:
3032:
3003:
2970:
2941:
2896:
2864:
2821:
2797:
2676:
2624:
2612:finite-dimensional
2593:
2567:
2537:
2461:
2394:
2372:
2334:
2285:
2261:
2215:
2165:
2143:
2131:Baer–Specker group
2111:
2075:The first summand
2063:
1933:
1895:{\displaystyle G,}
1892:
1854:
1839:
1814:
1520:
1499:
1481:
1420:
1400:
1378:
1356:
1336:
1309:
1276:
1249:
1218:
1203:
1172:
1158:It is a result of
1145:
1118:
1092:
1061:
1003:
983:
963:
909:
879:
859:
839:
815:
746:
720:
693:
621:
599:
563:
541:
519:
470:
446:
274:
230:
228:
201:
188:
156:
134:of circles in the
116:
95:
34:list of references
3533:{\displaystyle q}
3183:{\displaystyle k}
2854:
2781:
2659:
2627:{\displaystyle k}
2555:It is known that
2551:Higher dimensions
2288:{\displaystyle G}
2195:
2146:{\displaystyle 0}
1832:
1423:{\displaystyle G}
1359:{\displaystyle G}
1196:
1175:{\displaystyle G}
1006:{\displaystyle G}
986:{\displaystyle G}
882:{\displaystyle G}
862:{\displaystyle G}
842:{\displaystyle G}
758:fundamental group
756:has a nontrivial
670:Fundamental group
426:
384:
286:subspace topology
284:endowed with the
227:
187:
128:topological space
87:
86:
79:
3862:
3840:
3806:
3781:
3772:(4): 1585–1587,
3757:
3723:
3714:
3705:(5): 1515–1522,
3694:
3680:
3651:
3642:
3621:Cannon, James W.
3607:
3606:
3596:
3566:
3539:
3537:
3536:
3531:
3516:is a nontrivial
3515:
3513:
3512:
3507:
3502:
3494:
3493:
3488:
3479:
3478:
3459:
3457:
3456:
3451:
3430:
3428:
3427:
3422:
3408:
3407:
3380:
3378:
3377:
3372:
3367:
3361:
3356:
3331:
3329:
3328:
3323:
3318:
3317:
3312:
3303:
3302:
3286:
3284:
3283:
3278:
3258:
3256:
3255:
3250:
3222:
3220:
3219:
3214:
3189:
3187:
3186:
3181:
3169:
3167:
3166:
3161:
3140:
3138:
3137:
3132:
3130:
3129:
3124:
3105:
3103:
3102:
3097:
3095:
3094:
3083:
3070:
3068:
3067:
3062:
3060:
3059:
3054:
3041:
3039:
3038:
3033:
3031:
3030:
3025:
3012:
3010:
3009:
3004:
3002:
3001:
2996:
2979:
2977:
2976:
2971:
2969:
2968:
2963:
2950:
2948:
2947:
2942:
2913:
2905:
2903:
2902:
2897:
2895:
2894:
2889:
2873:
2871:
2870:
2865:
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2694:
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2594:
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2579:aspherical space
2576:
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2257:
2251:
2246:
2225:. Additionally,
2224:
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2211:
2203:
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2197:
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2188:
2174:
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2072:
2070:
2069:
2064:
2059:
2055:
2054:
2048:
2043:
2028:
2027:
2021:
2015:
2010:
1987:
1983:
1982:
1976:
1971:
1942:
1940:
1939:
1934:
1929:
1921:
1920:
1901:
1899:
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1125:
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1119:
1101:
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1064:{\displaystyle }
1062:
1057:
1056:
1035:
1034:
1012:
1010:
1009:
1004:
992:
990:
989:
984:
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888:
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868:
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824:
822:
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790:
782:
781:
755:
753:
752:
747:
745:
733:
729:
727:
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721:
719:
718:
702:
700:
699:
694:
661:
654:
650:
630:
628:
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622:
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608:
606:
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572:
570:
569:
564:
562:
550:
548:
547:
542:
540:
528:
526:
525:
520:
518:
479:
477:
476:
471:
469:
455:
453:
452:
447:
442:
438:
437:
436:
431:
427:
419:
409:
408:
396:
395:
390:
386:
385:
377:
360:
359:
354:
324:
319:
301:
283:
281:
280:
275:
239:
237:
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231:
229:
220:
210:
208:
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200:
196:
189:
180:
165:
163:
162:
157:
155:
154:
149:
125:
123:
122:
117:
115:
103:Hawaiian earring
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
3870:
3869:
3865:
3864:
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3812:Morgan, John W.
3692:
3616:
3614:Further reading
3611:
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3498:
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2702:
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2681:
2677:
2670:
2663:
2650:
2645:
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2642:
2639:
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2619:
2616:
2615:
2588:
2586:
2583:
2582:
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2515:
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2504:
2499:
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2439:
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2411:
2409:
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2389:
2387:
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2364:
2355:
2351:
2349:
2346:
2345:
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2312:
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2280:
2277:
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2253:
2247:
2236:
2230:
2227:
2226:
2207:
2198:
2187:
2186:
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2180:
2179:
2160:
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2138:
2135:
2134:
2103:
2097:
2086:
2080:
2077:
2076:
2050:
2044:
2033:
2023:
2022:
2017:
2011:
2000:
1995:
1991:
1978:
1972:
1961:
1956:
1952:
1950:
1947:
1946:
1925:
1916:
1912:
1910:
1907:
1906:
1884:
1881:
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1870:
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1794:
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1597:
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1567:
1555:
1551:
1541:
1537:
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1471:
1465:
1461:
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1373:
1371:
1368:
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1234:
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998:
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994:
978:
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854:
851:
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834:
831:
830:
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777:
773:
765:
762:
761:
741:
739:
736:
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731:
714:
710:
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705:
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679:
676:
675:
672:
656:
652:
648:
616:
614:
611:
610:
582:
579:
578:
558:
556:
553:
552:
536:
534:
531:
530:
514:
512:
509:
508:
497:one-dimensional
465:
463:
460:
459:
432:
418:
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391:
376:
369:
365:
364:
355:
350:
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326:
320:
309:
297:
295:
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245:
242:
241:
218:
216:
213:
212:
178:
177:
173:
171:
168:
167:
150:
145:
144:
142:
139:
138:
136:Euclidean plane
130:defined by the
111:
109:
106:
105:
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
3868:
3858:
3857:
3843:
3842:
3824:(3): 562–576,
3808:
3759:
3741:(1): 305–310,
3725:
3682:
3664:(2): 223–227,
3653:
3633:(3): 273–291,
3615:
3612:
3609:
3608:
3587:(2): 293–297.
3560:
3559:
3557:
3554:
3553:
3552:
3545:
3542:
3529:
3505:
3501:
3497:
3492:
3487:
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3449:
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3440:
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3398:
3395:
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3370:
3366:
3360:
3355:
3352:
3349:
3345:
3321:
3316:
3311:
3306:
3301:
3297:
3276:
3273:
3270:
3248:
3245:
3242:
3239:
3236:
3212:
3209:
3206:
3203:
3200:
3179:
3159:
3156:
3153:
3128:
3123:
3118:
3093:
3090:
3087:
3082:
3058:
3053:
3029:
3024:
3000:
2995:
2967:
2962:
2940:
2937:
2934:
2931:
2918:is given by a
2893:
2888:
2875:
2874:
2863:
2859:
2851:
2847:
2843:
2838:
2833:
2828:
2824:
2820:
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2800:
2796:
2791:
2786:
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2777:
2772:
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2753:
2748:
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2653:
2648:
2623:
2591:
2565:
2552:
2549:
2535:
2529:
2524:
2521:
2518:
2514:
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2459:
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2245:
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2239:
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2214:
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2201:
2194:
2191:
2163:
2142:
2123:direct product
2110:
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2100:
2095:
2092:
2089:
2085:
2062:
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2053:
2047:
2042:
2039:
2036:
2032:
2026:
2020:
2014:
2009:
2006:
2003:
1999:
1994:
1990:
1986:
1981:
1975:
1970:
1967:
1964:
1960:
1955:
1932:
1928:
1924:
1919:
1915:
1891:
1888:
1877:abelianisation
1869:
1866:
1851:
1847:
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1812:
1808:
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1797:
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1303:
1300:
1296:
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1246:
1242:
1215:
1211:
1207:
1202:
1199:
1171:
1142:
1138:
1117:
1114:
1111:
1089:
1085:
1060:
1055:
1052:
1049:
1046:
1042:
1038:
1033:
1030:
1026:
1022:
1002:
982:
962:
959:
956:
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947:
942:
938:
934:
931:
906:
902:
878:
858:
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814:
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808:
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802:
799:
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789:
785:
780:
776:
772:
769:
744:
717:
713:
692:
689:
686:
683:
671:
668:
633:covering space
619:
598:
595:
592:
589:
586:
561:
539:
517:
490:open intervals
468:
445:
441:
435:
430:
425:
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412:
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399:
394:
389:
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114:
85:
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42:external links
31:
29:
22:
15:
9:
6:
4:
3:
2:
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2609:
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2603:are trivial.
2580:
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2512:
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2073:
2060:
2056:
2040:
2037:
2034:
2030:
2007:
2004:
2001:
1997:
1992:
1988:
1984:
1968:
1965:
1962:
1958:
1953:
1944:
1917:
1913:
1905:
1889:
1886:
1878:
1874:
1865:
1849:
1845:
1841:
1836:
1833:
1810:
1806:
1803:
1798:
1795:
1785:
1781:
1772:
1769:
1759:
1755:
1743:
1739:
1727:
1723:
1714:
1711:
1701:
1697:
1688:
1685:
1675:
1671:
1659:
1655:
1643:
1639:
1632:
1627:
1624:
1614:
1610:
1601:
1598:
1588:
1584:
1572:
1568:
1556:
1552:
1545:
1542:
1538:
1512:
1509:
1504:
1500:
1494:
1491:
1486:
1482:
1476:
1472:
1466:
1462:
1448:
1445:
1442:
1438:
1417:
1353:
1331:
1327:
1304:
1301:
1298:
1294:
1271:
1267:
1244:
1240:
1213:
1209:
1205:
1200:
1197:
1188:
1187:inverse limit
1184:
1169:
1161:
1156:
1140:
1136:
1115:
1112:
1109:
1087:
1083:
1053:
1050:
1047:
1044:
1040:
1036:
1031:
1028:
1024:
1000:
980:
957:
954:
951:
948:
940:
936:
922:
919:generate the
904:
900:
890:
876:
856:
836:
828:
812:
803:
800:
797:
791:
778:
774:
770:
767:
759:
715:
711:
690:
687:
684:
681:
667:
665:
659:
651:-ball around
645:
641:
636:
634:
609:. Therefore,
593:
590:
587:
576:
506:
502:
498:
493:
491:
487:
483:
456:
443:
439:
433:
428:
423:
420:
415:
410:
405:
401:
397:
392:
387:
381:
378:
373:
370:
366:
361:
356:
346:
340:
337:
334:
327:
316:
313:
310:
306:
302:
289:
287:
271:
268:
265:
262:
259:
256:
253:
250:
247:
224:
221:
197:
193:
190:
184:
181:
174:
151:
137:
133:
129:
104:
100:
91:
81:
78:
70:
60:
56:
50:
49:
43:
39:
35:
30:
21:
20:
3821:
3815:
3779:math/0501482
3769:
3763:
3738:
3732:
3729:Eda, Katsuya
3702:
3696:
3686:Eda, Katsuya
3661:
3657:
3630:
3624:
3584:
3578:
3571:Milnor, John
3564:
3382:
3143:
2876:
2605:
2554:
2272:
2074:
1945:
1871:
1232:generators,
1157:
891:
826:
673:
657:
637:
494:
482:homeomorphic
457:
290:
166:with center
102:
96:
73:
64:
53:Please help
45:
3518:uncountable
2608:John Milnor
1873:Katsuya Eda
1346:. However,
1160:John Morgan
1128:, the loop
211:and radius
99:mathematics
59:introducing
3556:References
3259:-connected
3223:-connected
921:free group
664:open cover
458:The space
3413:−
3394:≡
3359:∞
3344:∏
3296:π
3272:≥
3241:−
3205:−
3155:≥
3117:Σ
2936:∞
2933:→
2910:union of
2908:countable
2816:⋯
2771:−
2731:∈
2712:…
2668:∈
2661:⋃
2528:∞
2513:⨁
2495:∞
2480:∏
2452:∞
2437:∏
2433:→
2325:∞
2310:∏
2306:→
2249:∞
2234:∏
2193:ˇ
2099:∞
2084:∏
2046:∞
2031:⨁
2013:∞
1998:∏
1989:⊕
1974:∞
1959:∏
1842:
1837:←
1807:…
1796:−
1782:ℓ
1770:−
1756:ℓ
1740:ℓ
1724:ℓ
1712:−
1698:ℓ
1686:−
1672:ℓ
1656:ℓ
1640:ℓ
1625:−
1611:ℓ
1599:−
1585:ℓ
1569:ℓ
1553:ℓ
1510:−
1501:ℓ
1492:−
1483:ℓ
1473:ℓ
1463:ℓ
1454:∞
1439:∏
1302:−
1206:
1201:←
1185:into the
1137:ℓ
1113:≥
1084:ℓ
1045:−
1029:−
961:⟩
955:≥
949:∣
937:ℓ
930:⟨
901:ℓ
889:is free.
775:π
712:ℓ
703:the loop
685:≥
640:wedge sum
374:−
362:∣
347:∈
322:∞
307:⋃
272:…
67:June 2017
3849:Category
3688:(2002),
3573:(1962).
3544:See also
3227:locally
2916:topology
3838:0868459
3804:2186111
3784:Bibcode
3755:1772189
3721:1879978
3678:2126731
3649:1775710
3603:0137110
2877:Hence,
2121:is the
760:
573:is not
501:compact
484:to the
126:is the
55:improve
3836:
3802:
3753:
3719:
3676:
3647:
3601:
3261:. For
3170:, the
2920:metric
2906:is a
2577:is an
1183:embeds
653:(0, 0)
101:, the
3774:arXiv
3693:(PDF)
2129:(the
132:union
40:, or
3442:>
3431:and
3383:For
3225:and
3144:For
644:rose
240:for
3826:doi
3792:doi
3743:doi
3707:doi
3703:130
3666:doi
3635:doi
3631:106
3589:doi
3401:mod
2275:of
1879:of
1834:lim
1286:to
1198:lim
577:at
480:is
97:In
3851::
3834:MR
3832:,
3822:53
3820:,
3800:MR
3798:,
3790:,
3782:,
3768:,
3751:MR
3749:,
3739:62
3737:,
3717:MR
3715:,
3701:,
3695:,
3674:MR
3672:,
3660:,
3645:MR
3643:,
3629:,
3599:MR
3597:.
3585:13
3583:.
3577:.
3540:.
3141:.
1864:.
660:/2
503:,
499:,
492:.
288::
44:,
36:,
3841:.
3828::
3807:.
3794::
3786::
3776::
3770:5
3758:.
3745::
3724:.
3709::
3681:.
3668::
3662:8
3652:.
3637::
3605:.
3591::
3528:q
3504:)
3500:Q
3496:;
3491:k
3486:H
3481:(
3476:q
3472:H
3448:,
3445:1
3439:q
3419:)
3416:1
3410:k
3405:(
3397:1
3391:q
3369:.
3365:Z
3354:1
3351:=
3348:i
3320:)
3315:k
3310:H
3305:(
3300:k
3275:2
3269:k
3247:)
3244:1
3238:k
3235:(
3211:)
3208:1
3202:k
3199:(
3178:k
3158:1
3152:k
3127:k
3122:H
3092:1
3089:+
3086:k
3081:H
3057:1
3052:H
3028:0
3023:H
2999:k
2994:R
2966:k
2961:H
2939:.
2930:n
2912:k
2892:k
2887:H
2862:.
2858:}
2850:2
2846:n
2842:1
2837:=
2832:2
2827:k
2823:x
2819:+
2813:+
2808:2
2803:1
2799:x
2795:+
2790:2
2785:)
2779:n
2776:1
2766:0
2762:x
2757:(
2752::
2747:1
2744:+
2741:k
2736:R
2728:)
2723:k
2719:x
2715:,
2709:,
2704:1
2700:x
2696:,
2691:0
2687:x
2683:(
2679:{
2672:N
2665:n
2657:=
2652:k
2647:H
2622:k
2590:H
2564:H
2534:Z
2523:1
2520:=
2517:i
2507:/
2501:Z
2490:1
2487:=
2484:i
2458:Z
2447:1
2444:=
2441:i
2430:)
2426:H
2422:(
2417:1
2413:H
2391:H
2370:)
2366:H
2362:(
2357:1
2353:H
2331:Z
2320:1
2317:=
2314:i
2303:G
2283:G
2259:,
2255:Z
2244:1
2241:=
2238:i
2213:)
2209:H
2205:(
2200:1
2190:H
2162:H
2141:0
2109:,
2105:Z
2094:1
2091:=
2088:i
2061:.
2057:)
2052:Z
2041:1
2038:=
2035:i
2025:/
2019:Z
2008:1
2005:=
2002:i
1993:(
1985:)
1980:Z
1969:1
1966:=
1963:i
1954:(
1931:)
1927:H
1923:(
1918:1
1914:H
1890:,
1887:G
1850:n
1846:F
1811:)
1804:,
1799:1
1792:]
1786:3
1778:[
1773:1
1766:]
1760:1
1752:[
1749:]
1744:3
1736:[
1733:]
1728:1
1720:[
1715:1
1708:]
1702:2
1694:[
1689:1
1682:]
1676:1
1668:[
1665:]
1660:2
1652:[
1649:]
1644:1
1636:[
1633:,
1628:1
1621:]
1615:2
1607:[
1602:1
1595:]
1589:1
1581:[
1578:]
1573:2
1565:[
1562:]
1557:1
1549:[
1546:,
1543:1
1539:(
1518:]
1513:1
1505:n
1495:1
1487:1
1477:n
1467:1
1459:[
1449:2
1446:=
1443:n
1418:G
1397:H
1375:H
1354:G
1332:n
1328:F
1305:1
1299:n
1295:F
1272:n
1268:F
1245:n
1241:F
1230:n
1214:n
1210:F
1170:G
1141:n
1116:1
1110:n
1088:n
1073:n
1059:]
1054:1
1051:+
1048:n
1041:2
1037:,
1032:n
1025:2
1021:[
1001:G
981:G
958:1
952:n
946:]
941:n
933:[
905:n
877:G
857:G
837:G
813:,
810:)
807:)
804:0
801:,
798:0
795:(
792:,
788:H
784:(
779:1
771:=
768:G
743:H
732:n
716:n
691:,
688:1
682:n
658:ε
649:ε
618:H
597:)
594:0
591:,
588:0
585:(
560:H
538:R
516:H
467:H
444:.
440:}
434:2
429:)
424:n
421:1
416:(
411:=
406:2
402:y
398:+
393:2
388:)
382:n
379:1
371:x
367:(
357:2
352:R
344:)
341:y
338:,
335:x
332:(
328:{
317:1
314:=
311:n
303:=
299:H
269:,
266:3
263:,
260:2
257:,
254:1
251:=
248:n
225:n
222:1
198:)
194:0
191:,
185:n
182:1
175:(
152:2
147:R
113:H
80:)
74:(
69:)
65:(
51:.
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