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is PL, since it is a simplicial map between a subdivision of |K| into two triangles and a subdivision of |L| into two triangles. This notion is an adaptation of the general notion of a
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As an example, let K be the ASC containing the sets {1,2},{2,3},{3,1} and their subsets, and let L be the ASC containing the set {4,5,6} and its subsets. Define a mapping
171:
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861:
Simplicial maps are determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes.
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367:. Isomorphic simplicial complexes are essentially "the same", up ro a renaming of the vertices. The existence of an isomorphism between L and K is usually denoted by
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523:, which is not a simplex in K. If we modify L by removing {4,5,6}, that is, L is the ASC containing only the sets {4,5},{5,6},{6,4} and their subsets, then
865:
1880:
is a simplicial map of K' into L'. Every simplicial map is PL, but the opposite is not true. For example, suppose |K| and |L| are two triangles, and let
663:
Equivalently, one can define a simplicial map as a function from the underlying space of K (the union of simplices in K) to the underlying space of L,
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A simplicial map between two ASCs induces a simplicial map between their geometric realizations (their underlying polyhedra) using
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The function f defined above is not an isomorphism since it is not bijective. If we modify the definition to
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such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex
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be a continuous map between the underlying polyhedra of simplicial complexes and let us write
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A simplicial map is defined in slightly different ways in different contexts.
1971:| is a PL mapping such that the simplicial mapping between the subdivisions,
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maps the one-dimensional simplex {1,2} to the zero-dimensional simplex {4}.
60:
660:. Note that this implies that vertices of K are mapped to vertices of L.
1728:
1693:{\displaystyle f({\text{st}}(v))\subseteq {\text{st}}(f_{\triangle }(v))}
33:
1153:
has a unique representation as a convex combination of the vertices,
2165:
Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.),
147:, that maps every simplex in K to a simplex in L. That is, for any
36:
always span a simplex. Simplicial maps can be used to approximate
63:
simplicial map such that both it and its inverse are simplicial.
2096:: Lectures on Topological Methods in Combinatorics and Geometry
239:({2,3})=f({3,1})={4,5} which is also a simplex in L, etc.
1932:
be a non-linear function that maps the leftmost half of |
653:{\displaystyle \operatorname {conv} (f(V(\sigma )))\in L}
32:, with the property that the images of the vertices of a
412:
is bijective but it is still not an isomorphism, since
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2098:(2nd ed.). Berlin-Heidelberg: Springer-Verlag.
1436:{\displaystyle |f|(x):=\sum _{i=0}^{k}a_{i}f(v_{i})}
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2130:Colin P. Rourke and Brian J. Sanderson (1982).
1447:| is a simplicial map of |K| into |L|; it is a
2173:, Amsterdam: North-Holland, pp. 219–259,
1063:be its support (the unique simplex containing
1067:in its interior), and denote the vertices of
8:
791:
719:to a simplex in L. That is, for any simplex
510:
492:
483:
465:
1620:{\displaystyle f_{\triangle }\colon K\to L}
1210:{\displaystyle x=\sum _{i=0}^{k}a_{i}v_{i}}
516:{\displaystyle f^{-1}(\{4,5,6\})=\{1,2,3\}}
2133:Introduction to Piecewise-Linear Topology
2018:
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1944:| linearly into the rightmostt half of |
2167:"Chapter 5 - Piecewise Linear Topology"
2042:
1936:| linearly into the leftmost half of |
1297:{\displaystyle \sum _{i=0}^{k}a_{i}=1}
858:. Every simplicial map is continuous.
343:is a simplicial map of L into K, then
235:({1,2})={4} which is a simplex in L,
7:
2083:
2081:
2079:
2077:
2048:
2046:
1743:Let K and L be two GSCs. A function
1940:|, and maps the rightmost half of |
1331:are the barycentric coordinates of
1126:{\displaystyle v_{0},\ldots ,v_{k}}
90:is a function from the vertices of
1673:
1600:
1546:{\displaystyle f\colon |K|\to |L|}
1007:defined as follows. For any point
14:
803:{\displaystyle f\vert _{\sigma }}
1733:simplicial approximation theorem
1731:to the map it approximates. See
868:. This can be defined precisely.
50:simplicial approximation theorem
773:{\displaystyle f(\sigma )\in L}
715:, that maps every simplex in K
231:is a simplicial mapping, since
201:{\displaystyle f(\sigma )\in L}
2171:Handbook of Geometric Topology
2058:Elements of Algebraic Topology
2026:{\displaystyle f:|K'|\to |L'|}
2019:
2006:
2002:
1998:
1985:
1918:
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1902:
1894:
1873:{\displaystyle f:|K'|\to |L'|}
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1727:A simplicial approximation is
1687:
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1588:of a vertex. A simplicial map
1577:{\displaystyle {\text{st}}(v)}
1571:
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1000:{\displaystyle |f|:|K|\to |L|}
993:
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981:
977:
969:
961:
953:
914:{\displaystyle f:V(K)\to V(L)}
908:
902:
896:
893:
887:
871:Let K, L be two ASCs, and let
761:
755:
701:
693:
689:
685:
677:
641:
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635:
629:
623:
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531:Geometric simplicial complexes
486:
462:
313:is bijective, and its inverse
189:
183:
140:{\displaystyle f:V(K)\to V(L)}
134:
128:
122:
119:
113:
1:
2136:. New York: Springer-Verlag.
2094:Using the Borsuk-Ulam Theorem
1802:if there exist a subdivision
262:is not bijective, it may map
81:abstract simplicial complexes
75:Abstract simplicial complexes
2110:Written in cooperation with
1925:{\displaystyle f:|K|\to |L|}
1788:{\displaystyle f:|K|\to |L|}
738:{\displaystyle \sigma \in K}
708:{\displaystyle f:|K|\to |L|}
597:{\displaystyle \sigma \in K}
537:geometric simplicial complex
166:{\displaystyle \sigma \in K}
48:; this is formalized by the
1243:{\displaystyle a_{i}\geq 0}
2232:
1467:is an isomorphism between
274:-dimensional simplices in
266:-dimensional simplices in
2142:10.1007/978-3-642-81735-9
1956:to simplicial complexes.
1954:piecewise-linear function
921:be a simplicial map. The
1702:simplicial approximation
1497:Simplicial approximation
1036:{\displaystyle x\in |K|}
571:{\displaystyle f:K\to L}
386:{\displaystyle K\cong L}
286:. In the above example,
1963:between two polyhedra |
1080:{\displaystyle \sigma }
1056:{\displaystyle \sigma }
866:barycentric coordinates
847:{\displaystyle \sigma }
2033:, is a homeomorphism.
2027:
1926:
1874:
1789:
1718:
1694:
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1578:
1547:
1437:
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828:
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774:
739:
709:
654:
598:
572:
517:
436:
435:{\displaystyle f^{-1}}
387:
365:simplicial isomorphism
357:
337:
336:{\displaystyle f^{-1}}
307:
256:
202:
167:
141:
57:simplicial isomorphism
2028:
1927:
1875:
1790:
1739:Piecewise-linear maps
1719:
1695:
1622:
1579:
1548:
1438:
1383:
1346:
1326:
1324:{\displaystyle a_{i}}
1299:
1257:
1245:
1212:
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1148:
1128:
1082:
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1975:
1884:
1822:
1810:, and a subdivision
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416:
371:
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317:
297:
246:
177:
151:
101:
38:continuous functions
30:simplicial complexes
2211:Simplicial homology
1463:| is injective; if
1449:continuous function
780:, and in addition,
535:Let K and L be two
527:is an isomorphism.
442:is not simplicial:
94:to the vertices of
79:Let K and L be two
2206:Algebraic topology
2060:. Westview Press.
2023:
1922:
1870:
1785:
1735:for more details.
1714:
1690:
1617:
1574:
1543:
1433:
1341:
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1143:
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568:
513:
432:
383:
353:
333:
303:
252:
198:
163:
137:
42:topological spaces
22:simplicial mapping
2180:978-0-444-82432-5
2151:978-3-540-11102-3
2116:Günter M. Ziegler
2105:978-3-540-00362-5
2067:978-0-201-62728-2
2054:Munkres, James R.
1717:{\displaystyle f}
1663:
1643:
1563:
1344:{\displaystyle x}
1146:{\displaystyle x}
938:{\displaystyle f}
827:{\displaystyle f}
356:{\displaystyle f}
306:{\displaystyle f}
255:{\displaystyle f}
2223:
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2187:
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2017:
2009:
2001:
1996:
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1961:PL homeomorphism
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1797:piecewise-linear
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923:affine extension
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2216:Simplicial sets
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856:linear function
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12:
11:
5:
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2198:
2197:
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2191:
2179:
2157:
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2122:
2112:Anders Björner
2104:
2089:Matoušek, Jiří
2073:
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546:is a function
541:simplicial map
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85:simplicial map
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18:simplicial map
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2214:
2212:
2209:
2207:
2204:
2203:
2201:
2182:
2176:
2172:
2168:
2161:
2158:
2153:
2147:
2143:
2139:
2135:
2134:
2126:
2123:
2120:, Section 4.3
2118:
2117:
2113:
2107:
2101:
2097:
2095:
2090:
2084:
2082:
2080:
2078:
2074:
2069:
2063:
2059:
2055:
2049:
2047:
2043:
2036:
2034:
2014:
2011:
1993:
1990:
1981:
1978:
1970:
1966:
1962:
1957:
1955:
1951:
1947:
1943:
1939:
1935:
1914:
1898:
1890:
1887:
1861:
1858:
1840:
1837:
1828:
1825:
1817:
1813:
1809:
1805:
1801:
1798:
1777:
1761:
1753:
1750:
1738:
1736:
1734:
1730:
1725:
1711:
1703:
1681:
1669:
1657:
1648:
1634:
1614:
1608:
1605:
1596:
1587:
1568:
1535:
1519:
1511:
1508:
1496:
1494:
1493:
1490:
1486:
1482:
1481:homeomorphism
1478:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1425:
1421:
1414:
1409:
1405:
1399:
1394:
1391:
1388:
1384:
1380:
1374:
1363:
1351:). We define
1338:
1316:
1312:
1291:
1288:
1283:
1279:
1273:
1268:
1265:
1262:
1258:
1237:
1234:
1229:
1225:
1202:
1198:
1192:
1188:
1182:
1177:
1174:
1171:
1167:
1163:
1160:
1140:
1118:
1114:
1110:
1107:
1104:
1099:
1095:
1074:
1066:
1050:
1025:
1017:
1014:
989:
973:
965:
957:
945:is a mapping
932:
924:
905:
899:
890:
884:
881:
878:
870:
867:
862:
859:
857:
841:
821:
813:
795:
787:
767:
764:
758:
752:
732:
729:
726:
718:
697:
681:
673:
670:
661:
647:
644:
632:
626:
620:
614:
611:
591:
588:
585:
565:
559:
556:
553:
545:
542:
538:
530:
528:
526:
507:
504:
501:
498:
495:
489:
480:
477:
474:
471:
468:
457:
454:
450:
427:
424:
420:
411:
407:
403:
399:
395:
380:
377:
374:
366:
350:
328:
325:
321:
300:
291:
289:
285:
281:
277:
273:
269:
265:
249:
240:
238:
234:
230:
226:
222:
218:
214:
210:
195:
192:
186:
180:
160:
157:
154:
131:
125:
116:
110:
107:
104:
97:
93:
89:
86:
82:
74:
72:
66:
64:
62:
58:
53:
51:
47:
43:
39:
35:
31:
27:
23:
20:(also called
19:
2184:, retrieved
2170:
2160:
2132:
2125:
2109:
2092:
2057:
1968:
1964:
1960:
1958:
1949:
1945:
1941:
1937:
1933:
1818:, such that
1815:
1811:
1807:
1803:
1799:
1796:
1742:
1726:
1701:
1500:
1492:
1488:
1484:
1476:
1472:
1468:
1464:
1460:
1452:
1444:
1133:. The point
1064:
922:
869:
863:
860:
716:
662:
543:
540:
539:es (GSC). A
534:
524:
409:
408:(3)=6, then
405:
401:
397:
394:
364:
363:is called a
292:
287:
283:
279:
275:
271:
267:
263:
241:
236:
232:
228:
227:(3)=5. Then
224:
220:
216:
212:
209:
95:
91:
87:
84:
78:
70:
56:
54:
46:triangulated
44:that can be
28:between two
21:
17:
15:
812:restriction
544:of K into L
88:of K into L
67:Definitions
2200:Categories
2186:2022-11-15
2037:References
1795:is called
1627:such that
2003:→
1907:→
1850:→
1770:→
1729:homotopic
1674:△
1658:⊆
1612:→
1606::
1601:△
1528:→
1512::
1483:between |
1457:injective
1385:∑
1259:∑
1235:≥
1168:∑
1108:…
1075:σ
1051:σ
1018:∈
982:→
897:→
842:σ
796:σ
765:∈
759:σ
730:∈
727:σ
690:→
645:∈
633:σ
615:
589:∈
586:σ
563:→
455:−
425:−
378:≅
326:−
193:∈
187:σ
158:∈
155:σ
123:→
83:(ASC). A
61:bijective
2091:(2007).
2056:(1995).
2015:′
1994:′
1948:|. Then
1862:′
1841:′
1584:for the
1475:, then |
1459:, then |
1443:. This |
717:linearly
278:for any
40:between
26:function
1967:| and |
1487:| and |
1479:| is a
854:) is a
404:(2)=5,
400:(1)=4,
223:(2)=4,
34:simplex
24:) is a
2177:
2148:
2102:
2064:
1043:, let
1814:' of
1806:' of
1451:. If
1304:(the
1217:with
810:(the
215:by:
59:is a
2175:ISBN
2146:ISBN
2114:and
2100:ISBN
2062:ISBN
1800:(PL)
1586:star
1501:Let
1471:and
1250:and
612:conv
219:(1)=
2138:doi
1704:to
1455:is
1087:by
925:of
834:to
814:of
604:,
293:If
270:to
242:If
52:.
2202::
2169:,
2144:.
2108:.
2076:^
2045:^
1959:A
1724:.
1662:st
1642:st
1562:st
1491:|.
1381::=
745:,
282:≤
276:L,
173:,
96:L,
55:A
16:A
2154:.
2140::
2070:.
2020:|
2012:L
2007:|
1999:|
1991:K
1986:|
1982::
1979:f
1969:L
1965:K
1950:f
1946:L
1942:K
1938:L
1934:K
1919:|
1915:L
1911:|
1903:|
1899:K
1895:|
1891::
1888:f
1867:|
1859:L
1854:|
1846:|
1838:K
1833:|
1829::
1826:f
1816:L
1812:L
1808:K
1804:K
1782:|
1778:L
1774:|
1766:|
1762:K
1758:|
1754::
1751:f
1712:f
1688:)
1685:)
1682:v
1679:(
1670:f
1666:(
1655:)
1652:)
1649:v
1646:(
1638:(
1635:f
1615:L
1609:K
1597:f
1572:)
1569:v
1566:(
1540:|
1536:L
1532:|
1524:|
1520:K
1516:|
1509:f
1489:L
1485:K
1477:f
1473:L
1469:K
1465:f
1461:f
1453:f
1445:f
1431:)
1426:i
1422:v
1418:(
1415:f
1410:i
1406:a
1400:k
1395:0
1392:=
1389:i
1378:)
1375:x
1372:(
1368:|
1364:f
1360:|
1339:x
1317:i
1313:a
1292:1
1289:=
1284:i
1280:a
1274:k
1269:0
1266:=
1263:i
1238:0
1230:i
1226:a
1203:i
1199:v
1193:i
1189:a
1183:k
1178:0
1175:=
1172:i
1164:=
1161:x
1141:x
1119:k
1115:v
1111:,
1105:,
1100:0
1096:v
1065:x
1030:|
1026:K
1022:|
1015:x
994:|
990:L
986:|
978:|
974:K
970:|
966::
962:|
958:f
954:|
933:f
909:)
906:L
903:(
900:V
894:)
891:K
888:(
885:V
882::
879:f
822:f
792:|
788:f
768:L
762:)
756:(
753:f
733:K
702:|
698:L
694:|
686:|
682:K
678:|
674::
671:f
648:L
642:)
639:)
636:)
630:(
627:V
624:(
621:f
618:(
592:K
566:L
560:K
557::
554:f
525:f
511:}
508:3
505:,
502:2
499:,
496:1
493:{
490:=
487:)
484:}
481:6
478:,
475:5
472:,
469:4
466:{
463:(
458:1
451:f
428:1
421:f
410:f
406:f
402:f
398:f
393:.
381:L
375:K
351:f
329:1
322:f
301:f
288:f
284:k
280:l
272:l
268:K
264:k
250:f
237:f
233:f
229:f
225:f
221:f
217:f
213:f
208:.
196:L
190:)
184:(
181:f
161:K
135:)
132:L
129:(
126:V
120:)
117:K
114:(
111:V
108::
105:f
92:K
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