2426:
2233:
1769:
2151:
2509:
1990:
1224:
145:
961:
788:
1525:
1417:
554:
468:
348:
2302:
1287:
2342:
1925:
1613:
1058:
2043:
1562:
1454:
1373:
1329:
1174:
1481:
626:
1854:
739:
382:
2259:
1795:
995:
911:
1881:
1658:
877:
842:
815:
1821:
1143:
1678:
2554:
2536:
2347:
2160:
1683:
1581:
2048:
1627:. Thus one often can use CW approximation to reduce a general statement to a simpler version that only concerns CW complexes.
2431:
688:
1930:
2528:
1623:. CW approximation, being a weak homotopy equivalence, induces isomorphisms on homology and cohomology groups of
299:
1179:
101:
920:
495:
417:
200:
191:. From an algebraic topological viewpoint, any map between CW-complexes can thus be taken to be cellular.
744:
298:. Then it is a general result of CW-complexes that any compact subspace of a CW-complex meets (that is,
1486:
1378:
513:
427:
320:
2264:
1249:
2307:
1890:
1586:
1036:
2006:
1530:
1422:
224:
1338:
1294:
271:
28:
1152:
695:, and patching these homotopies together, will finish the proof. For details, consult Hatcher.
286:, being the image of the characteristic map of the cell, and hence the image of the closure of
2532:
1459:
595:
36:
1826:
715:
361:
2238:
1774:
970:
886:
1859:
1636:
855:
820:
793:
1800:
1122:
1663:
1335:-connected, so it follows from the long exact sequence of homotopy groups for the pair
845:
709:
216:
17:
2548:
2522:
151:. The content of the cellular approximation theorem is then that any continuous map
1116:
997:
which consists of the base point only. That is, any such map is nullhomotopic.
302:
279:
2157:
The cellular approximation ensures then that adding (i+1)-cells doesn't affect
1631:
914:
76:
40:
708:
The cellular approximation theorem can be used to immediately calculate some
471:
172:
1096:. Use the homotopy extension property to extend this homotopy to all of
1018:
2421:{\displaystyle \pi _{i}(Z_{i+1}){\stackrel {\cong }{\to }}\pi _{i}(X)}
592:
to begin with, we can repeat this process finitely many times to make
1100:, and apply cellular approximation again to obtain a map cellular on
655:
is already cellular, and we thus obtain a homotopy (relative to the (
2228:{\displaystyle \pi _{k}(Z_{i}){\stackrel {\cong }{\to }}\pi _{k}(X)}
416:. It is then a technical, non-trivial result (see Hatcher) that the
187:, then we can furthermore choose the homotopy to be stationary on
1764:{\displaystyle (f_{i})_{*}\colon \pi _{k}(Z_{i})\to \pi _{k}(X)}
43:
can always be taken to be of a specific type. Concretely, if
2146:{\displaystyle \pi _{i+1}(X)/(f_{i})_{*}(\pi _{i+1}(Z_{i}))}
2304:
gets factored by the classes of the attachment mappings
1242:, any such map is homotopic to a map whose image is in
1230:
is homotopic to a cellular map of pairs, and since the
1088:
and use cellular approximation to obtain a homotopy of
2511:
is evident from the second step of the construction.
2434:
2350:
2310:
2267:
2241:
2163:
2051:
2009:
1933:
1893:
1862:
1829:
1803:
1777:
1686:
1666:
1639:
1589:
1533:
1489:
1462:
1425:
1381:
1341:
1297:
1252:
1182:
1155:
1125:
1039:
973:
923:
889:
858:
823:
796:
747:
718:
598:
516:
430:
364:
323:
104:
2504:{\displaystyle \pi _{i+1}(Z_{i+1})\to \pi _{i+1}(X)}
1999:
are attached by constant mappings and are mapped to
1883:by attaching (i+1)-cells that (for all basepoints)
1246:, and hence it is 0 in the relative homotopy group
963:is then homotopic to a map whose image lies in the
243:to a map which is cellular on the 0-skeleton of X.
2503:
2420:
2336:
2296:
2253:
2227:
2145:
2037:
1996:by the contraction of the corresponding spheroids)
1984:
1919:
1875:
1848:
1815:
1789:
1763:
1672:
1652:
1607:
1556:
1519:
1475:
1448:
1411:
1367:
1323:
1281:
1218:
1168:
1137:
1052:
989:
955:
905:
871:
836:
809:
782:
733:
620:
548:
462:
376:
342:
139:
1104:, but without violating the cellular property on
848:CW-structure, with one 0-cell each, and with one
305:) only finitely many cells of the complex. Thus
1985:{\displaystyle \pi _{i}(Z_{i})\to \pi _{i}(X)}
1630:CW approximation is constructed inducting on
506:, we can further homotope the restriction of
8:
215:. For the base case n=0, notice that every
350:to be a cell of highest dimension meeting
239:by a path, but this gives a homotopy from
2480:
2458:
2439:
2433:
2403:
2391:
2386:
2384:
2383:
2368:
2355:
2349:
2328:
2315:
2309:
2285:
2272:
2266:
2240:
2210:
2198:
2193:
2191:
2190:
2181:
2168:
2162:
2131:
2112:
2099:
2089:
2077:
2056:
2050:
2014:
2008:
1967:
1951:
1938:
1932:
1911:
1898:
1892:
1867:
1861:
1834:
1828:
1802:
1776:
1746:
1730:
1717:
1704:
1694:
1685:
1665:
1644:
1638:
1588:
1553:
1538:
1532:
1516:
1507:
1494:
1488:
1472:
1461:
1445:
1430:
1424:
1408:
1399:
1386:
1380:
1364:
1355:
1340:
1320:
1311:
1296:
1278:
1257:
1251:
1215:
1206:
1190:
1181:
1165:
1154:
1124:
1111:As a consequence, we have that a CW-pair
1049:
1038:
978:
972:
947:
934:
922:
894:
888:
863:
857:
828:
822:
801:
795:
765:
752:
746:
717:
609:
597:
540:
521:
515:
454:
435:
429:
363:
328:
322:
131:
115:
103:
1219:{\displaystyle (D^{i},\partial D^{i})\,}
691:to extend this to a homotopy on all of
392:, since in this case only cells of the
313:) meets at most finitely many cells of
140:{\displaystyle f(X^{n})\subseteq Y^{n}}
956:{\displaystyle f\colon S^{n}\to S^{k}}
1145:have dimension strictly greater than
235:can thus be connected to a 0-cell of
7:
179:is already cellular on a subcomplex
1199:
783:{\displaystyle \pi _{n}(S^{k})=0.}
679:to a map cellular on all cells of
659: − 1)-skeleton of
588:) met only finitely many cells of
254: − 1)-skeleton of
25:
1520:{\displaystyle \pi _{n}(X^{n})\,}
1412:{\displaystyle \pi _{i}(X^{n})\,}
647:, fixing cells of the subcomplex
639:We repeat this process for every
549:{\displaystyle X^{n-1}\cup e^{n}}
463:{\displaystyle X^{n-1}\cup e^{n}}
1001:Cellular approximation for pairs
343:{\displaystyle e^{k}\subseteq Y}
2297:{\displaystyle \pi _{i}(Z_{i})}
1576:one can construct a CW complex
1282:{\displaystyle \pi _{i}(X,A)\,}
1072:is homotopic to a cellular map
2555:Theorems in algebraic topology
2498:
2492:
2473:
2470:
2451:
2415:
2409:
2387:
2380:
2361:
2337:{\displaystyle S^{i}\to Z_{i}}
2321:
2291:
2278:
2222:
2216:
2194:
2187:
2174:
2140:
2137:
2124:
2105:
2096:
2082:
2074:
2068:
2032:
2026:
1979:
1973:
1960:
1957:
1944:
1920:{\displaystyle S^{i}\to Z_{i}}
1904:
1758:
1752:
1739:
1736:
1723:
1701:
1687:
1608:{\displaystyle f\colon Z\to X}
1599:
1550:
1544:
1513:
1500:
1442:
1436:
1405:
1392:
1361:
1342:
1317:
1298:
1275:
1263:
1212:
1183:
1053:{\displaystyle A\subseteq X\,}
940:
771:
758:
615:
602:
121:
108:
33:cellular approximation theorem
1:
2038:{\displaystyle \pi _{i+1}(X)}
1887:are attached by the mappings
1557:{\displaystyle \pi _{n}(X)\,}
1449:{\displaystyle \pi _{i}(X)\,}
1927:that generate the kernel of
211:is cellular on the skeleton
1368:{\displaystyle (X,X^{n})\,}
1324:{\displaystyle (X,X^{n})\,}
1291:We have in particular that
689:homotopy extension property
223:must contain a 0-cell. The
2571:
2529:Cambridge University Press
1823:(for any basepoint). Then
1375:that we have isomorphisms
207:, with the statement that
199:The proof can be given by
175:to a cellular map, and if
63:is a continuous map, then
1582:weak homotopy equivalence
1169:{\displaystyle i\leq n\,}
632:of dimension larger than
560:, with the property that
478:to a map missing a point
408:), so we may assume that
1476:{\displaystyle i<n\,}
1080:. To see this, restrict
671:) of the restriction of
621:{\displaystyle f(e^{n})}
246:Assume inductively that
2521:Hatcher, Allen (2005),
1849:{\displaystyle Z_{i+1}}
734:{\displaystyle n<k,}
388:is already cellular on
377:{\displaystyle k\leq n}
2505:
2422:
2344:of these cells giving
2338:
2298:
2255:
2254:{\displaystyle k<i}
2229:
2147:
2039:
1986:
1921:
1877:
1850:
1817:
1791:
1790:{\displaystyle k<i}
1765:
1674:
1654:
1609:
1558:
1521:
1477:
1450:
1413:
1369:
1325:
1283:
1220:
1170:
1139:
1054:
991:
990:{\displaystyle S^{k},}
957:
907:
906:{\displaystyle S^{k}.}
873:
838:
811:
784:
735:
622:
550:
464:
378:
344:
141:
51:are CW-complexes, and
18:Cellular approximation
2506:
2423:
2339:
2299:
2256:
2230:
2148:
2040:
1987:
1922:
1878:
1876:{\displaystyle Z_{i}}
1851:
1818:
1792:
1766:
1675:
1655:
1653:{\displaystyle Z_{i}}
1610:
1559:
1522:
1478:
1451:
1414:
1370:
1326:
1284:
1221:
1171:
1140:
1092:to a cellular map on
1055:
992:
958:
915:base-point preserving
908:
874:
872:{\displaystyle S^{n}}
839:
837:{\displaystyle S^{k}}
812:
810:{\displaystyle S^{n}}
785:
736:
683:of dimension at most
623:
551:
465:
379:
345:
163:between CW-complexes
142:
2432:
2348:
2308:
2265:
2239:
2161:
2049:
2007:
1931:
1891:
1860:
1827:
1801:
1775:
1684:
1664:
1637:
1587:
1531:
1487:
1460:
1423:
1379:
1339:
1295:
1250:
1180:
1153:
1123:
1037:
971:
921:
887:
856:
821:
794:
745:
716:
712:. In particular, if
704:Some homotopy groups
596:
576:, still relative to
514:
496:deformation retracts
490: − {
428:
362:
321:
250:is cellular on the (
102:
1992:(and are mapped to
1816:{\displaystyle k=i}
1771:are isomorphic for
1680:, so that the maps
1138:{\displaystyle X-A}
1033:, and the image of
294:is also compact in
2524:Algebraic topology
2501:
2428:. Surjectivity of
2418:
2334:
2294:
2251:
2225:
2143:
2035:
1982:
1917:
1873:
1846:
1813:
1787:
1761:
1670:
1650:
1605:
1554:
1517:
1473:
1446:
1409:
1365:
1321:
1279:
1216:
1166:
1135:
1119:, if all cells of
1050:
987:
953:
903:
869:
834:
807:
780:
731:
628:miss all cells of
618:
568:) misses the cell
546:
498:onto the subspace
472:homotoped relative
460:
374:
340:
137:
29:algebraic topology
2538:978-0-521-79540-1
2396:
2203:
1797:and are onto for
1673:{\displaystyle Z}
1615:that is called a
1483:and a surjection
687:. Using then the
317:, so we can take
16:(Redirected from
2562:
2541:
2510:
2508:
2507:
2502:
2491:
2490:
2469:
2468:
2450:
2449:
2427:
2425:
2424:
2419:
2408:
2407:
2398:
2397:
2395:
2390:
2385:
2379:
2378:
2360:
2359:
2343:
2341:
2340:
2335:
2333:
2332:
2320:
2319:
2303:
2301:
2300:
2295:
2290:
2289:
2277:
2276:
2260:
2258:
2257:
2252:
2234:
2232:
2231:
2226:
2215:
2214:
2205:
2204:
2202:
2197:
2192:
2186:
2185:
2173:
2172:
2152:
2150:
2149:
2144:
2136:
2135:
2123:
2122:
2104:
2103:
2094:
2093:
2081:
2067:
2066:
2044:
2042:
2041:
2036:
2025:
2024:
1991:
1989:
1988:
1983:
1972:
1971:
1956:
1955:
1943:
1942:
1926:
1924:
1923:
1918:
1916:
1915:
1903:
1902:
1882:
1880:
1879:
1874:
1872:
1871:
1855:
1853:
1852:
1847:
1845:
1844:
1822:
1820:
1819:
1814:
1796:
1794:
1793:
1788:
1770:
1768:
1767:
1762:
1751:
1750:
1735:
1734:
1722:
1721:
1709:
1708:
1699:
1698:
1679:
1677:
1676:
1671:
1659:
1657:
1656:
1651:
1649:
1648:
1617:CW approximation
1614:
1612:
1611:
1606:
1572:For every space
1568:CW approximation
1563:
1561:
1560:
1555:
1543:
1542:
1526:
1524:
1523:
1518:
1512:
1511:
1499:
1498:
1482:
1480:
1479:
1474:
1455:
1453:
1452:
1447:
1435:
1434:
1418:
1416:
1415:
1410:
1404:
1403:
1391:
1390:
1374:
1372:
1371:
1366:
1360:
1359:
1330:
1328:
1327:
1322:
1316:
1315:
1288:
1286:
1285:
1280:
1262:
1261:
1225:
1223:
1222:
1217:
1211:
1210:
1195:
1194:
1175:
1173:
1172:
1167:
1144:
1142:
1141:
1136:
1059:
1057:
1056:
1051:
996:
994:
993:
988:
983:
982:
966:
962:
960:
959:
954:
952:
951:
939:
938:
912:
910:
909:
904:
899:
898:
882:
878:
876:
875:
870:
868:
867:
851:
843:
841:
840:
835:
833:
832:
816:
814:
813:
808:
806:
805:
789:
787:
786:
781:
770:
769:
757:
756:
740:
738:
737:
732:
627:
625:
624:
619:
614:
613:
555:
553:
552:
547:
545:
544:
532:
531:
469:
467:
466:
461:
459:
458:
446:
445:
412: >
383:
381:
380:
375:
349:
347:
346:
341:
333:
332:
146:
144:
143:
138:
136:
135:
120:
119:
21:
2570:
2569:
2565:
2564:
2563:
2561:
2560:
2559:
2545:
2544:
2539:
2520:
2517:
2476:
2454:
2435:
2430:
2429:
2399:
2364:
2351:
2346:
2345:
2324:
2311:
2306:
2305:
2281:
2268:
2263:
2262:
2237:
2236:
2206:
2177:
2164:
2159:
2158:
2127:
2108:
2095:
2085:
2052:
2047:
2046:
2010:
2005:
2004:
1963:
1947:
1934:
1929:
1928:
1907:
1894:
1889:
1888:
1863:
1858:
1857:
1830:
1825:
1824:
1799:
1798:
1773:
1772:
1742:
1726:
1713:
1700:
1690:
1682:
1681:
1662:
1661:
1640:
1635:
1634:
1585:
1584:
1570:
1534:
1529:
1528:
1503:
1490:
1485:
1484:
1458:
1457:
1426:
1421:
1420:
1395:
1382:
1377:
1376:
1351:
1337:
1336:
1307:
1293:
1292:
1290:
1253:
1248:
1247:
1202:
1186:
1178:
1177:
1176:, then any map
1151:
1150:
1121:
1120:
1035:
1034:
1003:
974:
969:
968:
964:
943:
930:
919:
918:
890:
885:
884:
880:
859:
854:
853:
849:
824:
819:
818:
797:
792:
791:
761:
748:
743:
742:
714:
713:
710:homotopy groups
706:
701:
605:
594:
593:
556:to a map, say,
536:
517:
512:
511:
450:
431:
426:
425:
360:
359:
324:
319:
318:
231:of a 0-cell of
197:
127:
111:
100:
99:
23:
22:
15:
12:
11:
5:
2568:
2566:
2558:
2557:
2547:
2546:
2543:
2542:
2537:
2516:
2513:
2500:
2497:
2494:
2489:
2486:
2483:
2479:
2475:
2472:
2467:
2464:
2461:
2457:
2453:
2448:
2445:
2442:
2438:
2417:
2414:
2411:
2406:
2402:
2394:
2389:
2382:
2377:
2374:
2371:
2367:
2363:
2358:
2354:
2331:
2327:
2323:
2318:
2314:
2293:
2288:
2284:
2280:
2275:
2271:
2250:
2247:
2244:
2224:
2221:
2218:
2213:
2209:
2201:
2196:
2189:
2184:
2180:
2176:
2171:
2167:
2155:
2154:
2142:
2139:
2134:
2130:
2126:
2121:
2118:
2115:
2111:
2107:
2102:
2098:
2092:
2088:
2084:
2080:
2076:
2073:
2070:
2065:
2062:
2059:
2055:
2034:
2031:
2028:
2023:
2020:
2017:
2013:
1997:
1981:
1978:
1975:
1970:
1966:
1962:
1959:
1954:
1950:
1946:
1941:
1937:
1914:
1910:
1906:
1901:
1897:
1870:
1866:
1856:is built from
1843:
1840:
1837:
1833:
1812:
1809:
1806:
1786:
1783:
1780:
1760:
1757:
1754:
1749:
1745:
1741:
1738:
1733:
1729:
1725:
1720:
1716:
1712:
1707:
1703:
1697:
1693:
1689:
1669:
1647:
1643:
1604:
1601:
1598:
1595:
1592:
1569:
1566:
1552:
1549:
1546:
1541:
1537:
1515:
1510:
1506:
1502:
1497:
1493:
1471:
1468:
1465:
1444:
1441:
1438:
1433:
1429:
1407:
1402:
1398:
1394:
1389:
1385:
1363:
1358:
1354:
1350:
1347:
1344:
1319:
1314:
1310:
1306:
1303:
1300:
1277:
1274:
1271:
1268:
1265:
1260:
1256:
1214:
1209:
1205:
1201:
1198:
1193:
1189:
1185:
1164:
1161:
1158:
1134:
1131:
1128:
1048:
1045:
1042:
1025:is a map from
1002:
999:
986:
981:
977:
950:
946:
942:
937:
933:
929:
926:
902:
897:
893:
866:
862:
831:
827:
804:
800:
779:
776:
773:
768:
764:
760:
755:
751:
730:
727:
724:
721:
705:
702:
700:
697:
617:
612:
608:
604:
601:
543:
539:
535:
530:
527:
524:
520:
457:
453:
449:
444:
441:
438:
434:
373:
370:
367:
339:
336:
331:
327:
217:path-component
196:
193:
134:
130:
126:
123:
118:
114:
110:
107:
67:is said to be
35:states that a
24:
14:
13:
10:
9:
6:
4:
3:
2:
2567:
2556:
2553:
2552:
2550:
2540:
2534:
2530:
2526:
2525:
2519:
2518:
2514:
2512:
2495:
2487:
2484:
2481:
2477:
2465:
2462:
2459:
2455:
2446:
2443:
2440:
2436:
2412:
2404:
2400:
2392:
2375:
2372:
2369:
2365:
2356:
2352:
2329:
2325:
2316:
2312:
2286:
2282:
2273:
2269:
2248:
2245:
2242:
2219:
2211:
2207:
2199:
2182:
2178:
2169:
2165:
2132:
2128:
2119:
2116:
2113:
2109:
2100:
2090:
2086:
2078:
2071:
2063:
2060:
2057:
2053:
2029:
2021:
2018:
2015:
2011:
2002:
1998:
1995:
1976:
1968:
1964:
1952:
1948:
1939:
1935:
1912:
1908:
1899:
1895:
1886:
1885:
1884:
1868:
1864:
1841:
1838:
1835:
1831:
1810:
1807:
1804:
1784:
1781:
1778:
1755:
1747:
1743:
1731:
1727:
1718:
1714:
1710:
1705:
1695:
1691:
1667:
1645:
1641:
1633:
1628:
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1622:
1618:
1602:
1596:
1593:
1590:
1583:
1579:
1575:
1567:
1565:
1547:
1539:
1535:
1508:
1504:
1495:
1491:
1469:
1466:
1463:
1439:
1431:
1427:
1400:
1396:
1387:
1383:
1356:
1352:
1348:
1345:
1334:
1312:
1308:
1304:
1301:
1272:
1269:
1266:
1258:
1254:
1245:
1241:
1237:
1234:-skeleton of
1233:
1229:
1207:
1203:
1196:
1191:
1187:
1162:
1159:
1156:
1148:
1132:
1129:
1126:
1118:
1114:
1109:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1046:
1043:
1040:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1000:
998:
984:
979:
975:
967:-skeleton of
948:
944:
935:
931:
927:
924:
916:
900:
895:
891:
864:
860:
847:
829:
825:
802:
798:
777:
774:
766:
762:
753:
749:
728:
725:
722:
719:
711:
703:
698:
696:
694:
690:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
637:
635:
631:
610:
606:
599:
591:
587:
583:
579:
575:
571:
567:
563:
559:
541:
537:
533:
528:
525:
522:
518:
509:
505:
501:
497:
493:
489:
485:
482: ∈
481:
477:
473:
455:
451:
447:
442:
439:
436:
432:
423:
419:
415:
411:
407:
403:
399:
396:-skeleton of
395:
391:
387:
371:
368:
365:
357:
353:
337:
334:
329:
325:
316:
312:
308:
304:
303:non-trivially
301:
297:
293:
289:
285:
281:
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273:
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261:
257:
253:
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226:
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218:
214:
210:
206:
202:
195:Idea of proof
194:
192:
190:
186:
182:
178:
174:
170:
166:
162:
158:
154:
150:
132:
128:
124:
116:
112:
105:
97:
93:
90:-skeleton of
89:
85:
81:
79:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
19:
2523:
2156:
2003:to generate
2000:
1993:
1629:
1624:
1620:
1616:
1577:
1573:
1571:
1332:
1243:
1239:
1238:sits inside
1235:
1231:
1227:
1146:
1112:
1110:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1064:sits inside
1061:
1030:
1026:
1022:
1017:be a map of
1014:
1010:
1006:
1004:
707:
699:Applications
692:
684:
680:
676:
672:
668:
664:
660:
656:
652:
648:
644:
640:
638:
633:
629:
589:
585:
581:
577:
573:
569:
565:
561:
557:
507:
503:
499:
491:
487:
483:
479:
475:
421:
413:
409:
405:
401:
397:
393:
389:
385:
355:
351:
314:
310:
306:
295:
291:
287:
283:
275:
267:
263:
259:
255:
251:
247:
245:
240:
236:
232:
228:
220:
212:
208:
204:
198:
188:
184:
180:
176:
168:
164:
160:
156:
152:
148:
95:
91:
87:
83:
77:
72:
68:
64:
60:
56:
52:
48:
44:
41:CW-complexes
32:
26:
1117:n-connected
1021:, that is,
418:restriction
2515:References
883:-cell for
852:-cell for
667:-cells of
384:, the map
300:intersects
258:, and let
98:, i.e. if
75:takes the
2478:π
2474:→
2437:π
2401:π
2393:≅
2388:→
2353:π
2322:→
2270:π
2208:π
2200:≅
2195:→
2166:π
2110:π
2101:∗
2054:π
2012:π
1965:π
1961:→
1936:π
1905:→
1744:π
1740:→
1715:π
1711::
1706:∗
1600:→
1594::
1536:π
1492:π
1428:π
1384:π
1255:π
1200:∂
1160:≤
1130:−
1044:⊆
941:→
928::
846:canonical
750:π
651:on which
643:-cell of
534:∪
526:−
448:∪
440:−
369:≤
335:⊆
266:-cell of
201:induction
173:homotopic
125:⊆
80:-skeleton
2549:Category
2261:, while
1456:for all
1019:CW-pairs
879:and one
663:and the
580:. Since
486:. Since
155: :
147:for all
94:for all
69:cellular
55: :
39:between
1632:skeleta
1068:. Then
470:can be
280:compact
272:closure
86:to the
2535:
1580:and a
1060:under
844:their
400:meets
358:). If
290:under
270:. The
262:be an
227:under
203:after
31:, the
1228:(X,A)
1149:: If
1113:(X,A)
1078:(Y,B)
1074:(X,A)
1015:(Y,B)
1011:(X,A)
790:Give
741:then
225:image
71:, if
2533:ISBN
2246:<
2235:for
2045:(or
1782:<
1467:<
1005:Let
917:map
913:Any
817:and
723:<
167:and
47:and
1660:of
1619:to
1331:is
1115:is
1084:to
1029:to
675:to
572:of
510:to
474:to
424:to
420:of
282:in
278:is
274:of
219:of
183:of
171:is
82:of
37:map
27:In
2551::
2531:,
2527:,
2153:).
1564:.
1108:.
778:0.
636:.
494:}
159:→
59:→
2499:)
2496:X
2493:(
2488:1
2485:+
2482:i
2471:)
2466:1
2463:+
2460:i
2456:Z
2452:(
2447:1
2444:+
2441:i
2416:)
2413:X
2410:(
2405:i
2381:)
2376:1
2373:+
2370:i
2366:Z
2362:(
2357:i
2330:i
2326:Z
2317:i
2313:S
2292:)
2287:i
2283:Z
2279:(
2274:i
2249:i
2243:k
2223:)
2220:X
2217:(
2212:k
2188:)
2183:i
2179:Z
2175:(
2170:k
2141:)
2138:)
2133:i
2129:Z
2125:(
2120:1
2117:+
2114:i
2106:(
2097:)
2091:i
2087:f
2083:(
2079:/
2075:)
2072:X
2069:(
2064:1
2061:+
2058:i
2033:)
2030:X
2027:(
2022:1
2019:+
2016:i
2001:X
1994:X
1980:)
1977:X
1974:(
1969:i
1958:)
1953:i
1949:Z
1945:(
1940:i
1913:i
1909:Z
1900:i
1896:S
1869:i
1865:Z
1842:1
1839:+
1836:i
1832:Z
1811:i
1808:=
1805:k
1785:i
1779:k
1759:)
1756:X
1753:(
1748:k
1737:)
1732:i
1728:Z
1724:(
1719:k
1702:)
1696:i
1692:f
1688:(
1668:Z
1646:i
1642:Z
1625:X
1621:X
1603:X
1597:Z
1591:f
1578:Z
1574:X
1551:)
1548:X
1545:(
1540:n
1527:→
1514:)
1509:n
1505:X
1501:(
1496:n
1470:n
1464:i
1443:)
1440:X
1437:(
1432:i
1419:→
1406:)
1401:n
1397:X
1393:(
1388:i
1362:)
1357:n
1353:X
1349:,
1346:X
1343:(
1333:n
1318:)
1313:n
1309:X
1305:,
1302:X
1299:(
1289:.
1276:)
1273:A
1270:,
1267:X
1264:(
1259:i
1244:A
1240:A
1236:X
1232:n
1226:→
1213:)
1208:i
1204:D
1197:,
1192:i
1188:D
1184:(
1163:n
1157:i
1147:n
1133:A
1127:X
1106:A
1102:X
1098:X
1094:A
1090:f
1086:A
1082:f
1076:→
1070:f
1066:B
1062:f
1047:X
1041:A
1031:Y
1027:X
1023:f
1013:→
1009::
1007:f
985:,
980:k
976:S
965:n
949:k
945:S
936:n
932:S
925:f
901:.
896:k
892:S
881:k
865:n
861:S
850:n
830:k
826:S
803:n
799:S
775:=
772:)
767:k
763:S
759:(
754:n
729:,
726:k
720:n
693:X
685:n
681:X
677:X
673:f
669:A
665:n
661:X
657:n
653:f
649:A
645:X
641:n
634:n
630:Y
616:)
611:n
607:e
603:(
600:f
590:Y
586:e
584:(
582:f
578:X
574:Y
570:e
566:e
564:(
562:g
558:g
542:n
538:e
529:1
523:n
519:X
508:f
504:e
502:-
500:Y
492:p
488:Y
484:e
480:p
476:X
456:n
452:e
443:1
437:n
433:X
422:f
414:n
410:k
406:e
404:(
402:f
398:Y
394:n
390:e
386:f
372:n
366:k
356:e
354:(
352:f
338:Y
330:k
326:e
315:Y
311:e
309:(
307:f
296:Y
292:f
288:e
284:X
276:e
268:X
264:n
260:e
256:X
252:n
248:f
241:f
237:Y
233:X
229:f
221:Y
213:X
209:f
205:n
189:A
185:X
181:A
177:f
169:Y
165:X
161:Y
157:X
153:f
149:n
133:n
129:Y
122:)
117:n
113:X
109:(
106:f
96:n
92:Y
88:n
84:X
78:n
73:f
65:f
61:Y
57:X
53:f
49:Y
45:X
20:)
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