2808:
804:
848:
826:
31:
2591:
2829:
2797:
2866:
2839:
2819:
1254:
1837:
2099:
2150:
of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of
241:
1934:
1441:
1112:
2124:-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting
1161:
1636:
442:
788:
704:
1562:
594:
1156:
1070:
1011:
959:
920:
664:
564:
728:
630:
408:
361:
337:
184:
160:
124:
295:
268:
1314:. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a
1678:
2869:
748:
448:. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a
2154:, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.
2169:
2163:
2325:
2249:
1945:
2340:
2264:
452:-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as
2503:
803:
2905:
2857:
2852:
2433:
2396:
2378:
2309:
475:). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of)
2847:
196:
2117:
2203:
2749:
2330:
2254:
1852:
1362:
847:
825:
2895:
2890:
2193:
2172:
is: given a finite simplicial complex, decide whether it is homeomorphic to a given geometric object. This problem is
1311:
302:
82:
1480:
1075:
1249:{\displaystyle \mathrm {Cl} {\big (}\mathrm {st} (S){\big )}\setminus \mathrm {st} {\big (}\mathrm {Cl} (S){\big )}}
2757:
2425:
974:
1567:
2900:
2828:
2556:
453:
417:
1645:
of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging
761:
2842:
2198:
669:
2777:
2772:
2698:
2575:
2563:
2536:
2496:
2619:
2546:
2295:
1509:
569:
85:. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a
2807:
1321:
which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a
1130:
1044:
985:
933:
894:
635:
535:
2767:
2719:
2693:
2541:
2352:
2291:
2276:
2213:
709:
611:
389:
367:. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any
342:
318:
165:
141:
105:
2818:
2614:
2173:
1273:
476:
69:(see illustration). Simplicial complexes should not be confused with the more abstract notion of a
1298:. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at
2812:
2762:
2683:
2551:
2531:
2140:
1279:
529:
273:
246:
2782:
1290:
directly, provided that consistent orientations are made of all simplices. The requirements of
2800:
2666:
2624:
2489:
2458:
2429:
2392:
2374:
2348:
2336:
2305:
2299:
2272:
2260:
2224:
1318:
605:
472:
47:
17:
2580:
2526:
1832:{\displaystyle F_{\Delta }(x-1)=h_{0}x^{d+1}+h_{1}x^{d}+h_{2}x^{d-1}+\cdots +h_{d}x+h_{d+1}}
190:
51:
30:
2443:
2406:
2639:
2634:
2439:
2402:
2218:
1303:
1291:
1119:
74:
1463: = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the
1282:, simplicial complexes are often useful for concrete calculations. For the definition of
486:
is used to refer to a simplex of a complex, not to be confused with a face of a simplex.
2729:
2661:
2461:
2151:
2112:-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial
1471:-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its
1283:
733:
70:
2108:-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this
2884:
2739:
2649:
2629:
2366:
2147:
1939:
We calculate the h-vector of the octahedron boundary (our first example) as follows:
1345:
1315:
1287:
510:
78:
2832:
2724:
2644:
2590:
2413:
2229:
514:
55:
2304:, Algorithms and Computation in Mathematics, vol. 25, Springer, p. 493,
305:, which loosely speaking is a simplicial complex without an associated geometry.
2822:
2734:
2417:
39:
2678:
2609:
2568:
2475:
1495:
1464:
1322:
1295:
368:
2146:
Simplicial complexes can be seen to have the same geometric structure as the
2703:
2466:
2208:
2136:
490:
2476:
Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk.
1475:-vector is (1, 18, 23, 8, 1). A complete characterization of the possible
2688:
2656:
2605:
2512:
2113:
1299:
1037:
is generally not a simplicial complex itself, so some authors define the
457:
59:
2094:{\displaystyle F(x-1)=(x-1)^{3}+6(x-1)^{2}+12(x-1)+8=x^{3}+3x^{2}+3x+1.}
1356:
1307:
135:
127:
63:
339:
is a simplicial complex where the largest dimension of any simplex in
410:
is a simplicial complex where every simplex of dimension less than
2485:
2333:: Lectures on Topological Methods in Combinatorics and Geometry
2257:: Lectures on Topological Methods in Combinatorics and Geometry
479:
this coincides with the meaning from polyhedral combinatorics.
1310:. That somewhat more concrete concept is there attributed to
467:
is a maximal simplex, i.e., any simplex in a complex that is
715:
686:
646:
617:
578:
546:
429:
395:
348:
324:
228:
171:
147:
111:
2481:
2301:
Triangulations: Structures for
Algorithms and Applications
1498:(written in decreasing order of exponents), we obtain the
471:
a face of any larger simplex. (Note the difference from a
236:{\displaystyle \sigma _{1},\sigma _{2}\in {\mathcal {K}}}
1286:
of a simplicial complex, one can read the corresponding
2131:
A complete characterization of all simplicial polytope
1948:
1855:
1681:
1570:
1512:
1365:
1164:
1133:
1078:
1047:
988:
936:
897:
764:
750:
in its relative interior. This simplex is called the
736:
712:
672:
638:
614:
572:
538:
509:
is sometimes used in a broader sense to denote a set
420:
392:
345:
321:
276:
249:
199:
168:
144:
108:
2391:(Reprint of the 1980 ed.), Mineola, NY: Dover,
2335:(2nd ed.). Berlin-Heidelberg: Springer-Verlag.
2259:(2nd ed.). Berlin-Heidelberg: Springer-Verlag.
1929:{\displaystyle (h_{0},h_{1},h_{2},\cdots ,h_{d+1}).}
2748:
2712:
2598:
2519:
2221: – 1 dimensional simplicial complex
1436:{\displaystyle (f_{0},f_{1},f_{2},\ldots ,f_{d+1})}
2093:
1928:
1831:
1630:
1556:
1435:
1248:
1150:
1106:
1064:
1005:
953:
914:
782:
742:
722:
698:
658:
624:
588:
558:
436:
402:
355:
331:
289:
262:
235:
178:
154:
118:
1641:Combinatorists are often quite interested in the
1479:-vectors of simplicial complexes is given by the
1107:{\displaystyle \mathrm {Cl} \ \mathrm {st} \ S}
1013:) is the union of the stars of each simplex in
2497:
1241:
1217:
1199:
1175:
8:
1294:lead to the use of more general spaces, the
583:
573:
1456:−1)-dimensional faces of Δ (by convention,
1334:
922:) is the smallest simplicial subcomplex of
532:of its simplices. It is usually denoted by
513:to a simplex, leading to the definition of
2865:
2838:
2504:
2490:
2482:
1631:{\displaystyle x^{4}+18x^{3}+23x^{2}+8x+1}
1355:of a simplicial d-complex Δ, which is the
81:counterpart to a simplicial complex is an
2070:
2054:
2014:
1986:
1947:
1908:
1889:
1876:
1863:
1854:
1817:
1801:
1776:
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1753:
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1724:
1714:
1686:
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1607:
1591:
1575:
1569:
1533:
1517:
1511:
1418:
1399:
1386:
1373:
1364:
1240:
1239:
1222:
1216:
1215:
1207:
1198:
1197:
1180:
1174:
1173:
1165:
1163:
1134:
1132:
1090:
1079:
1077:
1048:
1046:
989:
987:
937:
935:
898:
896:
763:
735:
714:
713:
711:
691:
685:
684:
679:
671:
651:
645:
644:
639:
637:
632:form a partition of its underlying space
616:
615:
613:
577:
576:
571:
551:
545:
544:
539:
537:
437:{\displaystyle \sigma \in {\mathcal {K}}}
428:
427:
419:
394:
393:
391:
347:
346:
344:
323:
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281:
275:
254:
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227:
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217:
204:
198:
170:
169:
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146:
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130:that satisfies the following conditions:
110:
109:
107:
1653:-polynomial of Δ. Formally, if we write
1302:of simplicial complexes as subspaces of
783:{\displaystyle \operatorname {supp} (x)}
501:-faces are sometimes referred to as its
29:
2241:
1330:
1326:
1306:made up of subsets, each of which is a
1204:
799:
2170:simplicial complex recognition problem
2164:Simplicial complex recognition problem
1502:of Δ. In our two examples above, the
699:{\displaystyle x\in |{\mathcal {K}}|}
7:
2135:-vectors is given by the celebrated
961:is obtained by repeatedly adding to
1687:
1557:{\displaystyle x^{3}+6x^{2}+12x+8}
1226:
1223:
1211:
1208:
1184:
1181:
1169:
1166:
1138:
1135:
1094:
1091:
1083:
1080:
1052:
1049:
993:
990:
941:
938:
902:
899:
706:, there is exactly one simplex in
589:{\displaystyle \|{\mathcal {K}}\|}
25:
371:or higher-dimensional simplices.
2864:
2837:
2827:
2817:
2806:
2796:
2795:
2589:
1494:-complex Δ as coefficients of a
1260:minus the stars of all faces of
1151:{\displaystyle \mathrm {Lk} \ S}
1065:{\displaystyle \mathrm {St} \ S}
1006:{\displaystyle \mathrm {st} \ S}
954:{\displaystyle \mathrm {Cl} \ S}
915:{\displaystyle \mathrm {Cl} \ S}
876:be a collection of simplices in
872:be a simplicial complex and let
846:
824:
802:
659:{\displaystyle |{\mathcal {K}}|}
559:{\displaystyle |{\mathcal {K}}|}
528:of a simplicial complex, is the
73:appearing in modern simplicial
2387:Maunder, Charles R.F. (1996),
2204:Causal dynamical triangulation
2038:
2026:
2011:
1998:
1983:
1970:
1964:
1952:
1920:
1856:
1704:
1692:
1430:
1366:
1236:
1230:
1194:
1188:
1114:the closure of the star of S.
965:each face of every simplex in
926:that contains each simplex in
777:
771:
723:{\displaystyle {\mathcal {K}}}
692:
680:
652:
640:
625:{\displaystyle {\mathcal {K}}}
552:
540:
403:{\displaystyle {\mathcal {K}}}
356:{\displaystyle {\mathcal {K}}}
332:{\displaystyle {\mathcal {K}}}
301:See also the definition of an
179:{\displaystyle {\mathcal {K}}}
155:{\displaystyle {\mathcal {K}}}
119:{\displaystyle {\mathcal {K}}}
1:
2331:Using the Borsuk-Ulam Theorem
2255:Using the Borsuk-Ulam Theorem
18:Facet of a simplicial complex
2347:Written in cooperation with
2271:Written in cooperation with
2120:). In general, however, the
456:and provide a definition of
87:geometric simplicial complex
2194:Abstract simplicial complex
2180:-dimensional manifolds for
1668:-polynomial of Δ, then the
1256:. It is the closed star of
1025:is the set of simplices in
303:abstract simplicial complex
290:{\displaystyle \sigma _{2}}
263:{\displaystyle \sigma _{1}}
83:abstract simplicial complex
2922:
2758:Banach fixed-point theorem
2426:Cambridge University Press
2161:
2118:Dehn–Sommerville equations
1271:
414:is a face of some simplex
2791:
2587:
489:For a simplicial complex
67:-dimensional counterparts
2906:Triangulation (geometry)
1649: − 1 into the
1490:-vector of a simplicial
497:-dimensional space, the
2199:Barycentric subdivision
2128:-vector is (1, 3, −2).
1335:Hilton & Wylie 1967
1033:as a face. The star of
1017:. For a single simplex
796:Closure, star, and link
524:, sometimes called the
34:A simplicial 3-complex.
2813:Mathematics portal
2713:Metrics and properties
2699:Second-countable space
2158:Computational problems
2095:
1930:
1833:
1632:
1558:
1506:-polynomials would be
1481:Kruskal–Katona theorem
1437:
1250:
1152:
1108:
1066:
1007:
955:
916:
784:
744:
724:
700:
660:
626:
590:
560:
438:
404:
357:
333:
291:
264:
237:
180:
156:
120:
35:
2096:
1931:
1834:
1633:
1559:
1438:
1251:
1153:
1109:
1067:
1008:
956:
917:
785:
745:
725:
701:
661:
627:
591:
561:
444:of dimension exactly
439:
405:
358:
334:
292:
265:
238:
193:of any two simplices
181:
157:
121:
33:
2768:Invariance of domain
2720:Euler characteristic
2694:Bundle (mathematics)
2462:"Simplicial complex"
2214:Loop quantum gravity
2143:, Billera, and Lee.
1946:
1853:
1679:
1568:
1510:
1363:
1162:
1131:
1076:
1045:
986:
934:
895:
762:
734:
710:
670:
636:
612:
608:of all simplices in
570:
536:
477:simplicial polytopes
418:
390:
343:
319:
274:
247:
197:
166:
142:
106:
2778:Tychonoff's theorem
2773:Poincaré conjecture
2527:General (point-set)
1274:Simplicial homology
482:Sometimes the term
473:"face" of a simplex
2896:Algebraic topology
2891:Topological spaces
2763:De Rham cohomology
2684:Polyhedral complex
2674:Simplicial complex
2459:Weisstein, Eric W.
2389:Algebraic Topology
2371:Algebraic Topology
2292:De Loera, Jesús A.
2091:
1926:
1829:
1628:
1554:
1452:is the number of (
1433:
1280:algebraic topology
1268:Algebraic topology
1246:
1148:
1104:
1062:
1003:
951:
912:
780:
740:
720:
696:
656:
622:
606:relative interiors
586:
556:
434:
400:
353:
329:
287:
260:
243:is a face of both
233:
176:
152:
138:of a simplex from
116:
101:simplicial complex
44:simplicial complex
36:
2878:
2877:
2667:fundamental group
2367:Spanier, Edwin H.
2353:Günter M. Ziegler
2342:978-3-540-00362-5
2296:Santos, Francisco
2277:Günter M. Ziegler
2266:978-3-540-00362-5
1346:Combinatorialists
1319:topological space
1144:
1100:
1089:
1058:
999:
947:
908:
743:{\displaystyle x}
666:: for each point
189:2. The non-empty
16:(Redirected from
2913:
2868:
2867:
2841:
2840:
2831:
2821:
2811:
2810:
2799:
2798:
2593:
2506:
2499:
2492:
2483:
2472:
2471:
2446:
2414:Hilton, Peter J.
2409:
2383:
2358:
2356:
2322:
2316:
2314:
2294:; Rambau, Jörg;
2288:
2282:
2280:
2246:
2100:
2098:
2097:
2092:
2075:
2074:
2059:
2058:
2019:
2018:
1991:
1990:
1935:
1933:
1932:
1927:
1919:
1918:
1894:
1893:
1881:
1880:
1868:
1867:
1846:-vector of Δ is
1838:
1836:
1835:
1830:
1828:
1827:
1806:
1805:
1787:
1786:
1771:
1770:
1758:
1757:
1748:
1747:
1735:
1734:
1719:
1718:
1691:
1690:
1638:, respectively.
1637:
1635:
1634:
1629:
1612:
1611:
1596:
1595:
1580:
1579:
1563:
1561:
1560:
1555:
1538:
1537:
1522:
1521:
1442:
1440:
1439:
1434:
1429:
1428:
1404:
1403:
1391:
1390:
1378:
1377:
1348:often study the
1255:
1253:
1252:
1247:
1245:
1244:
1229:
1221:
1220:
1214:
1203:
1202:
1187:
1179:
1178:
1172:
1157:
1155:
1154:
1149:
1142:
1141:
1113:
1111:
1110:
1105:
1098:
1097:
1087:
1086:
1071:
1069:
1068:
1063:
1056:
1055:
1012:
1010:
1009:
1004:
997:
996:
960:
958:
957:
952:
945:
944:
921:
919:
918:
913:
906:
905:
862:
856:
850:
840:
834:
828:
818:
812:
806:
789:
787:
786:
781:
749:
747:
746:
741:
729:
727:
726:
721:
719:
718:
705:
703:
702:
697:
695:
690:
689:
683:
665:
663:
662:
657:
655:
650:
649:
643:
631:
629:
628:
623:
621:
620:
595:
593:
592:
587:
582:
581:
565:
563:
562:
557:
555:
550:
549:
543:
522:underlying space
443:
441:
440:
435:
433:
432:
409:
407:
406:
401:
399:
398:
362:
360:
359:
354:
352:
351:
338:
336:
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328:
327:
296:
294:
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288:
286:
285:
269:
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266:
261:
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209:
208:
185:
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125:
123:
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117:
115:
114:
27:Mathematical set
21:
2921:
2920:
2916:
2915:
2914:
2912:
2911:
2910:
2901:Simplicial sets
2881:
2880:
2879:
2874:
2805:
2787:
2783:Urysohn's lemma
2744:
2708:
2594:
2585:
2557:low-dimensional
2515:
2510:
2457:
2456:
2453:
2436:
2422:Homology Theory
2412:
2399:
2386:
2381:
2365:
2362:
2361:
2343:
2324:
2323:
2319:
2312:
2290:
2289:
2285:
2267:
2248:
2247:
2243:
2238:
2219:Polygonal chain
2190:
2166:
2160:
2152:sphere packings
2116:(these are the
2066:
2050:
2010:
1982:
1944:
1943:
1904:
1885:
1872:
1859:
1851:
1850:
1813:
1797:
1772:
1762:
1749:
1739:
1720:
1710:
1682:
1677:
1676:
1659:
1603:
1587:
1571:
1566:
1565:
1529:
1513:
1508:
1507:
1462:
1451:
1414:
1395:
1382:
1369:
1361:
1360:
1343:
1304:Euclidean space
1292:homotopy theory
1284:homology groups
1276:
1270:
1160:
1159:
1129:
1128:
1074:
1073:
1043:
1042:
984:
983:
932:
931:
893:
892:
864:
858:
854:
851:
842:
836:
832:
829:
820:
814:
810:
807:
798:
760:
759:
732:
731:
708:
707:
668:
667:
634:
633:
610:
609:
602:
568:
567:
534:
533:
416:
415:
388:
387:
341:
340:
317:
316:
277:
272:
271:
250:
245:
244:
213:
200:
195:
194:
164:
163:
140:
139:
104:
103:
97:
75:homotopy theory
28:
23:
22:
15:
12:
11:
5:
2919:
2917:
2909:
2908:
2903:
2898:
2893:
2883:
2882:
2876:
2875:
2873:
2872:
2862:
2861:
2860:
2855:
2850:
2835:
2825:
2815:
2803:
2792:
2789:
2788:
2786:
2785:
2780:
2775:
2770:
2765:
2760:
2754:
2752:
2746:
2745:
2743:
2742:
2737:
2732:
2730:Winding number
2727:
2722:
2716:
2714:
2710:
2709:
2707:
2706:
2701:
2696:
2691:
2686:
2681:
2676:
2671:
2670:
2669:
2664:
2662:homotopy group
2654:
2653:
2652:
2647:
2642:
2637:
2632:
2622:
2617:
2612:
2602:
2600:
2596:
2595:
2588:
2586:
2584:
2583:
2578:
2573:
2572:
2571:
2561:
2560:
2559:
2549:
2544:
2539:
2534:
2529:
2523:
2521:
2517:
2516:
2511:
2509:
2508:
2501:
2494:
2486:
2480:
2479:
2473:
2452:
2451:External links
2449:
2448:
2447:
2434:
2410:
2397:
2384:
2379:
2360:
2359:
2349:Anders Björner
2341:
2326:Matoušek, Jiří
2317:
2310:
2283:
2273:Anders Björner
2265:
2250:Matoušek, Jiří
2240:
2239:
2237:
2234:
2233:
2232:
2227:
2225:Tucker's lemma
2222:
2216:
2211:
2206:
2201:
2196:
2189:
2186:
2162:Main article:
2159:
2156:
2102:
2101:
2090:
2087:
2084:
2081:
2078:
2073:
2069:
2065:
2062:
2057:
2053:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2017:
2013:
2009:
2006:
2003:
2000:
1997:
1994:
1989:
1985:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1937:
1936:
1925:
1922:
1917:
1914:
1911:
1907:
1903:
1900:
1897:
1892:
1888:
1884:
1879:
1875:
1871:
1866:
1862:
1858:
1840:
1839:
1826:
1823:
1820:
1816:
1812:
1809:
1804:
1800:
1796:
1793:
1790:
1785:
1782:
1779:
1775:
1769:
1765:
1761:
1756:
1752:
1746:
1742:
1738:
1733:
1730:
1727:
1723:
1717:
1713:
1709:
1706:
1703:
1700:
1697:
1694:
1689:
1685:
1664:) to mean the
1657:
1627:
1624:
1621:
1618:
1615:
1610:
1606:
1602:
1599:
1594:
1590:
1586:
1583:
1578:
1574:
1553:
1550:
1547:
1544:
1541:
1536:
1532:
1528:
1525:
1520:
1516:
1460:
1447:
1432:
1427:
1424:
1421:
1417:
1413:
1410:
1407:
1402:
1398:
1394:
1389:
1385:
1381:
1376:
1372:
1368:
1342:
1339:
1272:Main article:
1269:
1266:
1243:
1238:
1235:
1232:
1228:
1225:
1219:
1213:
1210:
1206:
1201:
1196:
1193:
1190:
1186:
1183:
1177:
1171:
1168:
1147:
1140:
1137:
1103:
1096:
1093:
1085:
1082:
1061:
1054:
1051:
1041:of S (denoted
1021:, the star of
1002:
995:
992:
950:
943:
940:
911:
904:
901:
866:
865:
852:
845:
843:
830:
823:
821:
808:
801:
797:
794:
779:
776:
773:
770:
767:
739:
717:
694:
688:
682:
678:
675:
654:
648:
642:
619:
601:
598:
585:
580:
575:
554:
548:
542:
454:triangulations
431:
426:
423:
397:
350:
326:
299:
298:
284:
280:
257:
253:
230:
225:
220:
216:
212:
207:
203:
187:
173:
149:
113:
96:
93:
71:simplicial set
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2918:
2907:
2904:
2902:
2899:
2897:
2894:
2892:
2889:
2888:
2886:
2871:
2863:
2859:
2856:
2854:
2851:
2849:
2846:
2845:
2844:
2836:
2834:
2830:
2826:
2824:
2820:
2816:
2814:
2809:
2804:
2802:
2794:
2793:
2790:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2755:
2753:
2751:
2747:
2741:
2740:Orientability
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2717:
2715:
2711:
2705:
2702:
2700:
2697:
2695:
2692:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2672:
2668:
2665:
2663:
2660:
2659:
2658:
2655:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2627:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2607:
2604:
2603:
2601:
2597:
2592:
2582:
2579:
2577:
2576:Set-theoretic
2574:
2570:
2567:
2566:
2565:
2562:
2558:
2555:
2554:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2537:Combinatorial
2535:
2533:
2530:
2528:
2525:
2524:
2522:
2518:
2514:
2507:
2502:
2500:
2495:
2493:
2488:
2487:
2484:
2477:
2474:
2469:
2468:
2463:
2460:
2455:
2454:
2450:
2445:
2441:
2437:
2435:0-521-09422-4
2431:
2427:
2423:
2419:
2415:
2411:
2408:
2404:
2400:
2398:0-486-69131-4
2394:
2390:
2385:
2382:
2380:0-387-94426-5
2376:
2372:
2368:
2364:
2363:
2357:, Section 4.3
2355:
2354:
2350:
2344:
2338:
2334:
2332:
2327:
2321:
2318:
2313:
2311:9783642129711
2307:
2303:
2302:
2297:
2293:
2287:
2284:
2281:, Section 4.3
2279:
2278:
2274:
2268:
2262:
2258:
2256:
2251:
2245:
2242:
2235:
2231:
2228:
2226:
2223:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2192:
2191:
2187:
2185:
2183:
2179:
2175:
2171:
2165:
2157:
2155:
2153:
2149:
2148:contact graph
2144:
2142:
2138:
2134:
2129:
2127:
2123:
2119:
2115:
2111:
2107:
2088:
2085:
2082:
2079:
2076:
2071:
2067:
2063:
2060:
2055:
2051:
2047:
2044:
2041:
2035:
2032:
2029:
2023:
2020:
2015:
2007:
2004:
2001:
1995:
1992:
1987:
1979:
1976:
1973:
1967:
1961:
1958:
1955:
1949:
1942:
1941:
1940:
1923:
1915:
1912:
1909:
1905:
1901:
1898:
1895:
1890:
1886:
1882:
1877:
1873:
1869:
1864:
1860:
1849:
1848:
1847:
1845:
1824:
1821:
1818:
1814:
1810:
1807:
1802:
1798:
1794:
1791:
1788:
1783:
1780:
1777:
1773:
1767:
1763:
1759:
1754:
1750:
1744:
1740:
1736:
1731:
1728:
1725:
1721:
1715:
1711:
1707:
1701:
1698:
1695:
1683:
1675:
1674:
1673:
1671:
1667:
1663:
1656:
1652:
1648:
1644:
1639:
1625:
1622:
1619:
1616:
1613:
1608:
1604:
1600:
1597:
1592:
1588:
1584:
1581:
1576:
1572:
1551:
1548:
1545:
1542:
1539:
1534:
1530:
1526:
1523:
1518:
1514:
1505:
1501:
1497:
1493:
1489:
1486:By using the
1484:
1482:
1478:
1474:
1470:
1466:
1459:
1455:
1450:
1446:
1425:
1422:
1419:
1415:
1411:
1408:
1405:
1400:
1396:
1392:
1387:
1383:
1379:
1374:
1370:
1358:
1354:
1352:
1347:
1341:Combinatorics
1340:
1338:
1336:
1332:
1328:
1324:
1320:
1317:
1313:
1309:
1305:
1301:
1297:
1293:
1289:
1288:chain complex
1285:
1281:
1275:
1267:
1265:
1263:
1259:
1233:
1191:
1145:
1126:
1122:
1121:
1115:
1101:
1059:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1000:
981:
977:
976:
970:
968:
964:
948:
929:
925:
909:
890:
886:
881:
879:
875:
871:
861:
849:
844:
839:
827:
822:
817:
805:
800:
795:
793:
792:
774:
768:
765:
757:
753:
737:
676:
673:
607:
599:
597:
531:
527:
523:
518:
516:
512:
508:
504:
500:
496:
492:
487:
485:
480:
478:
474:
470:
466:
461:
459:
455:
451:
447:
424:
421:
413:
385:
381:
377:
372:
370:
366:
315:
313:
306:
304:
282:
278:
255:
251:
223:
218:
214:
210:
205:
201:
192:
188:
137:
133:
132:
131:
129:
102:
94:
92:
91:
88:
84:
80:
79:combinatorial
77:. The purely
76:
72:
68:
66:
61:
57:
56:line segments
53:
49:
45:
41:
32:
19:
2870:Publications
2735:Chern number
2725:Betti number
2673:
2608: /
2599:Key concepts
2547:Differential
2465:
2424:, New York:
2421:
2418:Wylie, Shaun
2388:
2373:, Springer,
2370:
2346:
2329:
2320:
2300:
2286:
2270:
2253:
2244:
2230:Simplex tree
2181:
2177:
2167:
2145:
2132:
2130:
2125:
2121:
2109:
2105:
2103:
1938:
1843:
1841:
1670:h-polynomial
1669:
1665:
1661:
1654:
1650:
1646:
1642:
1640:
1503:
1500:f-polynomial
1499:
1491:
1487:
1485:
1476:
1472:
1468:
1457:
1453:
1448:
1444:
1350:
1349:
1344:
1331:Maunder 1996
1327:Spanier 1966
1296:CW complexes
1277:
1261:
1257:
1124:
1118:
1116:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
979:
973:
971:
966:
962:
927:
923:
888:
884:
882:
877:
873:
869:
867:
859:
837:
815:
791:
758:and denoted
755:
751:
603:
525:
521:
519:
515:cell complex
511:homeomorphic
506:
502:
498:
494:
488:
483:
481:
468:
464:
462:
449:
445:
411:
383:
379:
375:
373:
364:
311:
309:
307:
300:
191:intersection
126:is a set of
100:
98:
90:
86:
64:
62:, and their
50:composed of
43:
37:
2833:Wikiversity
2750:Key results
2174:undecidable
1467:, then its
1039:closed star
730:containing
505:. The term
382:simplicial
380:homogeneous
310:simplicial
162:is also in
95:Definitions
40:mathematics
2885:Categories
2679:CW complex
2620:Continuity
2610:Closed set
2569:cohomology
2236:References
1496:polynomial
1465:octahedron
1323:polyhedron
1312:Alexandrov
1029:that have
813:and their
369:tetrahedra
2858:geometric
2853:algebraic
2704:Cobordism
2640:Hausdorff
2635:connected
2552:Geometric
2542:Continuum
2532:Algebraic
2467:MathWorld
2209:Delta set
2137:g-theorem
2033:−
2005:−
1977:−
1959:−
1899:⋯
1792:⋯
1781:−
1699:−
1688:Δ
1409:…
1359:sequence
1205:∖
1158:) equals
1127:(denoted
982:(denoted
891:(denoted
811:simplices
769:
677:∈
584:‖
574:‖
458:polytopes
425:∈
422:σ
386:-complex
279:σ
252:σ
224:∈
215:σ
202:σ
134:1. Every
128:simplices
60:triangles
2823:Wikibook
2801:Category
2689:Manifold
2657:Homotopy
2615:Interior
2606:Open set
2564:Homology
2513:Topology
2420:(1967),
2369:(1966),
2328:(2007).
2298:(2010),
2252:(2007).
2188:See also
2176:for any
2114:polytope
1842:and the
1672:of Δ is
1643:h-vector
1443:, where
1300:Polytope
857:and its
835:and its
491:embedded
314:-complex
2848:general
2650:uniform
2630:compact
2581:Digital
2444:0115161
2407:1402473
2141:Stanley
2104:So the
1357:integer
1353:-vector
1316:compact
1308:simplex
885:closure
816:closure
752:support
600:Support
526:carrier
363:equals
2843:Topics
2645:metric
2520:Fields
2442:
2432:
2405:
2395:
2377:
2339:
2308:
2263:
1143:
1099:
1088:
1057:
998:
946:
907:
855:vertex
833:vertex
52:points
2625:Space
2184:≥ 5.
1325:(see
1072:) as
530:union
503:cells
493:in a
465:facet
46:is a
2430:ISBN
2393:ISBN
2375:ISBN
2351:and
2337:ISBN
2306:ISBN
2275:and
2261:ISBN
2168:The
1564:and
1120:link
1117:The
975:star
972:The
883:The
868:Let
860:link
838:star
809:Two
766:supp
604:The
520:The
507:cell
484:face
376:pure
270:and
136:face
42:, a
2139:of
1337:).
1278:In
1123:of
978:of
887:of
754:of
566:or
469:not
450:non
378:or
48:set
38:In
2887::
2464:.
2440:MR
2438:,
2428:,
2416:;
2403:MR
2401:,
2345:.
2269:.
2089:1.
2024:12
1601:23
1585:18
1543:12
1483:.
1333:,
1329:,
1264:.
969:.
930:.
880:.
853:A
831:A
596:.
517:.
463:A
460:.
374:A
308:A
99:A
58:,
54:,
2505:e
2498:t
2491:v
2478:.
2470:.
2315:.
2182:d
2178:d
2133:h
2126:h
2122:h
2110:h
2106:h
2086:+
2083:x
2080:3
2077:+
2072:2
2068:x
2064:3
2061:+
2056:3
2052:x
2048:=
2045:8
2042:+
2039:)
2036:1
2030:x
2027:(
2021:+
2016:2
2012:)
2008:1
2002:x
1999:(
1996:6
1993:+
1988:3
1984:)
1980:1
1974:x
1971:(
1968:=
1965:)
1962:1
1956:x
1953:(
1950:F
1924:.
1921:)
1916:1
1913:+
1910:d
1906:h
1902:,
1896:,
1891:2
1887:h
1883:,
1878:1
1874:h
1870:,
1865:0
1861:h
1857:(
1844:h
1825:1
1822:+
1819:d
1815:h
1811:+
1808:x
1803:d
1799:h
1795:+
1789:+
1784:1
1778:d
1774:x
1768:2
1764:h
1760:+
1755:d
1751:x
1745:1
1741:h
1737:+
1732:1
1729:+
1726:d
1722:x
1716:0
1712:h
1708:=
1705:)
1702:1
1696:x
1693:(
1684:F
1666:f
1662:x
1660:(
1658:Δ
1655:F
1651:f
1647:x
1626:1
1623:+
1620:x
1617:8
1614:+
1609:2
1605:x
1598:+
1593:3
1589:x
1582:+
1577:4
1573:x
1552:8
1549:+
1546:x
1540:+
1535:2
1531:x
1527:6
1524:+
1519:3
1515:x
1504:f
1492:d
1488:f
1477:f
1473:f
1469:f
1461:0
1458:f
1454:i
1449:i
1445:f
1431:)
1426:1
1423:+
1420:d
1416:f
1412:,
1406:,
1401:2
1397:f
1393:,
1388:1
1384:f
1380:,
1375:0
1371:f
1367:(
1351:f
1262:S
1258:S
1242:)
1237:)
1234:S
1231:(
1227:l
1224:C
1218:(
1212:t
1209:s
1200:)
1195:)
1192:S
1189:(
1185:t
1182:s
1176:(
1170:l
1167:C
1146:S
1139:k
1136:L
1125:S
1102:S
1095:t
1092:s
1084:l
1081:C
1060:S
1053:t
1050:S
1035:S
1031:s
1027:K
1023:s
1019:s
1015:S
1001:S
994:t
991:s
980:S
967:S
963:S
949:S
942:l
939:C
928:S
924:K
910:S
903:l
900:C
889:S
878:K
874:S
870:K
863:.
841:.
819:.
790:.
778:)
775:x
772:(
756:x
738:x
716:K
693:|
687:K
681:|
674:x
653:|
647:K
641:|
618:K
579:K
553:|
547:K
541:|
499:k
495:k
446:k
430:K
412:k
396:K
384:k
365:k
349:K
325:K
312:k
297:.
283:2
256:1
229:K
219:2
211:,
206:1
186:.
172:K
148:K
112:K
89:.
65:n
20:)
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