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Cubical complex

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In algebraic topology, cubical complexes are often useful for concrete calculations. In particular, there is a definition of homology for cubical complexes that coincides with the
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are elementary intervals. Equivalently, an elementary cube is any translate of a unit cube
1141: 408: 8: 1378: 1111: 1582: 336:{\displaystyle Q=I_{1}\times I_{2}\times \cdots \times I_{d}\subsetneq \mathbf {R} ^{d}} 1576: 1546: 1526: 1447: 1437: 1315: 1295: 1178: 1106: 1098: 1042: 1014: 950: 930: 910: 886: 840: 820: 800: 776: 730: 710: 687: 667: 621: 243: 60: 1571: 1564: 1430: 1388: 1253: 1231: 1221: 1198: 1092: 1074: 72: 33: 1596: 1344: 1290: 1188: 83: 37: 1403: 1398: 448: 1586: 1654: 1493: 1425: 1193: 1164: 1648: 1503: 1413: 1393: 1202: 1078: 595: 87: 1235: 1488: 1408: 1354: 41: 1498: 17: 606:
Elementary intervals of length 0 (containing a single point) are called
1442: 1373: 1332: 594:) if it can be written as a union of elementary cubes (or possibly, is 64: 1467: 444: 53: 1452: 1420: 1369: 1276: 1216:
Kaczynski, Tomasz; Mischaikow, Konstantin; Mrozek, Marian (2004).
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of a cube is the number of nondegenerate intervals in
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is the finite product of elementary intervals, i.e.
1163:Werman, Michael; Wright, Matthew L. (2016-07-01). 1051: 1023: 1003: 959: 939: 919: 895: 875: 849: 829: 809: 785: 765: 739: 719: 696: 676: 656: 630: 578: 543: 517: 471: 436: 397: 335: 252: 229: 198: 123: 1646: 1138:"Introduction to Digital Topology Lecture Notes" 1165:"Intrinsic Volumes of Random Cubical Complexes" 1261: 518:{\displaystyle n,d\in \mathbf {N} \cup \{0\}} 1162: 512: 506: 579:{\displaystyle X\subseteq \mathbf {R} ^{d}} 199:{\displaystyle I=\quad {\text{or}}\quad I=} 1629: 1602: 1268: 1254: 1192: 1182: 398:{\displaystyle I_{1},I_{2},\ldots ,I_{d}} 684:is the largest dimension of any cube in 124:{\displaystyle I\subsetneq \mathbf {R} } 77: 1647: 1135: 601: 1249: 1170:Discrete & Computational Geometry 1062: 664:. The dimension of a cubical complex 90:to) 1-dimensional cubical complexes. 13: 14: 1681: 230:{\displaystyle l\in \mathbf {Z} } 1628: 1601: 1591: 1581: 1570: 1560: 1559: 1353: 1091: 566: 499: 472:{\displaystyle \mathbf {R} ^{d}} 459: 323: 223: 117: 1004:{\displaystyle \dim Q=\dim P-1} 174: 168: 59:. They are used analogously to 1209: 1156: 1129: 610:, while those of length 1 are 425: 412: 193: 181: 165: 147: 94: 1: 1122: 1275: 766:{\displaystyle Q\subseteq P} 7: 1084: 10: 1686: 1522:Banach fixed-point theorem 1066: 67:in the computation of the 1555: 1512: 1476: 1362: 1351: 1283: 1194:10.1007/s00454-016-9789-z 747:are elementary cubes and 57:-dimensional counterparts 876:{\displaystyle Q\neq P} 544:{\displaystyle n\leq d} 1670:Computational topology 1577:Mathematics portal 1477:Metrics and properties 1463:Second-countable space 1220:. New York: Springer. 1218:Computational Homology 1136:Kovalevsky, Vladimir. 1053: 1025: 1005: 961: 941: 921: 897: 877: 851: 831: 811: 787: 767: 741: 721: 698: 678: 658: 657:{\displaystyle \dim Q} 632: 580: 545: 519: 473: 438: 399: 337: 254: 231: 200: 125: 91: 1117:Abstract cell complex 1054: 1026: 1006: 962: 942: 922: 898: 878: 852: 832: 812: 788: 768: 742: 722: 699: 679: 659: 633: 581: 546: 520: 474: 439: 400: 338: 255: 232: 201: 126: 81: 1532:Invariance of domain 1484:Euler characteristic 1458:Bundle (mathematics) 1043: 1015: 971: 951: 931: 911: 887: 861: 841: 821: 801: 777: 751: 731: 711: 688: 668: 642: 622: 555: 529: 483: 454: 437:{\displaystyle ^{n}} 409: 350: 267: 244: 213: 138: 107: 61:simplicial complexes 1542:Tychonoff's theorem 1537:PoincarĂ© conjecture 1291:General (point-set) 1112:Simplicial homology 602:Related terminology 101:elementary interval 1665:Algebraic topology 1660:Topological spaces 1527:De Rham cohomology 1448:Polyhedral complex 1438:Simplicial complex 1107:Simplicial complex 1099:Mathematics portal 1063:Algebraic topology 1049: 1021: 1001: 957: 937: 917: 893: 873: 847: 827: 807: 783: 763: 737: 717: 694: 674: 654: 628: 576: 541: 515: 469: 434: 395: 333: 250: 227: 196: 121: 92: 73:topological spaces 1642: 1641: 1431:fundamental group 1075:singular homology 1052:{\displaystyle P} 1024:{\displaystyle Q} 960:{\displaystyle P} 940:{\displaystyle Q} 920:{\displaystyle P} 896:{\displaystyle Q} 850:{\displaystyle P} 830:{\displaystyle Q} 810:{\displaystyle P} 786:{\displaystyle Q} 740:{\displaystyle P} 720:{\displaystyle Q} 697:{\displaystyle X} 677:{\displaystyle X} 631:{\displaystyle Q} 253:{\displaystyle Q} 172: 30:Cartesian complex 1677: 1632: 1631: 1605: 1604: 1595: 1585: 1575: 1574: 1563: 1562: 1357: 1270: 1263: 1256: 1247: 1246: 1240: 1239: 1213: 1207: 1206: 1196: 1186: 1160: 1154: 1153: 1151: 1149: 1140:. Archived from 1133: 1101: 1096: 1095: 1069:Cubical homology 1058: 1056: 1055: 1050: 1030: 1028: 1027: 1022: 1010: 1008: 1007: 1002: 966: 964: 963: 958: 946: 944: 943: 938: 926: 924: 923: 918: 902: 900: 899: 894: 882: 880: 879: 874: 856: 854: 853: 848: 836: 834: 833: 828: 816: 814: 813: 808: 792: 790: 789: 784: 772: 770: 769: 764: 746: 744: 743: 738: 726: 724: 723: 718: 703: 701: 700: 695: 683: 681: 680: 675: 663: 661: 660: 655: 637: 635: 634: 629: 598:to such a set). 585: 583: 582: 577: 575: 574: 569: 550: 548: 547: 542: 524: 522: 521: 516: 502: 478: 476: 475: 470: 468: 467: 462: 443: 441: 440: 435: 433: 432: 404: 402: 401: 396: 394: 393: 375: 374: 362: 361: 342: 340: 339: 334: 332: 331: 326: 317: 316: 298: 297: 285: 284: 259: 257: 256: 251: 236: 234: 233: 228: 226: 205: 203: 202: 197: 173: 170: 130: 128: 127: 122: 120: 1685: 1684: 1680: 1679: 1678: 1676: 1675: 1674: 1645: 1644: 1643: 1638: 1569: 1551: 1547:Urysohn's lemma 1508: 1472: 1358: 1349: 1321:low-dimensional 1279: 1274: 1244: 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1501: 1496: 1494:Winding number 1491: 1486: 1480: 1478: 1474: 1473: 1471: 1470: 1465: 1460: 1455: 1450: 1445: 1440: 1435: 1434: 1433: 1428: 1426:homotopy group 1418: 1417: 1416: 1411: 1406: 1401: 1396: 1386: 1381: 1376: 1366: 1364: 1360: 1359: 1352: 1350: 1348: 1347: 1342: 1337: 1336: 1335: 1325: 1324: 1323: 1313: 1308: 1303: 1298: 1293: 1287: 1285: 1281: 1280: 1273: 1272: 1265: 1258: 1250: 1242: 1241: 1226: 1208: 1155: 1127: 1126: 1124: 1121: 1120: 1119: 1114: 1109: 1103: 1102: 1086: 1083: 1067:Main article: 1064: 1061: 1048: 1020: 1000: 997: 994: 991: 988: 985: 982: 979: 976: 956: 936: 916: 892: 872: 869: 866: 846: 826: 806: 782: 762: 759: 756: 736: 716: 693: 673: 653: 650: 647: 627: 603: 600: 573: 568: 563: 560: 540: 537: 534: 514: 511: 508: 505: 501: 497: 494: 491: 488: 466: 461: 431: 427: 423: 420: 417: 414: 392: 388: 384: 381: 378: 373: 369: 365: 360: 356: 344: 343: 330: 325: 320: 315: 311: 307: 304: 301: 296: 292: 288: 283: 279: 275: 272: 249: 225: 221: 218: 207: 206: 195: 192: 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1264: 1259: 1257: 1252: 1251: 1248: 1237: 1233: 1229: 1227:9780387215976 1223: 1219: 1212: 1204: 1200: 1195: 1190: 1185: 1180: 1177:(1): 93–113. 1176: 1172: 1171: 1166: 1159: 1144:on 2020-02-23 1143: 1139: 1132: 1128: 1118: 1115: 1113: 1110: 1108: 1105: 1104: 1100: 1094: 1089: 1082: 1080: 1076: 1070: 1060: 1046: 1038: 1034: 1018: 998: 995: 992: 989: 986: 983: 980: 977: 974: 954: 947:is a face of 934: 914: 906: 890: 870: 867: 864: 844: 837:is a face of 824: 804: 796: 780: 760: 757: 754: 734: 714: 705: 691: 671: 651: 648: 645: 625: 617: 613: 612:nondegenerate 609: 599: 597: 593: 589: 571: 561: 558: 538: 535: 532: 509: 503: 495: 492: 489: 486: 464: 450: 446: 429: 421: 418: 415: 390: 386: 382: 379: 376: 371: 367: 363: 358: 354: 328: 318: 313: 309: 305: 302: 299: 294: 290: 286: 281: 277: 273: 270: 263: 262: 261: 247: 240: 219: 216: 190: 187: 184: 178: 175: 162: 159: 156: 153: 150: 144: 141: 134: 133: 132: 113: 110: 102: 89: 85: 80: 76: 74: 70: 66: 62: 58: 56: 51: 47: 43: 42:line segments 39: 35: 31: 27: 24:(also called 23: 19: 1634:Publications 1499:Chern number 1489:Betti number 1372: / 1363:Key concepts 1311:Differential 1217: 1211: 1174: 1168: 1158: 1148:November 30, 1146:. Retrieved 1142:the original 1131: 1072: 1037:primary face 1036: 1032: 904: 794: 706: 615: 611: 607: 605: 596:homeomorphic 591: 587: 345: 238: 208: 131:of the form 103:is a subset 100: 98: 88:homeomorphic 65:CW complexes 54: 52:, and their 36:composed of 29: 25: 21: 15: 1597:Wikiversity 1514:Key results 905:proper face 592:cubical set 95:Definitions 26:cubical set 18:mathematics 1649:Categories 1443:CW complex 1384:Continuity 1374:Closed set 1333:cohomology 1123:References 1079:computable 638:, denoted 608:degenerate 479:(for some 1622:geometric 1617:algebraic 1468:Cobordism 1404:Hausdorff 1399:connected 1316:Geometric 1306:Continuum 1296:Algebraic 1203:0179-5376 1184:1402.5367 1077:, but is 996:− 990:⁡ 978:⁡ 868:≠ 758:⊆ 649:⁡ 616:dimension 562:⊆ 551:). A set 536:≤ 504:∪ 496:∈ 380:… 319:⊊ 306:× 303:⋯ 300:× 287:× 220:∈ 209:for some 114:⊊ 1587:Wikibook 1565:Category 1453:Manifold 1421:Homotopy 1379:Interior 1370:Open set 1328:Homology 1277:Topology 1236:55897585 1085:See also 445:embedded 69:homology 1612:general 1414:uniform 1394:compact 1345:Digital 1011:, then 883:, then 773:, then 46:squares 32:) is a 1607:Topics 1409:metric 1284:Fields 1234:  1224:  1201:  614:. The 346:where 84:graphs 38:points 1655:Cubes 1389:Space 1179:arXiv 1033:facet 1031:is a 927:. If 903:is a 817:. If 793:is a 586:is a 525:with 237:. An 86:are ( 50:cubes 1232:OCLC 1222:ISBN 1199:ISSN 1150:2021 967:and 857:and 795:face 727:and 590:(or 82:All 63:and 28:and 20:, a 1189:doi 1039:of 1035:or 987:dim 975:dim 907:of 797:of 707:If 646:dim 447:in 99:An 71:of 34:set 16:In 1651:: 1230:. 1197:. 1187:. 1175:56 1173:. 1167:. 1081:. 1059:. 704:. 171:or 75:. 48:, 44:, 40:, 1269:e 1262:t 1255:v 1238:. 1205:. 1191:: 1181:: 1152:. 1047:P 1019:Q 999:1 993:P 984:= 981:Q 955:P 935:Q 915:P 891:Q 871:P 865:Q 845:P 825:Q 805:P 781:Q 761:P 755:Q 735:P 715:Q 692:X 672:X 652:Q 626:Q 572:d 567:R 559:X 539:d 533:n 513:} 510:0 507:{ 500:N 493:d 490:, 487:n 465:d 460:R 430:n 426:] 422:1 419:, 416:0 413:[ 391:d 387:I 383:, 377:, 372:2 368:I 364:, 359:1 355:I 329:d 324:R 314:d 310:I 295:2 291:I 282:1 278:I 274:= 271:Q 248:Q 224:Z 217:l 194:] 191:l 188:, 185:l 182:[ 179:= 176:I 166:] 163:1 160:+ 157:l 154:, 151:l 148:[ 145:= 142:I 118:R 111:I 55:n

Index

mathematics
set
points
line segments
squares
cubes
n-dimensional counterparts
simplicial complexes
CW complexes
homology
topological spaces

graphs
homeomorphic
embedded
Euclidean space
homeomorphic
Cubical homology
singular homology
computable
icon
Mathematics portal
Simplicial complex
Simplicial homology
Abstract cell complex
"Introduction to Digital Topology Lecture Notes"
the original
"Intrinsic Volumes of Random Cubical Complexes"
Discrete & Computational Geometry
arXiv

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