17:
502:
605:
313:
980:
896:
224:
1490:
656:
416:
704:
361:
1279:
1041:
1113:
167:
1366:
1326:
1520:
1402:
1166:
524:
248:
1432:
1226:
918:
830:
100:
73:
1630:
1607:
1583:
1563:
1206:
1186:
1133:
1061:
803:
766:
421:
607:
In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints
529:
835:
171:
1231:
257:
1687:
1667:
923:
1457:
610:
370:
661:
318:
1493:
985:
1404:
Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at
1447:
1069:
129:
1451:
115:
111:
1331:
1286:
1692:
1499:
1530:
1371:
1138:
507:
231:
1648:, a topological space resembling, but not the same as, a wedge sum of countably many circles
1407:
1211:
903:
808:
78:
51:
982:
that is, of identifying the equator down to a single point. Then addition of two elements
832:. Doing so results in two copies of the sphere, joined at the point that was the equator:
8:
251:
1612:
1592:
1568:
1548:
1191:
1171:
1118:
1046:
788:
770:
751:
745:
1682:
1663:
1542:
707:
33:
1645:
729:
1064:
364:
1676:
1640:
45:
1586:
1534:
741:
718:
714:
497:{\displaystyle \bigvee _{i\in I}X_{i}=\coprod _{i\in I}X_{i}\;/{\sim },}
1538:
16:
1443:
713:
The wedge sum is again a pointed space, and the binary operation is
782:
25:
600:{\displaystyle \left\{\left(p_{i},p_{j}\right):i,j\in I\right\}.}
774:, while a wedge product of arbitrary spheres is often called a
1434:
the point common to the wedge sum of the underlying spaces.
1533:
gives certain conditions (which are usually fulfilled for
785:
is to identify all of the points along the equator of an
308:{\displaystyle \left\{\left(x_{0},y_{0}\right)\right\}.}
48:(i.e. topological spaces with distinguished basepoints
1615:
1595:
1571:
1551:
1502:
1460:
1410:
1374:
1334:
1289:
1234:
1214:
1194:
1174:
1141:
1121:
1072:
1049:
988:
926:
906:
838:
811:
791:
754:
664:
613:
532:
510:
424:
373:
321:
260:
234:
174:
132:
81:
54:
1624:
1601:
1577:
1557:
1514:
1484:
1450:. Alternatively, the wedge sum can be seen as the
1426:
1396:
1360:
1320:
1273:
1220:
1200:
1180:
1160:
1127:
1107:
1055:
1035:
974:
912:
890:
824:
797:
760:
698:
650:
599:
518:
496:
410:
355:
307:
242:
218:
161:
94:
67:
1674:
732:, which is also often called the wedge product.
975:{\displaystyle \Psi :S^{n}\to S^{n}\vee S^{n},}
891:{\displaystyle S^{n}/{\sim }=S^{n}\vee S^{n}.}
219:{\displaystyle X\vee Y=(X\amalg Y)\;/{\sim },}
1485:{\displaystyle X\leftarrow \{\bullet \}\to Y}
1509:
1503:
1473:
1467:
651:{\displaystyle \left(p_{i}\right)_{i\in I},}
411:{\displaystyle \left(p_{i}\right)_{i\in I}.}
699:{\displaystyle \left(X_{i}\right)_{i\in I}}
526:is the equivalence closure of the relation
356:{\displaystyle \left(X_{i}\right)_{i\in I}}
1437:
1328:are maps which take a distinguished point
728:, but this is not the same concept as the
480:
202:
1274:{\displaystyle f+g=(f\vee g)\circ \Psi .}
515:
511:
418:The wedge sum of the family is given by:
239:
235:
1168:can be understood as the composition of
1036:{\displaystyle f,g\in \pi _{n}(X,x_{0})}
15:
1442:The wedge sum can be understood as the
1675:
724:Sometimes the wedge sum is called the
32:is a "one-point union" of a family of
1660:An Introduction to Algebraic Topology
13:
1265:
1215:
927:
907:
367:of pointed spaces with basepoints
14:
1704:
1108:{\displaystyle \pi _{n}(X,x_{0})}
740:The wedge sum of two circles is
162:{\displaystyle x_{0}\sim y_{0}:}
1662:, Springer, 2004, p. 153.
1545:of the wedge sum of two spaces
1494:category of topological spaces
1476:
1464:
1361:{\displaystyle s_{0}\in S^{n}}
1321:{\displaystyle f,g:S^{n}\to X}
1312:
1259:
1247:
1102:
1083:
1030:
1011:
943:
199:
187:
1:
1652:
1589:of the fundamental groups of
1525:
1515:{\displaystyle \{\bullet \}}
7:
1634:
1397:{\displaystyle x_{0}\in X.}
1135:at the distinguished point
735:
10:
1709:
1448:category of pointed spaces
1161:{\displaystyle x_{0}\in X}
768:circles is often called a
20:A wedge sum of two circles
1522:is any one-point space).
781:A common construction in
519:{\displaystyle \,\sim \,}
243:{\displaystyle \,\sim \,}
1688:Operations on structures
315:More generally, suppose
1438:Categorical description
721:(up to homeomorphism).
1626:
1603:
1579:
1559:
1516:
1486:
1428:
1427:{\displaystyle s_{0},}
1398:
1362:
1322:
1275:
1222:
1202:
1182:
1162:
1129:
1109:
1057:
1037:
976:
914:
892:
826:
799:
762:
700:
652:
601:
520:
498:
412:
357:
309:
244:
220:
163:
126:by the identification
96:
69:
21:
1627:
1604:
1580:
1560:
1517:
1487:
1429:
1399:
1363:
1323:
1276:
1223:
1221:{\displaystyle \Psi }
1203:
1183:
1163:
1130:
1110:
1058:
1038:
977:
915:
913:{\displaystyle \Psi }
893:
827:
825:{\displaystyle S^{n}}
800:
763:
701:
653:
602:
521:
499:
413:
358:
310:
245:
221:
164:
97:
95:{\displaystyle y_{0}}
70:
68:{\displaystyle x_{0}}
19:
1613:
1593:
1569:
1549:
1531:Van Kampen's theorem
1500:
1458:
1408:
1372:
1332:
1287:
1232:
1212:
1192:
1172:
1139:
1119:
1070:
1047:
986:
924:
904:
836:
809:
789:
752:
662:
611:
530:
508:
422:
371:
319:
258:
232:
172:
130:
79:
52:
748:. The wedge sum of
252:equivalence closure
102:) the wedge sum of
36:. Specifically, if
1625:{\displaystyle Y.}
1622:
1599:
1575:
1555:
1541:) under which the
1512:
1482:
1424:
1394:
1358:
1318:
1271:
1218:
1198:
1178:
1158:
1125:
1105:
1053:
1033:
972:
910:
888:
822:
795:
776:bouquet of spheres
771:bouquet of circles
758:
746:figure-eight space
696:
658:unless the spaces
648:
597:
516:
494:
469:
440:
408:
353:
305:
240:
216:
159:
92:
65:
34:topological spaces
22:
1602:{\displaystyle X}
1578:{\displaystyle Y}
1558:{\displaystyle X}
1543:fundamental group
1201:{\displaystyle g}
1181:{\displaystyle f}
1128:{\displaystyle X}
1056:{\displaystyle n}
798:{\displaystyle n}
761:{\displaystyle n}
454:
425:
1700:
1658:Rotman, Joseph.
1646:Hawaiian earring
1631:
1629:
1628:
1623:
1608:
1606:
1605:
1600:
1584:
1582:
1581:
1576:
1564:
1562:
1561:
1556:
1537:spaces, such as
1521:
1519:
1518:
1513:
1491:
1489:
1488:
1483:
1433:
1431:
1430:
1425:
1420:
1419:
1403:
1401:
1400:
1395:
1384:
1383:
1367:
1365:
1364:
1359:
1357:
1356:
1344:
1343:
1327:
1325:
1324:
1319:
1311:
1310:
1280:
1278:
1277:
1272:
1227:
1225:
1224:
1219:
1207:
1205:
1204:
1199:
1187:
1185:
1184:
1179:
1167:
1165:
1164:
1159:
1151:
1150:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1101:
1100:
1082:
1081:
1062:
1060:
1059:
1054:
1042:
1040:
1039:
1034:
1029:
1028:
1010:
1009:
981:
979:
978:
973:
968:
967:
955:
954:
942:
941:
919:
917:
916:
911:
897:
895:
894:
889:
884:
883:
871:
870:
858:
853:
848:
847:
831:
829:
828:
823:
821:
820:
804:
802:
801:
796:
767:
765:
764:
759:
730:exterior product
705:
703:
702:
697:
695:
694:
683:
679:
678:
657:
655:
654:
649:
644:
643:
632:
628:
627:
606:
604:
603:
598:
593:
589:
570:
566:
565:
564:
552:
551:
525:
523:
522:
517:
503:
501:
500:
495:
490:
485:
479:
478:
468:
450:
449:
439:
417:
415:
414:
409:
404:
403:
392:
388:
387:
362:
360:
359:
354:
352:
351:
340:
336:
335:
314:
312:
311:
306:
301:
297:
293:
292:
291:
279:
278:
254:of the relation
249:
247:
246:
241:
225:
223:
222:
217:
212:
207:
168:
166:
165:
160:
155:
154:
142:
141:
101:
99:
98:
93:
91:
90:
74:
72:
71:
66:
64:
63:
1708:
1707:
1703:
1702:
1701:
1699:
1698:
1697:
1693:Homotopy theory
1673:
1672:
1655:
1637:
1614:
1611:
1610:
1594:
1591:
1590:
1570:
1567:
1566:
1550:
1547:
1546:
1528:
1501:
1498:
1497:
1459:
1456:
1455:
1454:of the diagram
1440:
1415:
1411:
1409:
1406:
1405:
1379:
1375:
1373:
1370:
1369:
1352:
1348:
1339:
1335:
1333:
1330:
1329:
1306:
1302:
1288:
1285:
1284:
1233:
1230:
1229:
1213:
1210:
1209:
1193:
1190:
1189:
1173:
1170:
1169:
1146:
1142:
1140:
1137:
1136:
1120:
1117:
1116:
1096:
1092:
1077:
1073:
1071:
1068:
1067:
1048:
1045:
1044:
1024:
1020:
1005:
1001:
987:
984:
983:
963:
959:
950:
946:
937:
933:
925:
922:
921:
905:
902:
901:
879:
875:
866:
862:
854:
849:
843:
839:
837:
834:
833:
816:
812:
810:
807:
806:
790:
787:
786:
753:
750:
749:
738:
684:
674:
670:
666:
665:
663:
660:
659:
633:
623:
619:
615:
614:
612:
609:
608:
560:
556:
547:
543:
542:
538:
537:
533:
531:
528:
527:
509:
506:
505:
486:
481:
474:
470:
458:
445:
441:
429:
423:
420:
419:
393:
383:
379:
375:
374:
372:
369:
368:
341:
331:
327:
323:
322:
320:
317:
316:
287:
283:
274:
270:
269:
265:
261:
259:
256:
255:
233:
230:
229:
208:
203:
173:
170:
169:
150:
146:
137:
133:
131:
128:
127:
86:
82:
80:
77:
76:
59:
55:
53:
50:
49:
12:
11:
5:
1706:
1696:
1695:
1690:
1685:
1671:
1670:
1654:
1651:
1650:
1649:
1643:
1636:
1633:
1621:
1618:
1598:
1574:
1554:
1527:
1524:
1511:
1508:
1505:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1439:
1436:
1423:
1418:
1414:
1393:
1390:
1387:
1382:
1378:
1355:
1351:
1347:
1342:
1338:
1317:
1314:
1309:
1305:
1301:
1298:
1295:
1292:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1217:
1197:
1177:
1157:
1154:
1149:
1145:
1124:
1104:
1099:
1095:
1091:
1088:
1085:
1080:
1076:
1065:homotopy group
1052:
1032:
1027:
1023:
1019:
1016:
1013:
1008:
1004:
1000:
997:
994:
991:
971:
966:
962:
958:
953:
949:
945:
940:
936:
932:
929:
909:
887:
882:
878:
874:
869:
865:
861:
857:
852:
846:
842:
819:
815:
794:
757:
737:
734:
693:
690:
687:
682:
677:
673:
669:
647:
642:
639:
636:
631:
626:
622:
618:
596:
592:
588:
585:
582:
579:
576:
573:
569:
563:
559:
555:
550:
546:
541:
536:
514:
493:
489:
484:
477:
473:
467:
464:
461:
457:
453:
448:
444:
438:
435:
432:
428:
407:
402:
399:
396:
391:
386:
382:
378:
365:indexed family
350:
347:
344:
339:
334:
330:
326:
304:
300:
296:
290:
286:
282:
277:
273:
268:
264:
238:
215:
211:
206:
201:
198:
195:
192:
189:
186:
183:
180:
177:
158:
153:
149:
145:
140:
136:
116:disjoint union
112:quotient space
89:
85:
62:
58:
46:pointed spaces
9:
6:
4:
3:
2:
1705:
1694:
1691:
1689:
1686:
1684:
1681:
1680:
1678:
1669:
1668:0-387-96678-1
1665:
1661:
1657:
1656:
1647:
1644:
1642:
1641:Smash product
1639:
1638:
1632:
1619:
1616:
1596:
1588:
1572:
1552:
1544:
1540:
1536:
1532:
1523:
1506:
1495:
1479:
1470:
1461:
1453:
1449:
1445:
1435:
1421:
1416:
1412:
1391:
1388:
1385:
1380:
1376:
1368:to the point
1353:
1349:
1345:
1340:
1336:
1315:
1307:
1303:
1299:
1296:
1293:
1290:
1281:
1268:
1262:
1256:
1253:
1250:
1244:
1241:
1238:
1235:
1195:
1175:
1155:
1152:
1147:
1143:
1122:
1097:
1093:
1089:
1086:
1078:
1074:
1066:
1063:-dimensional
1050:
1025:
1021:
1017:
1014:
1006:
1002:
998:
995:
992:
989:
969:
964:
960:
956:
951:
947:
938:
934:
930:
898:
885:
880:
876:
872:
867:
863:
859:
855:
850:
844:
840:
817:
813:
792:
784:
779:
777:
773:
772:
755:
747:
743:
733:
731:
727:
726:wedge product
722:
720:
716:
711:
709:
691:
688:
685:
680:
675:
671:
667:
645:
640:
637:
634:
629:
624:
620:
616:
594:
590:
586:
583:
580:
577:
574:
571:
567:
561:
557:
553:
548:
544:
539:
534:
512:
491:
487:
482:
475:
471:
465:
462:
459:
455:
451:
446:
442:
436:
433:
430:
426:
405:
400:
397:
394:
389:
384:
380:
376:
366:
348:
345:
342:
337:
332:
328:
324:
302:
298:
294:
288:
284:
280:
275:
271:
266:
262:
253:
236:
226:
213:
209:
204:
196:
193:
190:
184:
181:
178:
175:
156:
151:
147:
143:
138:
134:
125:
121:
117:
113:
109:
105:
87:
83:
60:
56:
47:
43:
39:
35:
31:
27:
18:
1659:
1587:free product
1539:CW complexes
1535:well-behaved
1529:
1441:
1282:
899:
780:
775:
769:
742:homeomorphic
739:
725:
723:
712:
227:
123:
119:
107:
103:
41:
37:
29:
23:
1115:of a space
920:be the map
719:commutative
715:associative
708:homogeneous
1677:Categories
1653:References
1526:Properties
1507:∙
1477:→
1471:∙
1465:←
1444:coproduct
1386:∈
1346:∈
1313:→
1266:Ψ
1263:∘
1254:∨
1216:Ψ
1153:∈
1075:π
1003:π
999:∈
957:∨
944:→
928:Ψ
908:Ψ
873:∨
856:∼
689:∈
638:∈
584:∈
513:∼
488:∼
463:∈
456:∐
434:∈
427:⋁
398:∈
346:∈
237:∼
210:∼
194:⨿
179:∨
144:∼
30:wedge sum
1683:Topology
1635:See also
805:-sphere
783:homotopy
736:Examples
26:topology
1585:is the
1496:(where
1492:in the
1452:pushout
1446:in the
1043:of the
250:is the
114:of the
110:is the
1666:
1283:Here,
504:where
228:where
28:, the
1208:with
744:to a
363:is a
1664:ISBN
1609:and
1565:and
1188:and
900:Let
717:and
706:are
122:and
106:and
75:and
44:are
40:and
118:of
24:In
1679::
1228::
778:.
710:.
1620:.
1617:Y
1597:X
1573:Y
1553:X
1510:}
1504:{
1480:Y
1474:}
1468:{
1462:X
1422:,
1417:0
1413:s
1392:.
1389:X
1381:0
1377:x
1354:n
1350:S
1341:0
1337:s
1316:X
1308:n
1304:S
1300::
1297:g
1294:,
1291:f
1269:.
1260:)
1257:g
1251:f
1248:(
1245:=
1242:g
1239:+
1236:f
1196:g
1176:f
1156:X
1148:0
1144:x
1123:X
1103:)
1098:0
1094:x
1090:,
1087:X
1084:(
1079:n
1051:n
1031:)
1026:0
1022:x
1018:,
1015:X
1012:(
1007:n
996:g
993:,
990:f
970:,
965:n
961:S
952:n
948:S
939:n
935:S
931::
886:.
881:n
877:S
868:n
864:S
860:=
851:/
845:n
841:S
818:n
814:S
793:n
756:n
692:I
686:i
681:)
676:i
672:X
668:(
646:,
641:I
635:i
630:)
625:i
621:p
617:(
595:.
591:}
587:I
581:j
578:,
575:i
572::
568:)
562:j
558:p
554:,
549:i
545:p
540:(
535:{
492:,
483:/
476:i
472:X
466:I
460:i
452:=
447:i
443:X
437:I
431:i
406:.
401:I
395:i
390:)
385:i
381:p
377:(
349:I
343:i
338:)
333:i
329:X
325:(
303:.
299:}
295:)
289:0
285:y
281:,
276:0
272:x
267:(
263:{
214:,
205:/
200:)
197:Y
191:X
188:(
185:=
182:Y
176:X
157::
152:0
148:y
139:0
135:x
124:Y
120:X
108:Y
104:X
88:0
84:y
61:0
57:x
42:Y
38:X
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