17:
314:
1203:
1893:
1062:
150:
1552:
520:
130:
700:
1367:
1636:
852:
1973:
1591:
442:
1073:
896:
1501:
746:
625:
1935:
1771:
410:
1716:
1456:
1421:
783:
98:
803:
720:
645:
599:
571:
543:
462:
361:
337:
67:
1787:
960:
309:{\displaystyle CX=(X\times )\cup _{p}v\ =\ \varinjlim {\bigl (}(X\times )\hookleftarrow (X\times \{0\})\xrightarrow {p} v{\bigr )},}
1509:
2046:
2031:
467:
2093:
2085:
1594:
1555:
103:
2036:
661:
1297:
651:
for compact spaces when the latter is defined. However, the topological cone construction is more general.
2127:
1600:
828:
1198:{\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=(z-1)^{2}{\mbox{ and }}0\leq z\leq 1\}.}
1940:
1564:
415:
364:
857:
2003:
1477:
725:
604:
1976:
1901:
1737:
377:
1993:
1988:
1644:
1281:
2058:
8:
1378:
372:
74:
1438:
1403:
765:
80:
1253:
1235:
1210:
937:
788:
705:
630:
584:
578:
556:
528:
447:
346:
322:
52:
37:
2122:
2089:
2081:
2075:
2042:
1888:{\displaystyle (X\times )/(X\times \left\{0\right\}\cup \left\{x_{0}\right\}\times )}
1723:
1432:
813:). The considered spaces are compact, so we get the same result up to homeomorphism.
46:
2054:
1284:
since every point can be connected to the vertex point. Furthermore, every cone is
901:
The cone over two points {0, 1} is a "V" shape with endpoints at {0} and {1}.
368:
340:
1057:{\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=1{\mbox{ and }}z=0\}}
2008:
1393:
941:
905:
810:
655:
550:
77:
and then collapsing one of its end faces to a point. The cone of X is denoted by
1898:
where we take the basepoint of the reduced cone to be the equivalence class of
1459:
759:
648:
2116:
2071:
1774:
1389:
806:
546:
1998:
1285:
574:
2104:
1975:
becomes a based map. This construction also gives a functor, from the
1374:
822:
1423:
can be visualized as the collection of lines joining every point of
287:
1289:
1224:
912:
33:
1504:
1264:
923:
2039:: Lectures on Topological Methods in Combinatorics and Geometry
1242:
948:
2099:
1373:
The cone is used in algebraic topology precisely because it
919:), otherwise known as a 2-simplex (see the final example).
2080:
Cambridge
University Press, Cambridge, 2002. xii+544 pp.
1547:{\displaystyle C\colon \mathbf {Top} \to \mathbf {Top} }
16:
1168:
1036:
647:
itself. That is, the topological cone agrees with the
515:{\displaystyle p:{\bigl (}X\times \{0\}{\bigr )}\to v}
1943:
1904:
1790:
1740:
1647:
1603:
1567:
1512:
1480:
1441:
1406:
1300:
1076:
963:
860:
831:
791:
768:
728:
708:
664:
633:
607:
587:
559:
531:
470:
450:
418:
380:
349:
325:
153:
106:
83:
55:
2041:(2nd ed.). Berlin-Heidelberg: Springer-Verlag.
1427:
to a single point. However, this picture fails when
1400:can be embedded in Euclidean space), then the cone
367:to that point. In other words, it is the result of
1967:
1929:
1887:
1765:
1710:
1630:
1585:
1546:
1495:
1450:
1431:is not compact or not Hausdorff, as generally the
1415:
1361:
1197:
1056:
890:
846:
797:
777:
740:
714:
694:
639:
619:
593:
565:
537:
514:
456:
436:
404:
355:
331:
308:
124:
92:
61:
944:of classical geometry (hence the concept's name).
2114:
1937:. With this definition, the natural inclusion
501:
479:
298:
227:
1189:
1077:
1051:
964:
867:
861:
686:
680:
496:
490:
431:
425:
277:
271:
1209:This in turn is homeomorphic to the closed
627:such that these segments intersect only in
1106:
1068:is the curved surface of the solid cone:
993:
834:
2030:
125:{\displaystyle \operatorname {cone} (X)}
24:is in blue, and the collapsed end point
15:
1777:, there is a related construction, the
2115:
695:{\displaystyle CX\simeq X\star \{v\}=}
69:is intuitively obtained by stretching
1362:{\displaystyle h_{t}(x,s)=(x,(1-t)s)}
20:Cone of a circle. The original space
2026:
2024:
1248:is also homeomorphic to the closed (
343:(called the vertex of the cone) and
13:
14:
2139:
2021:
1631:{\displaystyle Cf\colon CX\to CY}
1540:
1537:
1534:
1526:
1523:
1520:
911:of the real line is a filled-in
847:{\displaystyle \mathbb {R} ^{2}}
654:The cone is a special case of a
1729:
1469:
1230:is homeomorphic to the closed (
1968:{\displaystyle x\mapsto (x,1)}
1962:
1950:
1947:
1924:
1905:
1882:
1879:
1867:
1823:
1815:
1812:
1800:
1791:
1760:
1741:
1705:
1696:
1690:
1684:
1678:
1675:
1663:
1660:
1657:
1648:
1619:
1586:{\displaystyle f\colon X\to Y}
1577:
1556:category of topological spaces
1530:
1484:
1462:than the set of lines joining
1356:
1350:
1338:
1329:
1323:
1311:
1158:
1145:
1098:
1080:
985:
967:
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280:
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232:
187:
184:
172:
163:
135:
119:
113:
1:
2037:Using the Borsuk-Ulam Theorem
2014:
1979:of pointed spaces to itself.
1722:where square brackets denote
1275:
915:(with one of the edges being
437:{\displaystyle X\times \{0\}}
2053:Written in cooperation with
891:{\displaystyle \{p\}\times }
7:
1982:
1496:{\displaystyle X\mapsto CX}
1288:to the vertex point by the
753:
741:{\displaystyle v\not \in X}
620:{\displaystyle v\not \in X}
10:
2144:
1930:{\displaystyle (x_{0},0)}
1766:{\displaystyle (X,x_{0})}
1381:of a contractible space.
405:{\displaystyle X\times }
2004:Mapping cone (topology)
1711:{\displaystyle (Cf)()=}
929:is a pyramid with base
1969:
1931:
1889:
1767:
1712:
1632:
1587:
1548:
1497:
1452:
1417:
1363:
1217:More general examples:
1199:
1058:
892:
848:
817:The cone over a point
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779:
742:
716:
696:
641:
621:
595:
567:
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516:
458:
438:
406:
357:
333:
310:
140:Formally, the cone of
126:
94:
63:
29:
1994:Suspension (topology)
1989:Cone (disambiguation)
1970:
1932:
1890:
1768:
1713:
1633:
1588:
1549:
1498:
1453:
1418:
1364:
1200:
1059:
893:
849:
825:is a line-segment in
800:
780:
743:
717:
697:
642:
622:
596:
568:
540:
517:
464:along the projection
459:
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358:
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311:
127:
95:
64:
19:
1941:
1902:
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1510:
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1404:
1298:
1074:
961:
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829:
789:
766:
758:Here we often use a
726:
722:with a single point
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631:
605:
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529:
468:
448:
416:
378:
347:
323:
151:
104:
81:
53:
2077:Algebraic topology.
1724:equivalence classes
1396:(essentially, when
1271:+ 1)-simplex.
601:to any fixed point
291:
2128:Algebraic topology
1965:
1927:
1885:
1763:
1708:
1628:
1583:
1544:
1493:
1451:{\displaystyle CX}
1448:
1416:{\displaystyle CX}
1413:
1359:
1195:
1172:
1054:
1040:
888:
844:
795:
778:{\displaystyle CX}
775:
738:
712:
692:
637:
617:
591:
563:
535:
512:
454:
434:
402:
353:
329:
306:
220:
122:
93:{\displaystyle CX}
90:
59:
38:algebraic topology
30:
2059:Günter M. Ziegler
2048:978-3-540-00362-5
1433:quotient topology
1259:The cone over an
1241:The cone over an
1223:The cone over an
1171:
1039:
798:{\displaystyle X}
715:{\displaystyle X}
640:{\displaystyle v}
594:{\displaystyle X}
581:of segments from
566:{\displaystyle X}
538:{\displaystyle X}
457:{\displaystyle v}
356:{\displaystyle p}
332:{\displaystyle v}
292:
213:
211:
205:
62:{\displaystyle X}
47:topological space
2135:
2109:
2064:
2062:
2028:
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1041:
1037:
1028:
1027:
1015:
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947:The cone over a
936:The cone over a
922:The cone over a
904:The cone over a
897:
895:
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889:
853:
851:
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845:
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837:
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338:
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231:
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131:
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68:
66:
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2143:
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2134:
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2132:
2113:
2112:
2098:
2068:
2067:
2049:
2029:
2022:
2017:
2009:Join (topology)
1985:
1942:
1939:
1938:
1912:
1908:
1903:
1900:
1899:
1854:
1850:
1846:
1832:
1818:
1789:
1786:
1785:
1754:
1750:
1739:
1736:
1735:
1732:
1646:
1643:
1642:
1638:is defined by
1602:
1599:
1598:
1566:
1563:
1562:
1533:
1519:
1511:
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1479:
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1405:
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1301:
1299:
1296:
1295:
1278:
1170: and
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1161:
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1123:
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1110:
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1075:
1072:
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1038: and
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1023:
1019:
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1006:
997:
992:
991:
962:
959:
958:
906:closed interval
859:
856:
855:
838:
833:
832:
830:
827:
826:
811:Euclidean space
805:is a non-empty
790:
787:
786:
767:
764:
763:
756:
727:
724:
723:
707:
704:
703:
663:
660:
659:
632:
629:
628:
606:
603:
602:
586:
583:
582:
558:
555:
554:
551:Euclidean space
545:is a non-empty
530:
527:
526:
500:
499:
478:
477:
469:
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465:
449:
446:
445:
417:
414:
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379:
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375:
348:
345:
344:
324:
321:
320:
297:
296:
226:
225:
212:
194:
190:
152:
149:
148:
144:is defined as:
138:
105:
102:
101:
82:
79:
78:
54:
51:
50:
12:
11:
5:
2141:
2131:
2130:
2125:
2111:
2110:
2096:
2066:
2065:
2055:Anders Björner
2047:
2032:Matoušek, Jiří
2019:
2018:
2016:
2013:
2012:
2011:
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2001:
1996:
1991:
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1911:
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1615:
1612:
1609:
1606:
1595:continuous map
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1331:
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1282:path-connected
1280:All cones are
1277:
1274:
1273:
1272:
1257:
1239:
1215:
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1207:
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952:
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902:
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887:
884:
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878:
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869:
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863:
841:
836:
794:
774:
771:
760:geometric cone
755:
752:
737:
734:
731:
711:
691:
688:
685:
682:
679:
676:
673:
670:
667:
649:geometric cone
636:
616:
613:
610:
590:
562:
553:, the cone on
534:
511:
508:
503:
498:
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492:
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6:
4:
3:
2:
2140:
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2118:
2107:
2106:
2101:
2097:
2095:
2094:0-521-79540-0
2091:
2087:
2086:0-521-79160-X
2083:
2079:
2078:
2073:
2072:Allen Hatcher
2070:
2069:
2063:, Section 4.3
2061:
2060:
2056:
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1782:
1780:
1776:
1775:pointed space
1755:
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1727:
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1382:
1380:
1377:a space as a
1376:
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1302:
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979:
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950:
946:
943:
940:is the solid
939:
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932:
928:
925:
921:
918:
914:
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36:, especially
35:
27:
23:
18:
2103:
2076:
2052:
2035:
1999:Desuspension
1897:
1781:, given by
1779:reduced cone
1778:
1733:
1730:Reduced cone
1721:
1558:
1473:
1470:Cone functor
1466:to a point.
1463:
1428:
1424:
1397:
1385:
1383:
1372:
1286:contractible
1279:
1268:
1260:
1249:
1243:
1231:
1225:
1218:
1216:
930:
926:
916:
908:
818:
809:subspace of
757:
749:
702:the join of
653:
575:homeomorphic
549:subspace of
524:
412:by its face
318:
141:
139:
70:
44:
41:
31:
28:is in green.
25:
21:
1252:+ 1)-
1234:+ 1)-
444:to a point
136:Definitions
2117:Categories
2105:PlanetMath
2015:References
1503:induces a
1276:Properties
365:projection
1948:↦
1865:×
1844:∪
1830:×
1798:×
1620:→
1611::
1578:→
1572::
1531:→
1517::
1485:↦
1394:Hausdorff
1345:−
1184:≤
1178:≤
1152:−
1117:∣
1102:∈
1004:∣
989:∈
871:×
823:real line
678:⋆
672:≃
507:→
488:×
423:×
385:×
369:attaching
269:×
260:↩
239:×
223:
218:→
192:∪
170:×
111:
2123:Topology
2034:(2007).
1983:See also
1977:category
1474:The map
1458:will be
1379:subspace
1290:homotopy
951:given by
913:triangle
754:Examples
733:∉
612:∉
373:cylinder
285:→
75:cylinder
34:topology
1597:, then
1554:on the
1505:functor
1390:compact
1267:is an (
1265:simplex
1228:-sphere
924:polygon
821:of the
807:compact
577:to the
547:compact
363:is the
100:or by
73:into a
2100:"Cone"
2092:
2084:
2045:
1375:embeds
949:circle
785:where
319:where
210:
204:
40:, the
1773:is a
1593:is a
1561:. If
1460:finer
1384:When
1246:-ball
579:union
341:point
339:is a
45:of a
2090:ISBN
2088:and
2082:ISBN
2057:and
2043:ISBN
1392:and
1254:ball
1236:ball
1211:disc
942:cone
938:disk
656:join
522:.
371:the
108:cone
42:cone
1734:If
1559:Top
1435:on
1388:is
573:is
525:If
215:lim
132:.
32:In
2119::
2102:.
2074:,
2051:.
2023:^
1726:.
854:,
658::
2108:.
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1960:1
1957:,
1954:x
1951:(
1945:x
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1914:0
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1682:=
1679:)
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1667:x
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1661:(
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1269:n
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1232:n
1226:n
1213:.
1193:.
1190:}
1187:1
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1175:0
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1146:(
1143:=
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1134:y
1130:+
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1107:R
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1096:z
1093:,
1090:y
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1084:x
1081:(
1078:{
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1033:1
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1025:2
1021:y
1017:+
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983:z
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865:p
862:{
840:2
835:R
819:p
793:X
773:X
770:C
762:(
748:.
736:X
730:v
710:X
690:=
687:}
684:v
681:{
675:X
669:X
666:C
635:v
615:X
609:v
589:X
561:X
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510:v
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497:}
494:0
491:{
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480:(
475::
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432:}
429:0
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397:1
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278:}
275:0
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158:X
155:C
142:X
120:)
117:X
114:(
88:X
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26:v
22:X
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