1007:
343:
17:
3687:
3823:
3406:
1031:. The vertices of the barycentric subdivision correspond to the faces of all dimensions of the original polytope. Two vertices are adjacent in the barycentric subdivision when they correspond to two faces of different dimensions with the lower-dimensional face included in the higher-dimensional face. The
971:
One can generalize the subdivision for simplicial complexes whose simplices are not all contained in a single simplex of maximal dimension, i.e. simplicial complexes that do not correspond geometrically to one simplex. This can be done by effectuating the steps described above simultaneously for
57:
to the original maps (see also simplicial approximation). In general, such an assignment requires a refinement of the given complex, meaning, one replaces bigger simplices by a union of smaller simplices. A standard way to effectuate such a refinement is the barycentric subdivision. Moreover,
1210:
1359:
3927:
48:
The barycentric subdivision is an operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations: The substitution allows to assign combinatorial invariants as the
3693:
529:. To define the subdivision, we will consider a simplex as a simplicial complex that contains only one simplex of maximal dimension, namely the simplex itself. The barycentric subdivision of a simplex can be defined inductively by its dimension.
3682:{\displaystyle \cdots \to H_{n+1}(X)\,{\xrightarrow {\partial _{*}}}\,H_{n}(A\cap B)\,{\xrightarrow {(i_{*},j_{*})}}\,H_{n}(A)\oplus H_{n}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{n}(X)\,{\xrightarrow {\partial _{*}}}\,H_{n-1}(A\cap B)\to \cdots }
2001:
The barycentric subdivision can be applied on whole simplicial complexes as in the simplicial approximation theorem or it can be used to subdivide geometric simplices. Therefore it is crucial for statements in singular homology theory, see
1022:
of any dimension, producing another convex polytope of the same dimension. In this version of barycentric subdivision, it is not necessary for the polytope to form a simplicial complex: it can have faces that are not simplices. This is the
1702:-dimensional-standard-simplex. In an analogous way as described for simplicial homology groups, barycentric subdivision can be interpreted as an endomorphism of singular chain complexes. Here again, there exists a subdivision operator
1508:
2609:
by its iterated barycentric subdivision. The theorem plays an important role for certain statements in algebraic topology in order to reduce the behavior of continuous maps on those of simplicial maps, as for instance in
1861:
4429:
527:
4347:
1105:
4162:
710:
3138:
4493:
may appear such that their image is not part of the subsets mentioned in the theorem. Analogously those can be understood as a sum of images of smaller simplices obtained by the barycentric subdivision.
4172:
Excision can be used to determine relative homology groups. It allows in certain cases to forget about subsets of topological spaces for their homology groups and therefore simplifies their computation:
2323:
2280:
1771:
2948:
2779:
1964:
116:
1653:
1587:
911:
1100:
803:
4490:
3982:
3831:
1803:
53:
to the spaces. One can ask if there is an analogous way to replace the continuous functions defined on the topological spaces by functions that are linear on the simplices and which are
3309:
2559:
2164:
1282:
3209:
3339:
The Mayer–Vietoris sequence is often used to compute singular homology groups and gives rise to inductive arguments in topology. The related statement can be formulated as follows:
3818:{\displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdots \to H_{0}(A)\oplus H_{0}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{0}(X)\to 0.}
1400:
337:
281:
202:
145:
4279:
4226:
2377:
2695:
1373:
groups one requires a map from the homology-group of the original simplicial complex to the groups of the subdivided complex. Indeed it can be shown that for any subdivision
425:
3065:
2607:
2583:
2450:
2212:
2188:
2064:
2040:
1552:
1424:
308:
252:
226:
169:
1235:
1991:
1889:
1680:
737:
600:
3372:
4455:
1589:
and such that the maps induces endomorphisms of chain complexes. Moreover, the induced map is an isomorphism: Subdivision does not change the homology of the complex.
4002:
3947:
3164:
1909:
1528:
1275:
966:
553:
373:
2850:
946:
3236:
2118:
2091:
3003:
2426:
829:
626:
4246:
4194:
4082:
4062:
4042:
4022:
3392:
3329:
3085:
3023:
2974:
2820:
2800:
2663:
2643:
2510:
2490:
2470:
2397:
2347:
2232:
1700:
1614:
1255:
990:
757:
573:
3087:
has a fixed-point. In the proof this is first shown only for simplicial maps and then generalized for any continuous functions via the approximation theorem.
1054:. The degree-6, degree-4, and degree-8 vertices of the disdyakis dodecahedron correspond to the vertices, edges, and square facets of the cube, respectively.
1429:
1001:
2234:
induces a map between the geometric realizations of the complexes. Each point in a geometric complex lies in the inner of exactly one simplex, its
1212:. One way to measure the mesh of a geometric, simplicial complex is to take the maximal diameter of the simplices contained in the complex. Let
4044:. This can be fixed using the subdivision operator: By considering the images of such maps as the sum of images of smaller simplices, lying in
1809:
4352:
1205:{\displaystyle \operatorname {diam} (\Delta )=\operatorname {max} {\Bigl \{}\|a-b\|_{\mathbb {R} ^{n}}\;{\Big |}\;a,b\in \Delta {\Bigr \}}}
430:
4284:
4619:
Langer, Joel C.; Singer, David A. (2010), "Reflections on the lemniscate of
Bernoulli: the forty-eight faces of a mathematical gem",
4087:
631:
3093:
2288:
2245:
2625:
is a useful tool to find out whether a continuous function admits fixed-points. This data is computed as follows: Suppose that
1705:
2857:
346:
Barycentric subdivision of a 2-simplex. The colored points added on the right are the barycenters of the simplexes on the left
2701:
1914:
82:
739:. The barycentric subdivision is then defined to be the geometric simplicial complex whose maximal simplices of dimension
1361:. Therefore, by applying barycentric subdivision sufficiently often, the largest edge can be made as small as desired.
1619:
1557:
834:
4773:
4718:
4694:
4547:
4523:
3922:{\displaystyle i:A\cap B\hookrightarrow A,\;j:A\cap B\hookrightarrow B,\;k:A\hookrightarrow X,\;l:B\hookrightarrow X}
1070:
1369:
For some statements in homology-theory one wishes to replace simplicial complexes by a subdivision. On the level of
762:
4463:
3955:
1776:
1354:{\displaystyle \operatorname {diam} (\Delta ')\leq \left({\frac {n}{n+1}}\right)\;\operatorname {diam} (\Delta )}
3241:
3952:
For the construction of singular homology groups one considers continuous maps defined on the standard simplex
2518:
2123:
3169:
4778:
4768:
4763:
4251:
4199:
2352:
2003:
63:
532:
For points, i.e. simplices of dimension 0, the barycentric subdivision is defined as the point itself.
58:
barycentric subdivision induces maps on homology groups and is helpful for computational concerns, see
2668:
4562:
Ewald, G.; Shephard, G. C. (1974), "Stellar subdivisions of boundary complexes of convex polytopes",
1376:
1006:
378:
313:
257:
178:
121:
36:
is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool in
3028:
2588:
2564:
2431:
2193:
2169:
2045:
2021:
1533:
1405:
289:
233:
207:
150:
1969:
1867:
1658:
715:
578:
3345:
1051:
1011:
992:-th skeleton of the simplicial complex. It allows effectuating the subdivision more than once.
4434:
1215:
4601:
3987:
3932:
3143:
2512:
by defining it on each simplex; there, it always exists, because simplices are contractible.
1894:
1513:
1260:
951:
538:
358:
2825:
916:
4640:
4583:
3214:
2096:
2069:
50:
2979:
2402:
8:
1370:
808:
605:
4231:
4179:
4067:
4047:
4027:
4007:
3377:
3314:
3070:
3008:
2959:
2805:
2785:
2648:
2628:
2495:
2475:
2455:
2382:
2332:
2217:
1685:
1599:
1240:
1047:
975:
742:
558:
37:
33:
342:
4714:
4690:
4543:
4519:
1593:
1503:{\displaystyle \lambda _{n}:C_{n}({\mathcal {K}})\rightarrow C_{n}({\mathcal {K'}})}
4628:
4571:
4164:
induces an isomorphism on homology which is needed to compare the homology groups.
2007:
1032:
59:
1002:
Schläfli orthoscheme § Characteristic simplex of the general regular polytope
4654:
4636:
4605:
4579:
1036:
1024:
1019:
339:
equal as sets and as topological spaces, only the simplicial structure changes.
3398:
1277:
obtained by the barycentric subdivision. Then, the following estimation holds:
1028:
4689:(in German) (2. ĂĽberarbeitete ed.), Stuttgart: B.G. Teubner, p. 81,
4632:
2515:
The simplicial approximation theorem guarantees for every continuous function
4757:
972:
every simplex of maximal dimension. The induction will then be based on the
2665:
are topological spaces that admit finite triangulations. A continuous map
1856:{\displaystyle \sum \varepsilon _{B_{\Delta }}\sigma \vert _{B_{\Delta }}}
16:
3090:
Now, Brouwer's fixpoint theorem is a special case of this statement. Let
2561:
the existence of a simplicial approximation at least after refinement of
4575:
4424:{\displaystyle H_{k}(X\setminus Z,A\setminus Z)\rightarrow H_{k}(X,A)}
3762:
3627:
3574:
3493:
3447:
2782:
between its simplicial homology groups with coefficients in a field
4609:
See p. 22, where the omnitruncation is described as a "flag graph".
1035:
of the barycentric subdivision are simplices, corresponding to the
522:{\displaystyle b_{\Delta }={\frac {1}{n+1}}(p_{0}+p_{1}+...+p_{n})}
54:
4342:{\displaystyle i:(X\setminus Z,A\setminus Z)\hookrightarrow (X,A)}
29:
2452:. If such an approximation exists, one can construct a homotopy
1426:
there is a unique sequence of maps between the homology groups
4157:{\displaystyle C_{n}(A)\oplus C_{n}(B)\hookrightarrow C_{n}(X)}
1018:
The operation of barycentric subdivision can be applied to any
995:
705:{\displaystyle \Delta _{i,1},\;\Delta _{i,2}...,\Delta _{i,n!}}
3133:{\displaystyle f:\mathbb {D} ^{n}\rightarrow \mathbb {D} ^{n}}
1993:. This map also induces an automorphism of chain complexes.
4713:(in German), Berlin/ Heidelberg/ New York, pp. 254 f,
4684:
1043:
2318:{\displaystyle g:{\mathcal {K}}\rightarrow {\mathcal {L}}}
2275:{\displaystyle f:{\mathcal {K}}\rightarrow {\mathcal {L}}}
1766:{\displaystyle \lambda _{n}:C_{n}(X)\rightarrow C_{n}(X)}
2943:{\displaystyle L_{K}(f)=\sum _{i}(-1)^{i}tr_{i}(f)\in K}
4666:
350:
4738:
4726:
2774:{\displaystyle f_{i}:H_{i}(X,K)\rightarrow H_{i}(Y,K)}
1959:{\displaystyle \varepsilon _{B_{\Delta }}\in \{1,-1\}}
1257:- dimensional simplex that comes from the covering of
628:
are already divided. Therefore, there exist simplices
111:{\displaystyle {\mathcal {S}}\subset \mathbb {R} ^{n}}
74:
20:
Iterate 1 to 4 barycentric subdivisions of 2-simplices
4466:
4437:
4355:
4287:
4254:
4234:
4202:
4182:
4090:
4070:
4050:
4030:
4010:
3990:
3958:
3935:
3834:
3696:
3409:
3380:
3348:
3317:
3244:
3217:
3172:
3146:
3096:
3073:
3031:
3011:
2982:
2962:
2860:
2828:
2808:
2788:
2704:
2671:
2651:
2631:
2591:
2567:
2521:
2498:
2478:
2458:
2434:
2405:
2385:
2355:
2335:
2291:
2248:
2220:
2196:
2172:
2126:
2099:
2072:
2048:
2024:
1972:
1917:
1897:
1870:
1812:
1779:
1708:
1688:
1661:
1622:
1602:
1560:
1536:
1516:
1432:
1408:
1379:
1285:
1263:
1243:
1218:
1108:
1073:
978:
954:
919:
837:
811:
765:
745:
718:
634:
608:
581:
561:
541:
433:
381:
361:
316:
292:
260:
236:
210:
181:
153:
124:
85:
3984:. An obstacle in the proof of the theorem are maps
4600:(Doctoral dissertation), Northeastern University,
4484:
4449:
4423:
4341:
4273:
4240:
4220:
4188:
4156:
4076:
4056:
4036:
4016:
3996:
3976:
3941:
3921:
3817:
3681:
3386:
3366:
3323:
3303:
3230:
3203:
3158:
3132:
3079:
3059:
3017:
2997:
2968:
2942:
2844:
2814:
2794:
2773:
2689:
2657:
2637:
2616:
2601:
2577:
2553:
2504:
2484:
2464:
2444:
2420:
2391:
2371:
2341:
2317:
2274:
2226:
2206:
2182:
2158:
2112:
2085:
2058:
2034:
1985:
1958:
1903:
1883:
1855:
1797:
1765:
1694:
1674:
1647:
1608:
1581:
1546:
1522:
1502:
1418:
1394:
1353:
1269:
1249:
1229:
1204:
1094:
984:
960:
940:
905:
823:
797:
751:
731:
704:
620:
594:
567:
547:
521:
419:
367:
331:
302:
275:
246:
220:
196:
163:
139:
110:
4598:Convex Polytopes and Tilings with Few Flag Orbits
1197:
1174:
1135:
427:, the barycenter is defined to be the point
4755:
4542:(in German), Menlo Park, Calif., pp. 85 f,
1648:{\displaystyle \sigma :\Delta ^{n}\rightarrow X}
1582:{\displaystyle \lambda (\Delta )\subset \Delta }
906:{\displaystyle i\in {0,...,n},\;j\in {1,...,n!}}
2214:. By affin-linear extension on the simplices,
1095:{\displaystyle \Delta \subset \mathbb {R} ^{n}}
1042:For instance, the barycentric subdivision of a
4561:
4518:(in German), Menlo Park, Calif., p. 96,
3005:, this number is the Euler characteristic of
798:{\displaystyle \Delta _{i,j}\cup b_{\Delta }}
118:be a geometric simplicial complex. A complex
4618:
4537:
4513:
4485:{\displaystyle \sigma :\Delta \rightarrow X}
3977:{\displaystyle \sigma :\Delta \rightarrow X}
3828:where we consider singular homology groups,
3025:. The fixpoint theorem states that whenever
2852:can be determined and their alternating sum
2066:be abstract simplicial complexes above sets
1953:
1938:
1837:
1798:{\displaystyle \sigma :\Delta \rightarrow X}
1153:
1140:
996:Barycentric subdivision of a convex polytope
4708:
2013:
4004:such that their image is nor contained in
3949:denotes the direct sum of abelian groups.
3903:
3884:
3859:
3334:
3304:{\displaystyle L_{K}(f)=tr_{0}(f)=1\neq 0}
1335:
1179:
1171:
870:
654:
3789:
3756:
3641:
3621:
3601:
3568:
3526:
3487:
3461:
3441:
3188:
3140:is an endomorphism of the unit-ball. For
3120:
3105:
1159:
1082:
98:
4709:Bredon, Glen E., Springer Verlag (ed.),
2554:{\displaystyle f:V_{K}\rightarrow V_{L}}
2159:{\displaystyle f:V_{K}\rightarrow V_{L}}
1005:
341:
15:
4744:
4732:
4672:
4652:
3374:an open cover of the topological space
3204:{\displaystyle H_{k}(\mathbb {D} ^{n})}
1014:, the barycentric subdivision of a cube
28:is a standard way to subdivide a given
4756:
4595:
1864:where the sum runs over all simplices
4509:
4507:
351:Barycentric subdivision of a simplex
32:into smaller ones. Its extension on
4274:{\displaystyle Z\subset A^{\circ }}
4221:{\displaystyle Z\subset A\subset X}
2372:{\displaystyle x\in {\mathcal {K}}}
1616:one considers continuous functions
75:Subdivision of simplicial complexes
13:
4473:
4459:Again, in singular homology, maps
3965:
3629:
3449:
2594:
2570:
2437:
2364:
2310:
2300:
2267:
2257:
2199:
2175:
2051:
2027:
1978:
1928:
1898:
1876:
1846:
1826:
1786:
1663:
1630:
1576:
1567:
1539:
1517:
1488:
1461:
1411:
1383:
1345:
1296:
1264:
1220:
1192:
1118:
1074:
955:
790:
767:
720:
684:
656:
636:
583:
542:
439:
362:
320:
295:
264:
254:is a finite union of simplices of
239:
213:
185:
156:
128:
88:
14:
4790:
4685:Ralph Stöcker, Heiner Zieschang,
4504:
4384:
4372:
4312:
4300:
2120:. A simplicial map is a function
4084:one can show that the inclusion
2802:. These are linear maps between
2690:{\displaystyle f:X\rightarrow Y}
2612:Lefschetz's fixed-point theorem.
1911:by barycentric subdivision, and
4196:be a topological space and let
3708:
3707:
3706:
3705:
3704:
3703:
3702:
3701:
3700:
3699:
3698:
3697:
2822:- vectorspaces, so their trace
2617:Lefschetz's fixed-point theorem
1996:
1891:that appear in the covering of
1402:of a finite simplicial complex
1395:{\displaystyle {\mathcal {K'}}}
420:{\displaystyle p_{0},...,p_{n}}
332:{\displaystyle {\mathcal {S'}}}
276:{\displaystyle {\mathcal {S'}}}
197:{\displaystyle {\mathcal {S'}}}
147:is said to be a subdivision of
140:{\displaystyle {\mathcal {S'}}}
4702:
4678:
4646:
4612:
4589:
4555:
4540:Elements of algebraic topology
4531:
4516:Elements of algebraic topology
4476:
4418:
4406:
4393:
4390:
4366:
4336:
4324:
4321:
4318:
4294:
4151:
4145:
4132:
4129:
4123:
4107:
4101:
3968:
3913:
3894:
3875:
3850:
3809:
3806:
3800:
3753:
3747:
3731:
3725:
3712:
3673:
3670:
3658:
3618:
3612:
3565:
3559:
3543:
3537:
3520:
3494:
3484:
3472:
3438:
3432:
3413:
3286:
3280:
3261:
3255:
3198:
3183:
3115:
3060:{\displaystyle L_{K}(f)\neq 0}
3048:
3042:
2931:
2925:
2903:
2893:
2877:
2871:
2768:
2756:
2743:
2740:
2728:
2681:
2602:{\displaystyle {\mathcal {K}}}
2578:{\displaystyle {\mathcal {K}}}
2538:
2445:{\displaystyle {\mathcal {L}}}
2415:
2409:
2305:
2262:
2207:{\displaystyle {\mathcal {L}}}
2183:{\displaystyle {\mathcal {K}}}
2143:
2059:{\displaystyle {\mathcal {L}}}
2035:{\displaystyle {\mathcal {K}}}
1789:
1760:
1754:
1741:
1738:
1732:
1639:
1596:groups of a topological space
1570:
1564:
1547:{\displaystyle {\mathcal {K}}}
1497:
1482:
1469:
1466:
1456:
1419:{\displaystyle {\mathcal {K}}}
1348:
1342:
1303:
1292:
1121:
1115:
932:
920:
516:
465:
303:{\displaystyle {\mathcal {S}}}
247:{\displaystyle {\mathcal {S}}}
221:{\displaystyle {\mathcal {S}}}
164:{\displaystyle {\mathcal {S}}}
1:
4497:
1057:
204:is contained in a simplex of
69:
43:
4621:Milan Journal of Mathematics
2585:, for instance by replacing
286:These conditions imply that
7:
4167:
3238:is always the identity, so
2166:which maps each simplex in
1986:{\displaystyle B_{\Delta }}
1884:{\displaystyle B_{\Delta }}
1675:{\displaystyle \Delta ^{n}}
1364:
759:are each a convex hulls of
732:{\displaystyle \Delta _{i}}
595:{\displaystyle \Delta _{i}}
535:Suppose then for a simplex
10:
4795:
1039:of the original polytope.
999:
4633:10.1007/s00032-010-0124-5
4596:Matteo, Nicholas (2015),
3367:{\displaystyle X=A\cup B}
4774:Triangulation (geometry)
4450:{\displaystyle k\geq 0.}
4349:induces an isomorphism
3166:all its homology groups
2327:simplicial approximation
2014:Simplicial approximation
1805:to a linear combination
1230:{\displaystyle \Delta '}
4653:Hatcher, Allen (2001),
3997:{\displaystyle \sigma }
3942:{\displaystyle \oplus }
3335:Mayer–Vietoris sequence
3159:{\displaystyle k\geq 1}
2004:Mayer–Vietoris sequence
1904:{\displaystyle \Delta }
1523:{\displaystyle \Delta }
1270:{\displaystyle \Delta }
1062:
961:{\displaystyle \Delta }
548:{\displaystyle \Delta }
368:{\displaystyle \Delta }
64:Mayer–Vietoris sequence
26:barycentric subdivision
4687:Algebraische Topologie
4486:
4451:
4425:
4343:
4275:
4242:
4222:
4190:
4158:
4078:
4058:
4038:
4018:
3998:
3978:
3943:
3923:
3819:
3683:
3388:
3368:
3325:
3305:
3232:
3205:
3160:
3134:
3081:
3061:
3019:
2999:
2970:
2944:
2846:
2845:{\displaystyle tr_{i}}
2816:
2796:
2775:
2697:induces homomorphisms
2691:
2659:
2639:
2603:
2579:
2555:
2506:
2486:
2466:
2446:
2422:
2393:
2373:
2343:
2319:
2276:
2228:
2208:
2184:
2160:
2114:
2087:
2060:
2036:
1987:
1960:
1905:
1885:
1857:
1799:
1767:
1696:
1676:
1649:
1610:
1583:
1548:
1524:
1504:
1420:
1396:
1355:
1271:
1251:
1231:
1206:
1096:
1052:disdyakis dodecahedron
1015:
1012:disdyakis dodecahedron
986:
962:
942:
941:{\displaystyle (n+1)!}
907:
825:
799:
753:
733:
706:
622:
596:
569:
549:
523:
421:
369:
347:
333:
304:
277:
248:
222:
198:
165:
141:
112:
21:
4711:Topology and Geometry
4564:Mathematische Annalen
4487:
4452:
4426:
4344:
4281:. Then the inclusion
4276:
4243:
4223:
4191:
4159:
4079:
4059:
4039:
4019:
3999:
3979:
3944:
3924:
3820:
3684:
3389:
3369:
3326:
3306:
3233:
3231:{\displaystyle f_{0}}
3206:
3161:
3135:
3082:
3062:
3020:
3000:
2971:
2945:
2847:
2817:
2797:
2776:
2692:
2660:
2640:
2604:
2580:
2556:
2507:
2487:
2467:
2447:
2423:
2394:
2374:
2344:
2320:
2277:
2229:
2209:
2185:
2161:
2115:
2113:{\displaystyle V_{L}}
2088:
2086:{\displaystyle V_{K}}
2061:
2037:
1988:
1961:
1906:
1886:
1858:
1800:
1768:
1697:
1677:
1650:
1611:
1584:
1549:
1525:
1505:
1421:
1397:
1356:
1272:
1252:
1232:
1207:
1102:a simplex and define
1097:
1009:
987:
963:
943:
908:
826:
800:
754:
734:
707:
623:
597:
570:
550:
524:
422:
370:
345:
334:
305:
278:
249:
223:
199:
166:
142:
113:
19:
4464:
4435:
4353:
4285:
4252:
4248:is closed such that
4232:
4200:
4180:
4088:
4068:
4048:
4028:
4008:
3988:
3956:
3933:
3832:
3694:
3407:
3378:
3346:
3315:
3242:
3215:
3170:
3144:
3094:
3071:
3029:
3009:
2998:{\displaystyle f=id}
2980:
2960:
2858:
2826:
2806:
2786:
2702:
2669:
2649:
2629:
2589:
2565:
2519:
2496:
2476:
2456:
2432:
2421:{\displaystyle f(x)}
2403:
2399:onto the support of
2383:
2353:
2349:if and only if each
2333:
2289:
2246:
2218:
2194:
2170:
2124:
2097:
2070:
2046:
2022:
1970:
1915:
1895:
1868:
1810:
1777:
1706:
1686:
1659:
1620:
1600:
1558:
1534:
1514:
1430:
1406:
1377:
1283:
1261:
1241:
1216:
1106:
1071:
976:
952:
917:
835:
809:
763:
743:
716:
632:
606:
579:
559:
539:
431:
379:
359:
314:
290:
258:
234:
208:
179:
151:
122:
83:
51:Euler characteristic
34:simplicial complexes
24:In mathematics, the
4779:Simplicial homology
3929:are embeddings and
3786:
3638:
3598:
3523:
3458:
1510:such that for each
1371:simplicial homology
948:simplices covering
913:, so there will be
824:{\displaystyle i,j}
621:{\displaystyle n-1}
4769:Geometric topology
4764:Algebraic topology
4656:Algebraic Topology
4576:10.1007/BF01344542
4538:James R. Munkres,
4514:James R. Munkres,
4482:
4447:
4421:
4339:
4271:
4238:
4228:be subsets, where
4218:
4186:
4154:
4074:
4054:
4034:
4014:
3994:
3974:
3939:
3919:
3815:
3679:
3384:
3364:
3321:
3301:
3228:
3201:
3156:
3130:
3077:
3057:
3015:
2995:
2966:
2940:
2892:
2842:
2812:
2792:
2771:
2687:
2655:
2635:
2599:
2575:
2551:
2502:
2482:
2462:
2442:
2418:
2389:
2369:
2339:
2315:
2272:
2224:
2204:
2190:onto a simplex in
2180:
2156:
2110:
2083:
2056:
2032:
1983:
1956:
1901:
1881:
1853:
1795:
1763:
1692:
1672:
1645:
1606:
1579:
1554:the maps fulfills
1544:
1520:
1500:
1416:
1392:
1351:
1267:
1247:
1227:
1202:
1092:
1048:regular octahedron
1016:
982:
958:
938:
903:
821:
795:
749:
729:
702:
618:
592:
565:
545:
519:
417:
375:spanned by points
365:
348:
329:
300:
273:
244:
218:
194:
161:
137:
108:
38:algebraic topology
22:
4675:, pp. 122 f.
4241:{\displaystyle Z}
4189:{\displaystyle X}
4077:{\displaystyle B}
4057:{\displaystyle A}
4037:{\displaystyle B}
4017:{\displaystyle A}
3787:
3639:
3599:
3524:
3459:
3387:{\displaystyle X}
3324:{\displaystyle f}
3080:{\displaystyle f}
3018:{\displaystyle K}
2969:{\displaystyle f}
2883:
2815:{\displaystyle K}
2795:{\displaystyle K}
2658:{\displaystyle Y}
2638:{\displaystyle X}
2505:{\displaystyle g}
2485:{\displaystyle f}
2465:{\displaystyle H}
2392:{\displaystyle g}
2342:{\displaystyle f}
2285:A simplicial map
2227:{\displaystyle f}
1695:{\displaystyle n}
1609:{\displaystyle X}
1594:singular homology
1329:
1250:{\displaystyle n}
985:{\displaystyle n}
752:{\displaystyle n}
568:{\displaystyle n}
463:
4786:
4748:
4742:
4736:
4730:
4724:
4723:
4706:
4700:
4699:
4682:
4676:
4670:
4664:
4663:
4661:
4650:
4644:
4643:
4616:
4610:
4608:
4593:
4587:
4586:
4559:
4553:
4552:
4535:
4529:
4528:
4511:
4491:
4489:
4488:
4483:
4456:
4454:
4453:
4448:
4430:
4428:
4427:
4422:
4405:
4404:
4365:
4364:
4348:
4346:
4345:
4340:
4280:
4278:
4277:
4272:
4270:
4269:
4247:
4245:
4244:
4239:
4227:
4225:
4224:
4219:
4195:
4193:
4192:
4187:
4163:
4161:
4160:
4155:
4144:
4143:
4122:
4121:
4100:
4099:
4083:
4081:
4080:
4075:
4063:
4061:
4060:
4055:
4043:
4041:
4040:
4035:
4023:
4021:
4020:
4015:
4003:
4001:
4000:
3995:
3983:
3981:
3980:
3975:
3948:
3946:
3945:
3940:
3928:
3926:
3925:
3920:
3824:
3822:
3821:
3816:
3799:
3798:
3788:
3785:
3784:
3772:
3771:
3758:
3746:
3745:
3724:
3723:
3688:
3686:
3685:
3680:
3657:
3656:
3640:
3637:
3636:
3623:
3611:
3610:
3600:
3597:
3596:
3584:
3583:
3570:
3558:
3557:
3536:
3535:
3525:
3519:
3518:
3506:
3505:
3489:
3471:
3470:
3460:
3457:
3456:
3443:
3431:
3430:
3393:
3391:
3390:
3385:
3373:
3371:
3370:
3365:
3331:has a fixpoint.
3330:
3328:
3327:
3322:
3310:
3308:
3307:
3302:
3279:
3278:
3254:
3253:
3237:
3235:
3234:
3229:
3227:
3226:
3210:
3208:
3207:
3202:
3197:
3196:
3191:
3182:
3181:
3165:
3163:
3162:
3157:
3139:
3137:
3136:
3131:
3129:
3128:
3123:
3114:
3113:
3108:
3086:
3084:
3083:
3078:
3066:
3064:
3063:
3058:
3041:
3040:
3024:
3022:
3021:
3016:
3004:
3002:
3001:
2996:
2975:
2973:
2972:
2967:
2954:Lefschetz number
2949:
2947:
2946:
2941:
2924:
2923:
2911:
2910:
2891:
2870:
2869:
2851:
2849:
2848:
2843:
2841:
2840:
2821:
2819:
2818:
2813:
2801:
2799:
2798:
2793:
2780:
2778:
2777:
2772:
2755:
2754:
2727:
2726:
2714:
2713:
2696:
2694:
2693:
2688:
2664:
2662:
2661:
2656:
2644:
2642:
2641:
2636:
2623:Lefschetz number
2608:
2606:
2605:
2600:
2598:
2597:
2584:
2582:
2581:
2576:
2574:
2573:
2560:
2558:
2557:
2552:
2550:
2549:
2537:
2536:
2511:
2509:
2508:
2503:
2491:
2489:
2488:
2483:
2471:
2469:
2468:
2463:
2451:
2449:
2448:
2443:
2441:
2440:
2427:
2425:
2424:
2419:
2398:
2396:
2395:
2390:
2378:
2376:
2375:
2370:
2368:
2367:
2348:
2346:
2345:
2340:
2325:is said to be a
2324:
2322:
2321:
2316:
2314:
2313:
2304:
2303:
2281:
2279:
2278:
2273:
2271:
2270:
2261:
2260:
2233:
2231:
2230:
2225:
2213:
2211:
2210:
2205:
2203:
2202:
2189:
2187:
2186:
2181:
2179:
2178:
2165:
2163:
2162:
2157:
2155:
2154:
2142:
2141:
2119:
2117:
2116:
2111:
2109:
2108:
2092:
2090:
2089:
2084:
2082:
2081:
2065:
2063:
2062:
2057:
2055:
2054:
2041:
2039:
2038:
2033:
2031:
2030:
1992:
1990:
1989:
1984:
1982:
1981:
1966:for all of such
1965:
1963:
1962:
1957:
1934:
1933:
1932:
1931:
1910:
1908:
1907:
1902:
1890:
1888:
1887:
1882:
1880:
1879:
1862:
1860:
1859:
1854:
1852:
1851:
1850:
1849:
1832:
1831:
1830:
1829:
1804:
1802:
1801:
1796:
1773:sending a chain
1772:
1770:
1769:
1764:
1753:
1752:
1731:
1730:
1718:
1717:
1701:
1699:
1698:
1693:
1681:
1679:
1678:
1673:
1671:
1670:
1654:
1652:
1651:
1646:
1638:
1637:
1615:
1613:
1612:
1607:
1588:
1586:
1585:
1580:
1553:
1551:
1550:
1545:
1543:
1542:
1529:
1527:
1526:
1521:
1509:
1507:
1506:
1501:
1496:
1495:
1494:
1481:
1480:
1465:
1464:
1455:
1454:
1442:
1441:
1425:
1423:
1422:
1417:
1415:
1414:
1401:
1399:
1398:
1393:
1391:
1390:
1389:
1360:
1358:
1357:
1352:
1334:
1330:
1328:
1314:
1302:
1276:
1274:
1273:
1268:
1256:
1254:
1253:
1248:
1236:
1234:
1233:
1228:
1226:
1211:
1209:
1208:
1203:
1201:
1200:
1178:
1177:
1170:
1169:
1168:
1167:
1162:
1139:
1138:
1101:
1099:
1098:
1093:
1091:
1090:
1085:
991:
989:
988:
983:
967:
965:
964:
959:
947:
945:
944:
939:
912:
910:
909:
904:
902:
866:
830:
828:
827:
822:
804:
802:
801:
796:
794:
793:
781:
780:
758:
756:
755:
750:
738:
736:
735:
730:
728:
727:
711:
709:
708:
703:
701:
700:
670:
669:
650:
649:
627:
625:
624:
619:
601:
599:
598:
593:
591:
590:
575:that its faces
574:
572:
571:
566:
554:
552:
551:
546:
528:
526:
525:
520:
515:
514:
490:
489:
477:
476:
464:
462:
448:
443:
442:
426:
424:
423:
418:
416:
415:
391:
390:
374:
372:
371:
366:
338:
336:
335:
330:
328:
327:
326:
309:
307:
306:
301:
299:
298:
282:
280:
279:
274:
272:
271:
270:
253:
251:
250:
245:
243:
242:
230:each simplex of
227:
225:
224:
219:
217:
216:
203:
201:
200:
195:
193:
192:
191:
175:each simplex of
170:
168:
167:
162:
160:
159:
146:
144:
143:
138:
136:
135:
134:
117:
115:
114:
109:
107:
106:
101:
92:
91:
4794:
4793:
4789:
4788:
4787:
4785:
4784:
4783:
4754:
4753:
4752:
4751:
4743:
4739:
4731:
4727:
4721:
4707:
4703:
4697:
4683:
4679:
4671:
4667:
4659:
4651:
4647:
4617:
4613:
4594:
4590:
4560:
4556:
4550:
4536:
4532:
4526:
4512:
4505:
4500:
4465:
4462:
4461:
4436:
4433:
4432:
4400:
4396:
4360:
4356:
4354:
4351:
4350:
4286:
4283:
4282:
4265:
4261:
4253:
4250:
4249:
4233:
4230:
4229:
4201:
4198:
4197:
4181:
4178:
4177:
4170:
4139:
4135:
4117:
4113:
4095:
4091:
4089:
4086:
4085:
4069:
4066:
4065:
4049:
4046:
4045:
4029:
4026:
4025:
4009:
4006:
4005:
3989:
3986:
3985:
3957:
3954:
3953:
3934:
3931:
3930:
3833:
3830:
3829:
3794:
3790:
3780:
3776:
3767:
3763:
3757:
3741:
3737:
3719:
3715:
3695:
3692:
3691:
3646:
3642:
3632:
3628:
3622:
3606:
3602:
3592:
3588:
3579:
3575:
3569:
3553:
3549:
3531:
3527:
3514:
3510:
3501:
3497:
3488:
3466:
3462:
3452:
3448:
3442:
3420:
3416:
3408:
3405:
3404:
3379:
3376:
3375:
3347:
3344:
3343:
3337:
3316:
3313:
3312:
3274:
3270:
3249:
3245:
3243:
3240:
3239:
3222:
3218:
3216:
3213:
3212:
3192:
3187:
3186:
3177:
3173:
3171:
3168:
3167:
3145:
3142:
3141:
3124:
3119:
3118:
3109:
3104:
3103:
3095:
3092:
3091:
3072:
3069:
3068:
3036:
3032:
3030:
3027:
3026:
3010:
3007:
3006:
2981:
2978:
2977:
2961:
2958:
2957:
2919:
2915:
2906:
2902:
2887:
2865:
2861:
2859:
2856:
2855:
2836:
2832:
2827:
2824:
2823:
2807:
2804:
2803:
2787:
2784:
2783:
2750:
2746:
2722:
2718:
2709:
2705:
2703:
2700:
2699:
2670:
2667:
2666:
2650:
2647:
2646:
2630:
2627:
2626:
2619:
2593:
2592:
2590:
2587:
2586:
2569:
2568:
2566:
2563:
2562:
2545:
2541:
2532:
2528:
2520:
2517:
2516:
2497:
2494:
2493:
2477:
2474:
2473:
2457:
2454:
2453:
2436:
2435:
2433:
2430:
2429:
2404:
2401:
2400:
2384:
2381:
2380:
2363:
2362:
2354:
2351:
2350:
2334:
2331:
2330:
2309:
2308:
2299:
2298:
2290:
2287:
2286:
2266:
2265:
2256:
2255:
2247:
2244:
2243:
2238:Consider now a
2219:
2216:
2215:
2198:
2197:
2195:
2192:
2191:
2174:
2173:
2171:
2168:
2167:
2150:
2146:
2137:
2133:
2125:
2122:
2121:
2104:
2100:
2098:
2095:
2094:
2077:
2073:
2071:
2068:
2067:
2050:
2049:
2047:
2044:
2043:
2026:
2025:
2023:
2020:
2019:
2016:
1999:
1977:
1973:
1971:
1968:
1967:
1927:
1923:
1922:
1918:
1916:
1913:
1912:
1896:
1893:
1892:
1875:
1871:
1869:
1866:
1865:
1845:
1841:
1840:
1836:
1825:
1821:
1820:
1816:
1811:
1808:
1807:
1778:
1775:
1774:
1748:
1744:
1726:
1722:
1713:
1709:
1707:
1704:
1703:
1687:
1684:
1683:
1666:
1662:
1660:
1657:
1656:
1633:
1629:
1621:
1618:
1617:
1601:
1598:
1597:
1592:To compute the
1559:
1556:
1555:
1538:
1537:
1535:
1532:
1531:
1515:
1512:
1511:
1487:
1486:
1485:
1476:
1472:
1460:
1459:
1450:
1446:
1437:
1433:
1431:
1428:
1427:
1410:
1409:
1407:
1404:
1403:
1382:
1381:
1380:
1378:
1375:
1374:
1367:
1318:
1313:
1309:
1295:
1284:
1281:
1280:
1262:
1259:
1258:
1242:
1239:
1238:
1219:
1217:
1214:
1213:
1196:
1195:
1173:
1172:
1163:
1158:
1157:
1156:
1152:
1134:
1133:
1107:
1104:
1103:
1086:
1081:
1080:
1072:
1069:
1068:
1065:
1060:
1020:convex polytope
1004:
998:
977:
974:
973:
953:
950:
949:
918:
915:
914:
877:
844:
836:
833:
832:
810:
807:
806:
789:
785:
770:
766:
764:
761:
760:
744:
741:
740:
723:
719:
717:
714:
713:
687:
683:
659:
655:
639:
635:
633:
630:
629:
607:
604:
603:
586:
582:
580:
577:
576:
560:
557:
556:
540:
537:
536:
510:
506:
485:
481:
472:
468:
452:
447:
438:
434:
432:
429:
428:
411:
407:
386:
382:
380:
377:
376:
360:
357:
356:
353:
319:
318:
317:
315:
312:
311:
294:
293:
291:
288:
287:
263:
262:
261:
259:
256:
255:
238:
237:
235:
232:
231:
212:
211:
209:
206:
205:
184:
183:
182:
180:
177:
176:
155:
154:
152:
149:
148:
127:
126:
125:
123:
120:
119:
102:
97:
96:
87:
86:
84:
81:
80:
77:
72:
46:
12:
11:
5:
4792:
4782:
4781:
4776:
4771:
4766:
4750:
4749:
4747:, p. 119.
4745:Hatcher (2001)
4737:
4735:, p. 149.
4733:Hatcher (2001)
4725:
4719:
4701:
4695:
4677:
4673:Hatcher (2001)
4665:
4645:
4627:(2): 643–682,
4611:
4588:
4554:
4548:
4530:
4524:
4502:
4501:
4499:
4496:
4481:
4478:
4475:
4472:
4469:
4446:
4443:
4440:
4420:
4417:
4414:
4411:
4408:
4403:
4399:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4363:
4359:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4268:
4264:
4260:
4257:
4237:
4217:
4214:
4211:
4208:
4205:
4185:
4169:
4166:
4153:
4150:
4147:
4142:
4138:
4134:
4131:
4128:
4125:
4120:
4116:
4112:
4109:
4106:
4103:
4098:
4094:
4073:
4053:
4033:
4013:
3993:
3973:
3970:
3967:
3964:
3961:
3938:
3918:
3915:
3912:
3909:
3906:
3902:
3899:
3896:
3893:
3890:
3887:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3858:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3826:
3825:
3814:
3811:
3808:
3805:
3802:
3797:
3793:
3783:
3779:
3775:
3770:
3766:
3761:
3755:
3752:
3749:
3744:
3740:
3736:
3733:
3730:
3727:
3722:
3718:
3714:
3711:
3689:
3678:
3675:
3672:
3669:
3666:
3663:
3660:
3655:
3652:
3649:
3645:
3635:
3631:
3626:
3620:
3617:
3614:
3609:
3605:
3595:
3591:
3587:
3582:
3578:
3573:
3567:
3564:
3561:
3556:
3552:
3548:
3545:
3542:
3539:
3534:
3530:
3522:
3517:
3513:
3509:
3504:
3500:
3496:
3492:
3486:
3483:
3480:
3477:
3474:
3469:
3465:
3455:
3451:
3446:
3440:
3437:
3434:
3429:
3426:
3423:
3419:
3415:
3412:
3399:exact sequence
3383:
3363:
3360:
3357:
3354:
3351:
3336:
3333:
3320:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3277:
3273:
3269:
3266:
3263:
3260:
3257:
3252:
3248:
3225:
3221:
3200:
3195:
3190:
3185:
3180:
3176:
3155:
3152:
3149:
3127:
3122:
3117:
3112:
3107:
3102:
3099:
3076:
3056:
3053:
3050:
3047:
3044:
3039:
3035:
3014:
2994:
2991:
2988:
2985:
2965:
2952:is called the
2939:
2936:
2933:
2930:
2927:
2922:
2918:
2914:
2909:
2905:
2901:
2898:
2895:
2890:
2886:
2882:
2879:
2876:
2873:
2868:
2864:
2839:
2835:
2831:
2811:
2791:
2770:
2767:
2764:
2761:
2758:
2753:
2749:
2745:
2742:
2739:
2736:
2733:
2730:
2725:
2721:
2717:
2712:
2708:
2686:
2683:
2680:
2677:
2674:
2654:
2634:
2618:
2615:
2596:
2572:
2548:
2544:
2540:
2535:
2531:
2527:
2524:
2501:
2481:
2461:
2439:
2417:
2414:
2411:
2408:
2388:
2366:
2361:
2358:
2338:
2312:
2307:
2302:
2297:
2294:
2269:
2264:
2259:
2254:
2251:
2223:
2201:
2177:
2153:
2149:
2145:
2140:
2136:
2132:
2129:
2107:
2103:
2080:
2076:
2053:
2029:
2015:
2012:
1998:
1995:
1980:
1976:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1930:
1926:
1921:
1900:
1878:
1874:
1848:
1844:
1839:
1835:
1828:
1824:
1819:
1815:
1794:
1791:
1788:
1785:
1782:
1762:
1759:
1756:
1751:
1747:
1743:
1740:
1737:
1734:
1729:
1725:
1721:
1716:
1712:
1691:
1669:
1665:
1644:
1641:
1636:
1632:
1628:
1625:
1605:
1578:
1575:
1572:
1569:
1566:
1563:
1541:
1519:
1499:
1493:
1490:
1484:
1479:
1475:
1471:
1468:
1463:
1458:
1453:
1449:
1445:
1440:
1436:
1413:
1388:
1385:
1366:
1363:
1350:
1347:
1344:
1341:
1338:
1333:
1327:
1324:
1321:
1317:
1312:
1308:
1305:
1301:
1298:
1294:
1291:
1288:
1266:
1246:
1225:
1222:
1199:
1194:
1191:
1188:
1185:
1182:
1176:
1166:
1161:
1155:
1151:
1148:
1145:
1142:
1137:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1089:
1084:
1079:
1076:
1064:
1061:
1059:
1056:
1029:omnitruncation
1025:dual operation
997:
994:
981:
957:
937:
934:
931:
928:
925:
922:
901:
898:
895:
892:
889:
886:
883:
880:
876:
873:
869:
865:
862:
859:
856:
853:
850:
847:
843:
840:
820:
817:
814:
792:
788:
784:
779:
776:
773:
769:
748:
726:
722:
699:
696:
693:
690:
686:
682:
679:
676:
673:
668:
665:
662:
658:
653:
648:
645:
642:
638:
617:
614:
611:
589:
585:
564:
544:
518:
513:
509:
505:
502:
499:
496:
493:
488:
484:
480:
475:
471:
467:
461:
458:
455:
451:
446:
441:
437:
414:
410:
406:
403:
400:
397:
394:
389:
385:
364:
355:For a simplex
352:
349:
325:
322:
297:
284:
283:
269:
266:
241:
228:
215:
190:
187:
158:
133:
130:
105:
100:
95:
90:
76:
73:
71:
68:
45:
42:
9:
6:
4:
3:
2:
4791:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4761:
4759:
4746:
4741:
4734:
4729:
4722:
4720:3-540-97926-3
4716:
4712:
4705:
4698:
4696:3-519-12226-X
4692:
4688:
4681:
4674:
4669:
4662:, p. 120
4658:
4657:
4649:
4642:
4638:
4634:
4630:
4626:
4622:
4615:
4607:
4603:
4599:
4592:
4585:
4581:
4577:
4573:
4569:
4565:
4558:
4551:
4549:0-201-04586-9
4545:
4541:
4534:
4527:
4525:0-201-04586-9
4521:
4517:
4510:
4508:
4503:
4495:
4492:
4479:
4470:
4467:
4457:
4444:
4441:
4438:
4415:
4412:
4409:
4401:
4397:
4387:
4381:
4378:
4375:
4369:
4361:
4357:
4333:
4330:
4327:
4315:
4309:
4306:
4303:
4297:
4291:
4288:
4266:
4262:
4258:
4255:
4235:
4215:
4212:
4209:
4206:
4203:
4183:
4174:
4165:
4148:
4140:
4136:
4126:
4118:
4114:
4110:
4104:
4096:
4092:
4071:
4051:
4031:
4011:
3991:
3971:
3962:
3959:
3950:
3936:
3916:
3910:
3907:
3904:
3900:
3897:
3891:
3888:
3885:
3881:
3878:
3872:
3869:
3866:
3863:
3860:
3856:
3853:
3847:
3844:
3841:
3838:
3835:
3812:
3803:
3795:
3791:
3781:
3777:
3773:
3768:
3764:
3759:
3750:
3742:
3738:
3734:
3728:
3720:
3716:
3709:
3690:
3676:
3667:
3664:
3661:
3653:
3650:
3647:
3643:
3633:
3624:
3615:
3607:
3603:
3593:
3589:
3585:
3580:
3576:
3571:
3562:
3554:
3550:
3546:
3540:
3532:
3528:
3515:
3511:
3507:
3502:
3498:
3490:
3481:
3478:
3475:
3467:
3463:
3453:
3444:
3435:
3427:
3424:
3421:
3417:
3410:
3403:
3402:
3401:
3400:
3395:
3381:
3361:
3358:
3355:
3352:
3349:
3340:
3332:
3318:
3298:
3295:
3292:
3289:
3283:
3275:
3271:
3267:
3264:
3258:
3250:
3246:
3223:
3219:
3193:
3178:
3174:
3153:
3150:
3147:
3125:
3110:
3100:
3097:
3088:
3074:
3054:
3051:
3045:
3037:
3033:
3012:
2992:
2989:
2986:
2983:
2963:
2955:
2950:
2937:
2934:
2928:
2920:
2916:
2912:
2907:
2899:
2896:
2888:
2884:
2880:
2874:
2866:
2862:
2853:
2837:
2833:
2829:
2809:
2789:
2781:
2765:
2762:
2759:
2751:
2747:
2737:
2734:
2731:
2723:
2719:
2715:
2710:
2706:
2684:
2678:
2675:
2672:
2652:
2632:
2624:
2614:
2613:
2546:
2542:
2533:
2529:
2525:
2522:
2513:
2499:
2479:
2472:transforming
2459:
2412:
2406:
2386:
2379:is mapped by
2359:
2356:
2336:
2328:
2295:
2292:
2284:
2252:
2249:
2241:
2237:
2221:
2151:
2147:
2138:
2134:
2130:
2127:
2105:
2101:
2078:
2074:
2011:
2009:
2005:
1994:
1974:
1950:
1947:
1944:
1941:
1935:
1924:
1919:
1872:
1863:
1842:
1833:
1822:
1817:
1813:
1792:
1783:
1780:
1757:
1749:
1745:
1735:
1727:
1723:
1719:
1714:
1710:
1689:
1667:
1642:
1634:
1626:
1623:
1603:
1595:
1590:
1573:
1561:
1491:
1477:
1473:
1451:
1447:
1443:
1438:
1434:
1386:
1372:
1362:
1339:
1336:
1331:
1325:
1322:
1319:
1315:
1310:
1306:
1299:
1289:
1286:
1278:
1244:
1223:
1189:
1186:
1183:
1180:
1164:
1149:
1146:
1143:
1130:
1127:
1124:
1112:
1109:
1087:
1077:
1055:
1053:
1049:
1045:
1040:
1038:
1034:
1030:
1026:
1021:
1013:
1008:
1003:
993:
979:
969:
935:
929:
926:
923:
899:
896:
893:
890:
887:
884:
881:
878:
874:
871:
867:
863:
860:
857:
854:
851:
848:
845:
841:
838:
818:
815:
812:
805:for one pair
786:
782:
777:
774:
771:
746:
724:
697:
694:
691:
688:
680:
677:
674:
671:
666:
663:
660:
651:
646:
643:
640:
615:
612:
609:
602:of dimension
587:
562:
555:of dimension
533:
530:
511:
507:
503:
500:
497:
494:
491:
486:
482:
478:
473:
469:
459:
456:
453:
449:
444:
435:
412:
408:
404:
401:
398:
395:
392:
387:
383:
344:
340:
323:
267:
229:
188:
174:
173:
172:
131:
103:
93:
67:
65:
61:
56:
52:
41:
39:
35:
31:
27:
18:
4740:
4728:
4710:
4704:
4686:
4680:
4668:
4655:
4648:
4624:
4620:
4614:
4597:
4591:
4567:
4563:
4557:
4539:
4533:
4515:
4460:
4458:
4175:
4171:
3951:
3827:
3397:There is an
3396:
3341:
3338:
3211:vanish, and
3089:
2953:
2951:
2854:
2698:
2622:
2620:
2611:
2514:
2326:
2282:
2239:
2235:
2017:
2000:
1997:Applications
1806:
1682:denotes the
1591:
1368:
1279:
1066:
1041:
1017:
970:
534:
531:
354:
285:
78:
47:
25:
23:
4024:neither in
4758:Categories
4606:1680014879
4498:References
2240:continuous
1058:Properties
1046:, or of a
1000:See also:
70:Definition
44:Motivation
4477:→
4474:Δ
4468:σ
4442:≥
4394:→
4385:∖
4373:∖
4322:↪
4313:∖
4301:∖
4267:∘
4259:⊂
4213:⊂
4207:⊂
4133:↪
4111:⊕
3992:σ
3969:→
3966:Δ
3960:σ
3937:⊕
3914:↪
3895:↪
3876:↪
3870:∩
3851:↪
3845:∩
3810:→
3782:∗
3774:−
3769:∗
3735:⊕
3713:→
3710:⋯
3677:⋯
3674:→
3665:∩
3651:−
3634:∗
3630:∂
3594:∗
3586:−
3581:∗
3547:⊕
3516:∗
3503:∗
3479:∩
3454:∗
3450:∂
3414:→
3411:⋯
3359:∪
3296:≠
3151:≥
3116:→
3052:≠
2935:∈
2897:−
2885:∑
2744:→
2682:→
2539:→
2360:∈
2306:→
2263:→
2144:→
1979:Δ
1948:−
1936:∈
1929:Δ
1920:ε
1899:Δ
1877:Δ
1847:Δ
1834:σ
1827:Δ
1818:ε
1814:∑
1790:→
1787:Δ
1781:σ
1742:→
1711:λ
1664:Δ
1640:→
1631:Δ
1624:σ
1577:Δ
1574:⊂
1568:Δ
1562:λ
1518:Δ
1470:→
1435:λ
1346:Δ
1340:
1307:≤
1297:Δ
1290:
1265:Δ
1221:Δ
1193:Δ
1190:∈
1154:‖
1147:−
1141:‖
1131:
1119:Δ
1113:
1078:⊂
1075:Δ
1050:, is the
956:Δ
875:∈
842:∈
831:for some
791:Δ
783:∪
768:Δ
721:Δ
712:covering
685:Δ
657:Δ
637:Δ
613:−
584:Δ
543:Δ
440:Δ
363:Δ
94:⊂
55:homotopic
4602:ProQuest
4570:: 7–16,
4431:for all
4168:Excision
3760:→
3625:→
3572:→
3491:→
3445:→
2236:support.
2008:excision
1492:′
1387:′
1365:Homology
1300:′
1224:′
324:′
268:′
189:′
132:′
60:Excision
4641:2781856
4584:0350623
30:simplex
4717:
4693:
4639:
4604:
4582:
4546:
4522:
1655:where
1237:be an
1033:facets
4660:(PDF)
3311:, so
2976:. If
2492:into
1067:Let
1037:flags
4715:ISBN
4691:ISBN
4544:ISBN
4520:ISBN
4176:Let
3342:Let
2645:and
2621:The
2242:map
2018:Let
2006:and
1337:diam
1287:diam
1110:diam
1063:Mesh
1044:cube
1010:The
310:and
79:Let
62:and
4629:doi
4572:doi
4568:210
4064:or
2956:of
2428:in
2329:of
1530:in
1128:max
1027:to
171:if
4760::
4637:MR
4635:,
4625:78
4623:,
4580:MR
4578:,
4566:,
4506:^
4445:0.
3813:0.
3394:.
3067:,
2093:,
2042:,
2010:.
968:.
66:.
40:.
4631::
4574::
4480:X
4471::
4439:k
4419:)
4416:A
4413:,
4410:X
4407:(
4402:k
4398:H
4391:)
4388:Z
4382:A
4379:,
4376:Z
4370:X
4367:(
4362:k
4358:H
4337:)
4334:A
4331:,
4328:X
4325:(
4319:)
4316:Z
4310:A
4307:,
4304:Z
4298:X
4295:(
4292::
4289:i
4263:A
4256:Z
4236:Z
4216:X
4210:A
4204:Z
4184:X
4152:)
4149:X
4146:(
4141:n
4137:C
4130:)
4127:B
4124:(
4119:n
4115:C
4108:)
4105:A
4102:(
4097:n
4093:C
4072:B
4052:A
4032:B
4012:A
3972:X
3963::
3917:X
3911:B
3908::
3905:l
3901:,
3898:X
3892:A
3889::
3886:k
3882:,
3879:B
3873:B
3867:A
3864::
3861:j
3857:,
3854:A
3848:B
3842:A
3839::
3836:i
3807:)
3804:X
3801:(
3796:0
3792:H
3778:l
3765:k
3754:)
3751:B
3748:(
3743:0
3739:H
3732:)
3729:A
3726:(
3721:0
3717:H
3671:)
3668:B
3662:A
3659:(
3654:1
3648:n
3644:H
3619:)
3616:X
3613:(
3608:n
3604:H
3590:l
3577:k
3566:)
3563:B
3560:(
3555:n
3551:H
3544:)
3541:A
3538:(
3533:n
3529:H
3521:)
3512:j
3508:,
3499:i
3495:(
3485:)
3482:B
3476:A
3473:(
3468:n
3464:H
3439:)
3436:X
3433:(
3428:1
3425:+
3422:n
3418:H
3382:X
3362:B
3356:A
3353:=
3350:X
3319:f
3299:0
3293:1
3290:=
3287:)
3284:f
3281:(
3276:0
3272:r
3268:t
3265:=
3262:)
3259:f
3256:(
3251:K
3247:L
3224:0
3220:f
3199:)
3194:n
3189:D
3184:(
3179:k
3175:H
3154:1
3148:k
3126:n
3121:D
3111:n
3106:D
3101::
3098:f
3075:f
3055:0
3049:)
3046:f
3043:(
3038:K
3034:L
3013:K
2993:d
2990:i
2987:=
2984:f
2964:f
2938:K
2932:)
2929:f
2926:(
2921:i
2917:r
2913:t
2908:i
2904:)
2900:1
2894:(
2889:i
2881:=
2878:)
2875:f
2872:(
2867:K
2863:L
2838:i
2834:r
2830:t
2810:K
2790:K
2769:)
2766:K
2763:,
2760:Y
2757:(
2752:i
2748:H
2741:)
2738:K
2735:,
2732:X
2729:(
2724:i
2720:H
2716::
2711:i
2707:f
2685:Y
2679:X
2676::
2673:f
2653:Y
2633:X
2595:K
2571:K
2547:L
2543:V
2534:K
2530:V
2526::
2523:f
2500:g
2480:f
2460:H
2438:L
2416:)
2413:x
2410:(
2407:f
2387:g
2365:K
2357:x
2337:f
2311:L
2301:K
2296::
2293:g
2283:.
2268:L
2258:K
2253::
2250:f
2222:f
2200:L
2176:K
2152:L
2148:V
2139:K
2135:V
2131::
2128:f
2106:L
2102:V
2079:K
2075:V
2052:L
2028:K
1975:B
1954:}
1951:1
1945:,
1942:1
1939:{
1925:B
1873:B
1843:B
1838:|
1823:B
1793:X
1784::
1761:)
1758:X
1755:(
1750:n
1746:C
1739:)
1736:X
1733:(
1728:n
1724:C
1720::
1715:n
1690:n
1668:n
1643:X
1635:n
1627::
1604:X
1571:)
1565:(
1540:K
1498:)
1489:K
1483:(
1478:n
1474:C
1467:)
1462:K
1457:(
1452:n
1448:C
1444::
1439:n
1412:K
1384:K
1349:)
1343:(
1332:)
1326:1
1323:+
1320:n
1316:n
1311:(
1304:)
1293:(
1245:n
1198:}
1187:b
1184:,
1181:a
1175:|
1165:n
1160:R
1150:b
1144:a
1136:{
1125:=
1122:)
1116:(
1088:n
1083:R
980:n
936:!
933:)
930:1
927:+
924:n
921:(
900:!
897:n
894:,
891:.
888:.
885:.
882:,
879:1
872:j
868:,
864:n
861:,
858:.
855:.
852:.
849:,
846:0
839:i
819:j
816:,
813:i
787:b
778:j
775:,
772:i
747:n
725:i
698:!
695:n
692:,
689:i
681:,
678:.
675:.
672:.
667:2
664:,
661:i
652:,
647:1
644:,
641:i
616:1
610:n
588:i
563:n
517:)
512:n
508:p
504:+
501:.
498:.
495:.
492:+
487:1
483:p
479:+
474:0
470:p
466:(
460:1
457:+
454:n
450:1
445:=
436:b
413:n
409:p
405:,
402:.
399:.
396:.
393:,
388:0
384:p
321:S
296:S
265:S
240:S
214:S
186:S
157:S
129:S
104:n
99:R
89:S
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
