164:
of dimension > 4: for simply-connected manifolds, the
Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same. The applications are to differentiable manifolds, PL manifolds and topological manifolds. The proofs were first obtained in the
1702:
1580:
1470:
2195:
1950:
1588:
2009:
471:
2593:
1784:
195:
theorem, which is a statement about simply connected manifolds, to non-simply connected manifolds, one must distinguish simple homotopy equivalences and non-simple homotopy equivalences. While an
1236:
660:
314:
144:
1849:
2267:
743:
997:
1372:
613:
399:
1289:
1082:
2658:
250:
2381:
215: > 4 is isomorphic to a cylinder (the corresponding homotopy equivalence can be taken to be a diffeomorphism, PL-isomorphism, or homeomorphism, respectively), the
2329:
2518:
2486:
2416:
53:
785:
2088:
1148:
1113:
904:
807:
510:
353:
2451:
2041:
1884:
93:
1479:
687:
1377:
173:
theory allowed much the same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of
2093:
1003:
is a generator of the cyclic group of order 5. This example is closely related to the existence of units of infinite order (in particular, the
787:
we have to show that any matrix can be written as a product of elementary matrices times a diagonal matrix; this follows easily from the fact that
2870:
2927:
2822:
576:
elements are 1 and there is at most one non-zero element not on the diagonal. The subgroup generated by elementary matrices is exactly the
564:); concretely, the group of invertible infinite matrices which differ from the identity matrix in only a finite number of coefficients. An
1697:{\displaystyle C_{\mathrm {odd} }:=\bigoplus _{n{\text{ odd}}}C_{n},\qquad C_{\mathrm {even} }:=\bigoplus _{n{\text{ even}}}C_{n}.}
1892:
1955:
408:
2523:
1710:
1174:
1296:
622:
271:
101:
2197:. Now we can apply the definition of the Whitehead torsion for a chain homotopy equivalence and obtain an element in
1793:
2200:
692:
2796:
913:
1318:
3003:
2962:
586:
372:
1241:
3066:
3061:
2721:
1040:
60:
2637:
229:
2910:
2342:
2296:
2914:
843:
2491:
2459:
2389:
26:
765:
2742:. The set of such equivalence classes form a group where the addition is given by taking union of
2046:
1122:
1087:
887:
790:
484:
3071:
1025:
322:
2421:
181:. The restriction to manifolds of dimension greater than four are due to the application of the
1159:
It is a well-known conjecture that the
Whitehead group of any torsion-free group should vanish.
2014:
1857:
1575:{\displaystyle (c_{*}+\gamma _{*})_{\mathrm {odd} }:C_{\mathrm {odd} }\to C_{\mathrm {even} }}
69:
2902:
1465:{\displaystyle c_{n+1}\circ \gamma _{n}+\gamma _{n-1}\circ c_{n}=\operatorname {id} _{C_{n}}}
178:
3032:
2991:
2950:
2891:
2843:
2786:
2669:
854:
669:
21:
521:
8:
2791:
2769:
2604:
1017:
537:
216:
3020:
2998:
2979:
2847:
2781:
2660:
vanishes. Moreover, for any element in the
Whitehead group there exists an h-cobordism
819:
580:, in other words the smallest normal subgroup such that the quotient by it is abelian.
147:
17:
2817:
867:
565:
317:
2941:
1007:) in the ring of integers of the cyclotomic field generated by fifth roots of unity.
853:; this is the essential ingredient of the 1969 Kirby–Siebenmann structure theory of
3012:
2971:
2936:
2879:
2851:
2831:
830:. This is quite hard to prove, but is important as it is used in the proof that an
810:
577:
158:
3028:
2987:
2946:
2887:
2839:
2883:
2675:
There exists a homotopy theoretic analogue of the s-cobordism theorem. Given a
910:. An example of a non-trivial unit in the group ring arises from the identity
839:
226:-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion
154:
2190:{\displaystyle C_{*}({\tilde {f}}):C_{*}({\tilde {X}})\to C_{*}({\tilde {Y}})}
3055:
2957:
2865:
907:
759:
573:
182:
166:
2907:
Foundational essays on topological manifolds, smoothings, and triangulations
2898:
2813:
1004:
876:
827:
569:
545:
174:
2922:
2616:
863:
189:
3045:
3024:
2983:
2835:
2809:
2676:
823:
513:
170:
56:
689:. Notice that this is the same as the quotient of the reduced K-group
846:
3016:
2975:
402:
161:
2668:
whose
Whitehead torsion is the considered element. The proofs use
1016:
is finitely generated, of rank equal to the number of irreducible
866:(or any subgroup of a braid group) is trivial. This was proved by
536:
is defined as the quotient of GL(A) by the subgroup generated by
2520:
be homotopy equivalences of finite connected CW-complexes. Then
1295:-chain complexes. We can assign to the homotopy equivalence its
2331:
be homotopy equivalences of finite connected CW-complexes. If
838:
is a product. It is also the key algebraic result used in the
662:
by the subgroup generated by elementary matrices, elements of
2760:. This group is natural isomorphic to the Whitehead group Wh(
1150:
modulo the group of "trivial units" generated by elements of
850:
835:
1945:{\displaystyle \tau (f)\in \operatorname {Wh} (\pi _{1}(Y))}
849:
of dimension at least 5 which are homotopy equivalent to a
2418:
is a homeomorphism of finite connected CW-complexes, then
222:
states that if the manifolds are not simply-connected, an
2004:{\displaystyle {\tilde {f}}:{\tilde {X}}\to {\tilde {Y}}}
762:
is trivial. Since the group ring of the trivial group is
466:{\displaystyle G\times \{\pm 1\}\to K_{1}(\mathbb {Z} )}
879:
is trivial if and only if it is of order 2, 3, 4, or 6.
2634:
if and only if the
Whitehead torsion of the inclusion
2588:{\displaystyle \tau (g\circ f)=g_{*}\tau (f)+\tau (g)}
884:
The
Whitehead group of the cyclic group of order 5 is
2640:
2526:
2494:
2462:
2424:
2392:
2345:
2299:
2203:
2096:
2049:
2017:
1958:
1895:
1860:
1796:
1779:{\displaystyle \tau (h_{*}):=\in {\tilde {K}}_{1}(R)}
1713:
1591:
1482:
1380:
1321:
1244:
1177:
1125:
1090:
1043:
916:
890:
793:
768:
695:
672:
625:
589:
487:
411:
375:
325:
274:
232:
104:
72:
29:
3046:
A description of
Whitehead torsion is in section two
2768:. The proof of this fact is similar to the proof of
1374:
be any chain contraction of the mapping cone, i.e.,
203:
between simply-connected closed connected manifolds
1231:{\displaystyle \tau (h_{*})\in {\tilde {K}}_{1}(R)}
169:, for differentiable manifolds. The development of
146:. These concepts are named after the mathematician
20:, a field within mathematics, the obstruction to a
2652:
2587:
2512:
2480:
2445:
2410:
2375:
2323:
2261:
2189:
2082:
2035:
2003:
1944:
1878:
1843:
1778:
1696:
1574:
1464:
1366:
1283:
1230:
1142:
1107:
1076:
991:
898:
834:-cobordism of dimension at least 6 whose ends are
801:
779:
737:
681:
654:
607:
504:
465:
393:
347:
308:
244:
138:
87:
47:
2682:, consider the set of all pairs of CW-complexes (
655:{\displaystyle \operatorname {GL} (\mathbb {Z} )}
3053:
2607:states for a closed connected oriented manifold
309:{\displaystyle \operatorname {Wh} (\pi _{1}(M))}
139:{\displaystyle \operatorname {Wh} (\pi _{1}(Y))}
2909:, Annals of Mathematics Studies, vol. 88,
2897:
2861:Graduate Text in Mathematics 10, Springer, 1973
2818:"The Whitehead group of a polynomial extension"
1886:of connected finite CW-complexes we define the
1844:{\displaystyle (c_{*}+\gamma _{*})_{\rm {odd}}}
153:The Whitehead torsion is important in applying
2871:Proceedings of the London Mathematical Society
2808:
2262:{\displaystyle {\tilde {K}}_{1}(\mathbb {Z} )}
738:{\displaystyle {\tilde {K}}_{1}(\mathbb {Z} )}
2928:Bulletin of the American Mathematical Society
2277:)). This is the Whitehead torsion τ(ƒ) ∈ Wh(π
1084:is isomorphic to the units of the group ring
992:{\displaystyle (1-t-t^{4})(1-t^{2}-t^{3})=1,}
1367:{\displaystyle \gamma _{*}:C_{*}\to C_{*+1}}
1311:) which is a contractible finite based free
427:
418:
2720:) are said to be equivalent, if there is a
2598:
2997:
2960:(1962), "On the structure of manifolds",
2940:
2227:
2051:
1127:
1092:
1058:
892:
795:
770:
719:
636:
489:
447:
1163:
1012:The Whitehead group of any finite group
264:of a connected CW-complex or a manifold
477:, ±1) to the invertible (1,1)-matrix (±
3054:
2921:
2864:
2698:is a homotopy equivalence. Two pairs (
2043:to the universal covering. It induces
608:{\displaystyle \operatorname {Wh} (G)}
394:{\displaystyle \operatorname {Wh} (G)}
2956:
2868:(1940), "The units of group-rings",
2823:Publications Mathématiques de l'IHÉS
1284:{\displaystyle h_{*}:D_{*}\to E_{*}}
583:In other words, the Whitehead group
548:of the finite-dimensional groups GL(
1077:{\displaystyle K_{1}(\mathbb {Z} )}
13:
2859:A course in simple homotopy theory
2653:{\displaystyle M\hookrightarrow W}
1835:
1832:
1829:
1657:
1654:
1651:
1648:
1604:
1601:
1598:
1566:
1563:
1560:
1557:
1542:
1539:
1536:
1521:
1518:
1515:
1028:. This was proved in 1965 by Bass.
255:
245:{\displaystyle M\hookrightarrow W}
14:
3083:
3039:
3001:(1950), "Simple homotopy types",
2376:{\displaystyle \tau (f)=\tau (g)}
1851:with respect to the given bases.
1238:for a chain homotopy equivalence
1115:under the determinant map, so Wh(
2324:{\displaystyle f,g\colon X\to Y}
1119:) is just the group of units of
1024:minus the number of irreducible
268:is equal to the Whitehead group
3004:American Journal of Mathematics
2963:American Journal of Mathematics
2942:10.1090/S0002-9904-1966-11484-2
1641:
2644:
2582:
2576:
2567:
2561:
2542:
2530:
2513:{\displaystyle g\colon Y\to Z}
2504:
2481:{\displaystyle f\colon X\to Y}
2472:
2434:
2428:
2411:{\displaystyle f\colon X\to Y}
2402:
2370:
2364:
2355:
2349:
2315:
2256:
2253:
2250:
2244:
2231:
2223:
2211:
2184:
2178:
2169:
2156:
2153:
2147:
2138:
2122:
2116:
2107:
2077:
2074:
2068:
2055:
2027:
1995:
1986:
1980:
1965:
1939:
1936:
1930:
1917:
1905:
1899:
1870:
1824:
1797:
1773:
1767:
1755:
1742:
1736:
1730:
1717:
1548:
1510:
1483:
1345:
1268:
1225:
1219:
1207:
1194:
1181:
1137:
1131:
1102:
1096:
1071:
1068:
1062:
1054:
1037:is a finite cyclic group then
977:
945:
942:
917:
732:
729:
723:
715:
703:
649:
646:
640:
632:
602:
596:
499:
493:
460:
457:
451:
443:
430:
388:
382:
342:
336:
303:
300:
294:
281:
236:
133:
130:
124:
111:
82:
76:
48:{\displaystyle f\colon X\to Y}
39:
1:
2925:(1966), "Whitehead torsion",
2802:
2797:Wall's finiteness obstruction
2690:) such that the inclusion of
2288:
2090:-chain homotopy equivalences
906:. This was proved in 1940 by
822:is trivial, a 1964 result of
780:{\displaystyle \mathbb {Z} ,}
2083:{\displaystyle \mathbb {Z} }
1143:{\displaystyle \mathbb {Z} }
1108:{\displaystyle \mathbb {Z} }
899:{\displaystyle \mathbb {Z} }
802:{\displaystyle \mathbb {Z} }
505:{\displaystyle \mathbb {Z} }
185:for removing double points.
7:
2775:
2722:simple homotopy equivalence
2386:Topological invariance: If
1854:For a homotopy equivalence
1476:. We obtain an isomorphism
758:The Whitehead group of the
752:
348:{\displaystyle \pi _{1}(M)}
95:which is an element in the
61:simple homotopy equivalence
10:
3088:
2911:Princeton University Press
2615: > 4 that an
2446:{\displaystyle \tau (f)=0}
2915:University of Tokyo Press
2456:Composition formula: Let
2293:Homotopy invariance: Let
862:The Whitehead group of a
818:The Whitehead group of a
2884:10.1112/plms/s2-46.1.231
2599:Geometric interpretation
2036:{\displaystyle f:X\to Y}
1879:{\displaystyle f:X\to Y}
1026:rational representations
857:of dimension at least 5.
88:{\displaystyle \tau (f)}
1168:At first we define the
2654:
2589:
2514:
2482:
2447:
2412:
2377:
2325:
2263:
2191:
2084:
2037:
2005:
1946:
1880:
1845:
1780:
1698:
1576:
1466:
1368:
1285:
1232:
1144:
1109:
1078:
993:
900:
875:The Whitehead group a
803:
781:
739:
683:
656:
609:
506:
467:
395:
349:
310:
246:
140:
89:
49:
2756:with common subspace
2670:handle decompositions
2655:
2626:and another manifold
2590:
2515:
2483:
2448:
2413:
2378:
2326:
2264:
2192:
2085:
2038:
2006:
1947:
1881:
1846:
1781:
1699:
1577:
1467:
1369:
1291:of finite based free
1286:
1233:
1164:The Whitehead torsion
1145:
1110:
1079:
994:
901:
870:and Sayed K. Roushon.
855:topological manifolds
804:
782:
740:
684:
682:{\displaystyle \pm 1}
657:
610:
507:
468:
401:is defined to be the
396:
350:
311:
247:
179:Laurent C. Siebenmann
141:
90:
50:
2787:Reidemeister torsion
2764:) of the CW-complex
2638:
2524:
2492:
2460:
2422:
2390:
2343:
2339:are homotopic, then
2297:
2269:which we map to Wh(π
2201:
2094:
2047:
2015:
1956:
1893:
1858:
1794:
1711:
1589:
1480:
1378:
1319:
1315:-chain complex. Let
1242:
1175:
1123:
1088:
1041:
1018:real representations
914:
888:
791:
766:
693:
670:
623:
587:
572:: one such that all
485:
409:
373:
323:
272:
230:
188:In generalizing the
102:
70:
27:
22:homotopy equivalence
2999:Whitehead, J. H. C.
2903:Siebenmann, Laurent
2792:s-Cobordism theorem
2770:s-cobordism theorem
2605:s-cobordism theorem
619:is the quotient of
538:elementary matrices
3067:Algebraic K-theory
3062:Geometric topology
2836:10.1007/BF02684690
2782:Algebraic K-theory
2650:
2585:
2510:
2478:
2443:
2408:
2373:
2321:
2259:
2187:
2080:
2033:
2001:
1942:
1876:
1841:
1776:
1694:
1680:
1627:
1572:
1462:
1364:
1281:
1228:
1140:
1105:
1074:
989:
896:
842:classification of
826:, Alex Heller and
820:free abelian group
799:
777:
735:
679:
652:
605:
520:. Recall that the
502:
463:
391:
345:
306:
242:
220:-cobordism theorem
148:J. H. C. Whitehead
136:
85:
45:
18:geometric topology
2913:Princeton, N.J.;
2214:
2181:
2150:
2119:
1998:
1983:
1968:
1888:Whitehead torsion
1790:is the matrix of
1758:
1677:
1666:
1624:
1613:
1210:
1170:Whitehead torsion
868:F. Thomas Farrell
706:
566:elementary matrix
318:fundamental group
65:Whitehead torsion
3079:
3035:
2994:
2953:
2944:
2918:
2894:
2854:
2812:; Heller, Alex;
2659:
2657:
2656:
2651:
2630:is trivial over
2594:
2592:
2591:
2586:
2557:
2556:
2519:
2517:
2516:
2511:
2487:
2485:
2484:
2479:
2452:
2450:
2449:
2444:
2417:
2415:
2414:
2409:
2382:
2380:
2379:
2374:
2330:
2328:
2327:
2322:
2268:
2266:
2265:
2260:
2243:
2242:
2230:
2222:
2221:
2216:
2215:
2207:
2196:
2194:
2193:
2188:
2183:
2182:
2174:
2168:
2167:
2152:
2151:
2143:
2137:
2136:
2121:
2120:
2112:
2106:
2105:
2089:
2087:
2086:
2081:
2067:
2066:
2054:
2042:
2040:
2039:
2034:
2010:
2008:
2007:
2002:
2000:
1999:
1991:
1985:
1984:
1976:
1970:
1969:
1961:
1952:as follows. Let
1951:
1949:
1948:
1943:
1929:
1928:
1885:
1883:
1882:
1877:
1850:
1848:
1847:
1842:
1840:
1839:
1838:
1822:
1821:
1809:
1808:
1785:
1783:
1782:
1777:
1766:
1765:
1760:
1759:
1751:
1729:
1728:
1703:
1701:
1700:
1695:
1690:
1689:
1679:
1678:
1675:
1662:
1661:
1660:
1637:
1636:
1626:
1625:
1622:
1609:
1608:
1607:
1581:
1579:
1578:
1573:
1571:
1570:
1569:
1547:
1546:
1545:
1526:
1525:
1524:
1508:
1507:
1495:
1494:
1471:
1469:
1468:
1463:
1461:
1460:
1459:
1458:
1441:
1440:
1428:
1427:
1409:
1408:
1396:
1395:
1373:
1371:
1370:
1365:
1363:
1362:
1344:
1343:
1331:
1330:
1290:
1288:
1287:
1282:
1280:
1279:
1267:
1266:
1254:
1253:
1237:
1235:
1234:
1229:
1218:
1217:
1212:
1211:
1203:
1193:
1192:
1149:
1147:
1146:
1141:
1130:
1114:
1112:
1111:
1106:
1095:
1083:
1081:
1080:
1075:
1061:
1053:
1052:
998:
996:
995:
990:
976:
975:
963:
962:
941:
940:
905:
903:
902:
897:
895:
844:piecewise linear
811:Euclidean domain
808:
806:
805:
800:
798:
786:
784:
783:
778:
773:
744:
742:
741:
736:
722:
714:
713:
708:
707:
699:
688:
686:
685:
680:
661:
659:
658:
653:
639:
614:
612:
611:
606:
578:derived subgroup
511:
509:
508:
503:
492:
472:
470:
469:
464:
450:
442:
441:
400:
398:
397:
392:
366:is a group, the
354:
352:
351:
346:
335:
334:
315:
313:
312:
307:
293:
292:
251:
249:
248:
243:
159:simply connected
145:
143:
142:
137:
123:
122:
94:
92:
91:
86:
54:
52:
51:
46:
3087:
3086:
3082:
3081:
3080:
3078:
3077:
3076:
3052:
3051:
3042:
3017:10.2307/2372133
2976:10.2307/2372978
2805:
2778:
2755:
2748:
2737:
2730:
2715:
2704:
2639:
2636:
2635:
2601:
2552:
2548:
2525:
2522:
2521:
2493:
2490:
2489:
2461:
2458:
2457:
2423:
2420:
2419:
2391:
2388:
2387:
2344:
2341:
2340:
2298:
2295:
2294:
2291:
2280:
2272:
2238:
2234:
2226:
2217:
2206:
2205:
2204:
2202:
2199:
2198:
2173:
2172:
2163:
2159:
2142:
2141:
2132:
2128:
2111:
2110:
2101:
2097:
2095:
2092:
2091:
2062:
2058:
2050:
2048:
2045:
2044:
2016:
2013:
2012:
2011:be the lift of
1990:
1989:
1975:
1974:
1960:
1959:
1957:
1954:
1953:
1924:
1920:
1894:
1891:
1890:
1859:
1856:
1855:
1828:
1827:
1823:
1817:
1813:
1804:
1800:
1795:
1792:
1791:
1761:
1750:
1749:
1748:
1724:
1720:
1712:
1709:
1708:
1685:
1681:
1674:
1670:
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588:
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540:. The group GL(
527:
488:
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368:Whitehead group
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262:Whitehead group
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165:early 1960s by
118:
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97:Whitehead group
71:
68:
67:
28:
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24:
12:
11:
5:
3085:
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3072:Surgery theory
3069:
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3050:
3049:
3041:
3040:External links
3038:
3037:
3036:
2995:
2970:(3): 387–399,
2958:Smale, Stephen
2954:
2935:(3): 358–426,
2919:
2895:
2866:Higman, Graham
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908:Graham Higman
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760:trivial group
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629:
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574:main diagonal
571:
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563:
559:
555:
551:
547:
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539:
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531:
523:
519:
515:
496:
480:
476:
473:which sends (
454:
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239:
233:
225:
221:
219:
214:
211:of dimension
210:
206:
202:
198:
194:
192:
186:
184:
183:Whitney trick
180:
176:
172:
168:
167:Stephen Smale
163:
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127:
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115:
108:
105:
98:
79:
73:
66:
62:
58:
42:
36:
33:
30:
23:
19:
3008:
3002:
2967:
2961:
2932:
2926:
2923:Milnor, John
2906:
2875:
2869:
2858:
2827:
2821:
2765:
2761:
2757:
2750:
2743:
2739:
2738:relative to
2732:
2725:
2717:
2710:
2706:
2699:
2695:
2691:
2687:
2683:
2679:
2674:
2665:
2661:
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2627:
2623:
2619:
2612:
2608:
2602:
2455:
2385:
2336:
2332:
2292:
2282:
2274:
1887:
1853:
1787:
1706:
1473:
1312:
1297:mapping cone
1292:
1169:
1167:
1151:
1116:
1034:
1021:
1013:
1005:golden ratio
1000:
877:cyclic group
831:
828:Richard Swan
746:
663:
616:
582:
570:transvection
561:
557:
553:
549:
546:direct limit
541:
533:
532:) of a ring
529:
517:
478:
474:
367:
363:
361:
356:
265:
261:
259:
223:
217:
212:
208:
204:
200:
196:
190:
187:
175:Robion Kirby
152:
96:
64:
57:CW-complexes
15:
3011:(1): 1–57,
2878:: 231–248,
2810:Bass, Hyman
2617:h-cobordism
864:braid group
615:of a group
405:of the map
199:-cobordism
3056:Categories
2857:Cohen, M.
2803:References
2677:CW-complex
2289:Properties
1707:We define
1676: even
824:Hyman Bass
568:here is a
514:group ring
252:vanishes.
193:-cobordism
171:handlebody
55:of finite
2830:: 61–79,
2645:↪
2574:τ
2559:τ
2554:∗
2537:∘
2528:τ
2505:→
2499::
2473:→
2467::
2426:τ
2403:→
2397::
2362:τ
2347:τ
2316:→
2310::
2236:π
2212:~
2179:~
2165:∗
2157:→
2148:~
2134:∗
2117:~
2103:∗
2060:π
2028:→
1996:~
1987:→
1981:~
1966:~
1922:π
1915:
1909:∈
1897:τ
1871:→
1819:∗
1815:γ
1806:∗
1756:~
1746:∈
1726:∗
1715:τ
1668:⨁
1623: odd
1615:⨁
1549:→
1505:∗
1501:γ
1492:∗
1430:∘
1422:−
1415:γ
1402:γ
1398:∘
1354:∗
1346:→
1341:∗
1328:∗
1324:γ
1277:∗
1269:→
1264:∗
1251:∗
1208:~
1198:∈
1190:∗
1179:τ
965:−
952:−
930:−
924:−
847:manifolds
704:~
674:±
630:
594:
544:) is the
431:→
422:±
416:×
380:
328:π
286:π
279:
237:↪
162:manifolds
116:π
109:
74:τ
40:→
34::
2905:(1977),
2816:(1964),
2776:See also
2724:between
2622:between
1786:, where
1472:for all
753:Examples
481:). Here
403:cokernel
59:being a
3033:0035437
3025:2372133
2992:0153022
2984:2372978
2951:0196736
2917:, Tokyo
2892:0002137
2852:4649786
2844:0174605
2709:) and (
1154:and −1.
556:) → GL(
522:K-group
512:is the
316:of the
157:to non-
63:is its
3031:
3023:
2990:
2982:
2949:
2890:
2850:
2842:
1582:with
999:where
3021:JSTOR
2980:JSTOR
2874:, 2,
2848:S2CID
2694:into
2664:over
851:torus
809:is a
2749:and
2731:and
2603:The
2335:and
2285:)).
836:tori
666:and
560:+1,
260:The
207:and
177:and
3013:doi
2972:doi
2937:doi
2880:doi
2832:doi
1033:If
1020:of
745:by
516:of
362:If
355:of
16:In
3058::
3029:MR
3027:,
3019:,
3009:72
3007:,
2988:MR
2986:,
2978:,
2968:84
2966:,
2947:MR
2945:,
2933:72
2931:,
2901:;
2888:MR
2886:,
2876:46
2846:,
2840:MR
2838:,
2828:22
2826:,
2820:,
2772:.
2716:,
2705:,
2686:,
2672:.
2595:.
2488:,
2453:.
2383:.
1912:Wh
1734::=
1664::=
1611::=
1447:id
1307:(h
749:.
627:GL
591:Wh
552:,
377:Wh
359:.
276:Wh
150:.
106:Wh
3048:.
3015::
2974::
2939::
2882::
2834::
2766:A
2762:A
2758:A
2754:2
2751:X
2747:1
2744:X
2740:A
2736:2
2733:X
2729:1
2726:X
2718:A
2714:2
2711:X
2707:A
2703:1
2700:X
2696:X
2692:A
2688:A
2684:X
2680:A
2666:M
2662:W
2648:W
2642:M
2632:M
2628:N
2624:M
2620:W
2613:n
2609:M
2583:)
2580:g
2577:(
2571:+
2568:)
2565:f
2562:(
2550:g
2546:=
2543:)
2540:f
2534:g
2531:(
2508:Z
2502:Y
2496:g
2476:Y
2470:X
2464:f
2441:0
2438:=
2435:)
2432:f
2429:(
2406:Y
2400:X
2394:f
2371:)
2368:g
2365:(
2359:=
2356:)
2353:f
2350:(
2337:g
2333:f
2319:Y
2313:X
2307:g
2304:,
2301:f
2283:Y
2281:(
2279:1
2275:Y
2273:(
2271:1
2257:)
2254:]
2251:)
2248:Y
2245:(
2240:1
2232:[
2228:Z
2224:(
2219:1
2209:K
2185:)
2176:Y
2170:(
2161:C
2154:)
2145:X
2139:(
2130:C
2126::
2123:)
2114:f
2108:(
2099:C
2078:]
2075:)
2072:Y
2069:(
2064:1
2056:[
2052:Z
2031:Y
2025:X
2022::
2019:f
1993:Y
1978:X
1972::
1963:f
1940:)
1937:)
1934:Y
1931:(
1926:1
1918:(
1906:)
1903:f
1900:(
1874:Y
1868:X
1865::
1862:f
1836:d
1833:d
1830:o
1825:)
1811:+
1802:c
1798:(
1788:A
1774:)
1771:R
1768:(
1763:1
1753:K
1743:]
1740:A
1737:[
1731:)
1722:h
1718:(
1692:.
1687:n
1683:C
1672:n
1658:n
1655:e
1652:v
1649:e
1644:C
1639:,
1634:n
1630:C
1619:n
1605:d
1602:d
1599:o
1594:C
1567:n
1564:e
1561:v
1558:e
1553:C
1543:d
1540:d
1537:o
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