36:
3711:. He showed that these spaces possess alternative infinite loop space structures which are in fact better from the following point of view: Recall that there is a surgery obstruction map from normal invariants to the L-group. With the above described groups structure on the normal invariants this map is NOT a homomorphism. However, with the group structure from Sullivan's theorem it becomes a homomorphism in the categories
1123:
However, it turns out that it is very difficult to decide whether it is possible to make a homotopy equivalence out of the map by means of surgery, whereas the same question is much easier when the map comes with the extra structure of a normal map. Therefore, in the classical surgery approach to our
2718:
with normal bundles. Recall that a (smooth) manifold has a unique tangent bundle and a unique stable normal bundle. But a finite
Poincaré complex does not possess such a unique bundle. Nevertheless, it possesses a substitute - a unique in some sense spherical fibration - the so-called Spivak normal
2448:
This is of course an almost trivial observation, but it is important because it turns out that there is an effective theory which answers question 1.' and also an effective theory which answers question 1. provided the answer to 1.' is yes. Similarly for questions 2. and 2.' Notice also that we can
1070:, and try to do surgery on it to make a homotopy equivalence out of it. Notice the following: Since our starting map was arbitrarily chosen, and surgery always produces cobordant maps, this procedure has to be performed (in the worst case) for all cobordism classes of maps
3325:
is a loop space and in fact an infinite loop space so the normal invariants are a zeroth cohomology group of an extraordinary cohomology theory defined by that infinite loop space. Note that similar ideas apply in the other categories of manifolds and one has bijections
230:
There are two equivalent definitions of normal maps, depending on whether one uses normal bundles or tangent bundles of manifolds. Hence it is possible to switch between the definitions which turns out to be quite convenient.
2357:
1647:. Then it can be represented by an embedding (or immersion) whose normal bundle is stably trivial. This observation is important since surgery is only possible on embeddings with a trivial normal bundle. For example, if
217:
on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings
1309:
3068:
2739:
is homotopy equivalent to a manifold then the spherical fibration associated to the pullback of the normal bundle of that manifold is isomorphic to the Spivak normal fibration. So it follows that if
1815:
Notice that this new approach makes it necessary to classify the bordism classes of normal maps, which are the normal invariants. Contrarily to cobordism classes of maps, the normal invariants are a
2948:
3120:
2776:
2489:
1389:
786:
3539:
2990:
2213:
2048:
1970:
3458:
210:. Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants.
3833:, ICTP Lecture Notes Series 9, Band 1, of the school "High-dimensional manifold theory" in Trieste, May/June 2001, Abdus Salam International Centre for Theoretical Physics, Trieste 1-224
2164:
3255:
2867:
2094:
1534:
3386:
1645:
1783:
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2400:
965:
552:
2893:
3295:
2716:
2667:
2634:
2601:
2565:
1858:
1596:
1154:
660:
304:
2532:
1718:
65:
3017:
369:
3744:
3160:
1809:
1094:
1028:
3709:
3638:
2245:
2002:
173:
3675:
3604:
826:
723:
409:
335:
3770:
3573:
3323:
2829:
2806:
3180:
2737:
2120:
1921:
1901:
1881:
1685:
1665:
1554:
1482:
1454:
1409:
1198:
1178:
1118:
1068:
1048:
889:
869:
849:
806:
743:
700:
680:
624:
604:
576:
476:
456:
436:
389:
264:
1819:. Its coefficients are known in the case of topological manifolds. For the case of smooth manifolds, the coefficients of the theory are much more complicated.
2099:
Notice that if the answer to these questions should be positive then it is a necessary condition that the answer to the following two questions is positive
2250:
191:
to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by
3915:
3867:
3772:. His theorem also links these new group structures to the well-known cohomology theories: the singular cohomology and real K-theory.
1206:
87:
3162:
are known in certain low-dimensions and are non-trivial which suggests the possibility that the above condition can fail for some
3022:
109:
131:
endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular,
192:
2895:
and which corresponds to taking the associated spherical fibration of a vector bundle. In fact we have a fibration sequence
3889:
2898:
2779:
3073:
1811:. Therefore, surgery on normal maps can always be done below the middle dimension. This is not true for arbitrary maps.
48:
2742:
2455:
1314:
751:
58:
52:
44:
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2953:
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2007:
1929:
3391:
2125:
69:
3198:
2834:
2053:
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3332:
1724:. On the other hand, every stably trivial normal bundle of such an embedding is automatically trivial, since
1605:
3941:
1599:
1727:
3893:
3845:
2405:
2362:
1461:
894:
481:
2872:
3267:
2688:
2639:
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2573:
2537:
1830:
1562:
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342:
136:
3898:
3850:
1127:
633:
277:
583:
243:
188:
176:
117:
3859:
2497:
105:
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are mutually not homotopy equivalent and hence one obtains three different cohomology theories.
1690:
1100:: therefore the cobordism classes of such maps are computable at least in theory for all spaces
1096:. This kind of cobordism theory is a homology theory whose coefficients have been calculated by
151:
matching the fundamental classes and preserving normal bundle information. If the dimension of
3911:
3863:
2995:
347:
124:
3903:
3855:
3811:
3714:
3128:
2669:
is much more accessible from the point of view of algebraic topology as is explained below.
1788:
1073:
1007:
3925:
3877:
3795:
3680:
3609:
2218:
1975:
158:
3921:
3873:
3791:
3787:
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3578:
3182:. There are in fact such finite Poincaré complexes, and the first example was obtained by
1721:
811:
708:
394:
320:
3749:
3550:
3300:
1411:
induces an isomorphism of fundamental groups and use homology with local coefficients in
3190:, yielding thus an example of a Poincaré complex not homotopy equivalent to a manifold.
2811:
2788:
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2722:
2105:
1906:
1886:
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1183:
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1103:
1053:
1033:
874:
854:
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791:
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728:
685:
665:
609:
589:
561:
461:
441:
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374:
249:
214:
113:
3888:, Mathematical Surveys and Monographs, vol. 69 (2nd ed.), Providence, R.I.:
3935:
3816:
1156:(suppose there exists any), and performs surgery on it. This has several advantages:
180:
3802:
Gitler, Samule; Stasheff, James D. (November 1965), "The first exotic class of BF",
3187:
140:
3839:
17:
3193:
2.' Relativizing the above considerations one obtains an (unnatural) bijection
2352:{\displaystyle (F,B)\colon (W,M_{0},M_{1})\to (X\times I,X\times 0,X\times 1)}
971:
Two normal maps are equivalent if there exists a normal bordism between them.
579:
239:
1557:
199:
3825:
1860:. Recall that the main goal of surgery theory is to answer the questions:
2603:
is really a first step in trying to understand the surgery structure set
1004:
A naive surgery approach to this question would be: start with some map
3907:
143:
collapse map, which is equivalent to there being a map from a manifold
2685:-dimensional Poincaré complex. It is useful to use the definition of
1484:
is a homotopy equivalence if and only if the surgery kernel is zero.
2778:
then the Spivak normal fibration has a bundle reduction. By the
1097:
831:
similarly as above it is required that the fundamental class of
1488:
The bundle data implies the following: Suppose that an element
1304:{\displaystyle K_{*}(M)=ker(f_{*}\colon H_{*}(M)\to H_{*}(X))}
29:
3470:
3398:
3338:
3273:
3204:
2831:
the classifying space for stable vector bundles and the map
2748:
2694:
2645:
2636:
which is the main goal in surgery theory. The point is that
2612:
2579:
2543:
2461:
1836:
2785:
This can be formulated in terms of homotopy theory. Recall
1827:
There are two reasons why it is important to study the set
3063:{\displaystyle {\tilde {\nu }}_{X}\colon X\rightarrow BO}
1160:
The map being of degree one implies that the homology of
2808:
the classifying space for stable spherical fibrations,
3070:. This is equivalent to requiring that the composition
3752:
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3683:
3652:
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3581:
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3394:
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3303:
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3168:
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3076:
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2956:
2950:. The Spivak normal fibration is classified by a map
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2010:
1978:
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161:
3844:, Oxford Mathematical Monographs, Clarendon Press,
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3011:
2992:. It has a vector bundle reduction if and only if
2984:
2943:{\displaystyle BO\rightarrow BG\rightarrow B(G/O)}
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329:
298:
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167:
3115:{\displaystyle X\rightarrow BG\rightarrow B(G/O)}
626:(with respect to the tangent bundle) consists of
3297:a structure of an abelian group since the space
2771:{\displaystyle {\mathcal {N}}(X)\neq \emptyset }
2484:{\displaystyle {\mathcal {N}}(X)\neq \emptyset }
1384:{\displaystyle H_{*}(M)=K_{*}(M)\oplus H_{*}(X)}
781:{\displaystyle \tau _{M}\oplus \varepsilon ^{k}}
57:but its sources remain unclear because it lacks
3534:{\displaystyle {\mathcal {N}}^{TOP}(X)\cong .}
2985:{\displaystyle \nu _{X}\colon X\rightarrow BG}
2208:{\displaystyle f_{i}\colon M_{i}\rightarrow X}
2043:{\displaystyle h\colon M_{0}\rightarrow M_{1}}
1965:{\displaystyle f_{i}\colon M_{i}\rightarrow X}
1720:is homotopic to an embedding by a theorem of
980:Surgery on maps versus surgery on normal maps
8:
3453:{\displaystyle {\mathcal {N}}^{PL}(X)\cong }
414:usually the normal map is supposed to be of
175:5 there is then only the algebraic topology
418:. That means that the fundamental class of
2159:{\displaystyle (f,b)\colon M\rightarrow X}
1180:splits as a direct sum of the homology of
3897:
3860:10.1093/acprof:oso/9780198509240.001.0001
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88:Learn how and when to remove this message
3250:{\displaystyle {\mathcal {N}}(X)\cong .}
2862:{\displaystyle J\colon BO\rightarrow BG}
2089:{\displaystyle f_{1}\circ h\simeq f_{0}}
1529:{\displaystyle \alpha \in \pi _{p+1}(f)}
3381:{\displaystyle {\mathcal {N}}(X)\cong }
2719:fibration. This has a property that if
1640:{\displaystyle f\circ \phi :S^{p}\to X}
1124:question, one starts with a normal map
582:whose cellular chain complex satisfies
242:whose cellular chain complex satisfies
3827:A basic introduction to surgery theory
1823:Normal invariants versus structure set
112:which is of fundamental importance in
3784:Surgery on simply-connected manifolds
1778:{\displaystyle \pi _{p}(BO,BO_{k})=0}
7:
2438:{\displaystyle \partial _{1}F=f_{1}}
2395:{\displaystyle \partial _{0}F=f_{0}}
2169:2.' Given two homotopy equivalences
2102:1.' Given a finite Poincaré complex
960:{\displaystyle f_{*}()=\in H_{n}(X)}
547:{\displaystyle f_{*}()=\in H_{n}(X)}
1926:2. Given two homotopy equivalences
1863:1. Given a finite Poincaré complex
1667:is less than half the dimension of
745:, and a stable map from the stable
2888:{\displaystyle O\hookrightarrow G}
2869:which is induced by the inclusion
2765:
2478:
2410:
2367:
25:
3544:It is well known that the spaces
3290:{\displaystyle {\mathcal {N}}(X)}
3125:Note that the homotopy groups of
2711:{\displaystyle {\mathcal {N}}(X)}
2662:{\displaystyle {\mathcal {N}}(X)}
2629:{\displaystyle {\mathcal {S}}(X)}
2596:{\displaystyle {\mathcal {N}}(X)}
2560:{\displaystyle {\mathcal {N}}(X)}
2449:phrase the questions as follows:
2122:is there a degree one normal map
1903:-manifold homotopy equivalent to
1853:{\displaystyle {\mathcal {N}}(X)}
1591:{\displaystyle \phi :S^{p}\to M}
1536:(the relative homotopy group of
996:homotopy-equivalent to a closed
34:
3841:Algebraic and Geometric Surgery
3525:
3499:
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3487:
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1956:
1847:
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1631:
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1523:
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1328:
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1273:
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1220:
1149:{\displaystyle f\colon M\to X}
1140:
1080:
1014:
954:
948:
932:
926:
920:
917:
911:
908:
655:{\displaystyle f\colon M\to X}
646:
541:
535:
519:
513:
507:
504:
498:
495:
299:{\displaystyle f\colon M\to X}
290:
1:
3890:American Mathematical Society
27:Concept in geometric topology
3886:Surgery on compact manifolds
3817:10.1016/0040-9383(65)90010-8
3646:Sullivan analyzed the cases
2780:Pontrjagin-Thom construction
2247:is there a normal cobordism
871:to the fundamental class of
558:2. Given a Poincaré complex
458:to the fundamental class of
341:, and a stable map from the
234:1. Given a Poincaré complex
2782:the converse is also true.
2527:{\displaystyle f_{0}=f_{1}}
1556:) can be represented by an
135:has a good candidate for a
3958:
3264:The above bijection gives
2004:is there a diffeomorphism
1713:{\displaystyle S^{p}\to X}
1602:) with a null-homotopy of
220:with trivial normal bundle
213:It is possible to perform
202:classes of normal maps on
3782:Browder, William (1972),
3838:Ranicki, Andrew (2002),
3012:{\displaystyle \nu _{X}}
1391:. (Here we suppose that
988:Is the Poincaré complex
984:Consider the question:
364:{\displaystyle \nu _{M}}
43:This article includes a
3884:Wall, C. T. C. (1999),
3824:LĂĽck, Wolfgang (2002),
851:should be mapped under
438:should be mapped under
72:more precise citations.
3766:
3740:
3739:{\displaystyle CAT=PL}
3705:
3671:
3634:
3600:
3569:
3535:
3454:
3382:
3319:
3291:
3251:
3176:
3156:
3155:{\displaystyle B(G/O)}
3116:
3064:
3013:
2986:
2944:
2889:
2863:
2825:
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2772:
2733:
2712:
2663:
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2209:
2160:
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2090:
2044:
1998:
1966:
1917:
1897:
1877:
1854:
1805:
1804:{\displaystyle k>p}
1779:
1714:
1681:
1661:
1641:
1598:(or more generally an
1592:
1550:
1530:
1478:
1450:
1405:
1385:
1305:
1194:
1174:
1150:
1114:
1090:
1089:{\displaystyle M\to X}
1064:
1044:
1024:
1023:{\displaystyle M\to X}
975:Role in surgery theory
961:
885:
865:
845:
822:
802:
782:
739:
719:
696:
682:-dimensional manifold
676:
656:
620:
600:
586:) of formal dimension
572:
548:
472:
452:
432:
405:
385:
365:
331:
310:-dimensional manifold
300:
260:
246:) of formal dimension
169:
123:(more geometrically a
3767:
3741:
3706:
3704:{\displaystyle G/TOP}
3672:
3635:
3633:{\displaystyle G/TOP}
3601:
3570:
3536:
3455:
3383:
3320:
3292:
3252:
3177:
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3117:
3065:
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2713:
2664:
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2529:
2486:
2440:
2397:
2354:
2242:
2240:{\displaystyle i=0,1}
2210:
2161:
2117:
2091:
2045:
1999:
1997:{\displaystyle i=0,1}
1967:
1918:
1898:
1878:
1855:
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823:
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783:
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621:
601:
573:
549:
473:
453:
433:
406:
386:
366:
332:
301:
261:
170:
168:{\displaystyle \geq }
3786:, Berlin, New York:
3750:
3715:
3681:
3670:{\displaystyle G/PL}
3650:
3610:
3599:{\displaystyle G/PL}
3579:
3551:
3464:
3392:
3333:
3301:
3268:
3260:Different categories
3199:
3166:
3129:
3074:
3023:
2996:
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2873:
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2789:
2743:
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2607:
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2106:
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2008:
1976:
1930:
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1887:
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1691:
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1606:
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1540:
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1415:
1395:
1315:
1207:
1184:
1164:
1128:
1104:
1074:
1054:
1034:
1008:
992:of formal dimension
895:
875:
855:
835:
821:{\displaystyle \xi }
812:
792:
752:
729:
718:{\displaystyle \xi }
709:
686:
666:
634:
610:
590:
562:
482:
462:
442:
422:
404:{\displaystyle \xi }
395:
375:
348:
343:stable normal bundle
330:{\displaystyle \xi }
321:
278:
250:
159:
137:stable normal bundle
3765:{\displaystyle TOP}
3568:{\displaystyle G/O}
3318:{\displaystyle G/O}
3122:is null-homotopic.
1462:Whitehead's theorem
1030:from some manifold
189:homotopy equivalent
177:surgery obstruction
127:), a normal map on
3762:
3736:
3701:
3667:
3630:
3596:
3565:
3531:
3450:
3378:
3315:
3287:
3247:
3172:
3152:
3112:
3060:
3009:
2982:
2940:
2885:
2859:
2824:{\displaystyle BO}
2821:
2801:{\displaystyle BG}
2798:
2768:
2729:
2708:
2659:
2626:
2593:
2557:
2524:
2481:
2435:
2392:
2349:
2237:
2205:
2156:
2112:
2086:
2040:
1994:
1962:
1913:
1893:
1873:
1850:
1801:
1775:
1710:
1677:
1657:
1637:
1588:
1546:
1526:
1474:
1446:
1401:
1381:
1301:
1200:and the so-called
1190:
1170:
1146:
1110:
1086:
1060:
1040:
1020:
957:
881:
861:
841:
818:
798:
778:
735:
715:
692:
672:
652:
616:
606:, a normal map on
596:
568:
544:
468:
448:
428:
401:
381:
361:
327:
296:
266:, a normal map on
256:
165:
106:geometric topology
100:In mathematics, a
45:list of references
3917:978-0-8218-0942-6
3869:978-0-19-850924-0
3175:{\displaystyle X}
3036:
2732:{\displaystyle X}
2115:{\displaystyle X}
1916:{\displaystyle X}
1896:{\displaystyle n}
1876:{\displaystyle X}
1817:cohomology theory
1680:{\displaystyle X}
1660:{\displaystyle p}
1549:{\displaystyle f}
1477:{\displaystyle f}
1449:{\displaystyle Z}
1404:{\displaystyle f}
1193:{\displaystyle X}
1173:{\displaystyle M}
1113:{\displaystyle X}
1063:{\displaystyle X}
1043:{\displaystyle M}
884:{\displaystyle X}
864:{\displaystyle f}
844:{\displaystyle M}
801:{\displaystyle M}
738:{\displaystyle X}
695:{\displaystyle M}
675:{\displaystyle n}
662:from some closed
619:{\displaystyle X}
599:{\displaystyle n}
571:{\displaystyle X}
471:{\displaystyle X}
451:{\displaystyle f}
431:{\displaystyle M}
384:{\displaystyle M}
306:from some closed
259:{\displaystyle n}
208:normal invariants
98:
97:
90:
18:Normal invariants
16:(Redirected from
3949:
3928:
3908:10.1090/surv/069
3901:
3880:
3853:
3834:
3832:
3820:
3819:
3798:
3771:
3769:
3768:
3763:
3745:
3743:
3742:
3737:
3710:
3708:
3707:
3702:
3691:
3676:
3674:
3673:
3668:
3660:
3639:
3637:
3636:
3631:
3620:
3605:
3603:
3602:
3597:
3589:
3574:
3572:
3571:
3566:
3561:
3540:
3538:
3537:
3532:
3515:
3486:
3485:
3474:
3473:
3459:
3457:
3456:
3451:
3440:
3411:
3410:
3402:
3401:
3387:
3385:
3384:
3379:
3371:
3342:
3341:
3324:
3322:
3321:
3316:
3311:
3296:
3294:
3293:
3288:
3277:
3276:
3256:
3254:
3253:
3248:
3237:
3208:
3207:
3181:
3179:
3178:
3173:
3161:
3159:
3158:
3153:
3145:
3121:
3119:
3118:
3113:
3105:
3069:
3067:
3066:
3061:
3044:
3043:
3038:
3037:
3029:
3018:
3016:
3015:
3010:
3008:
3007:
2991:
2989:
2988:
2983:
2966:
2965:
2949:
2947:
2946:
2941:
2933:
2894:
2892:
2891:
2886:
2868:
2866:
2865:
2860:
2830:
2828:
2827:
2822:
2807:
2805:
2804:
2799:
2777:
2775:
2774:
2769:
2752:
2751:
2738:
2736:
2735:
2730:
2717:
2715:
2714:
2709:
2698:
2697:
2668:
2666:
2665:
2660:
2649:
2648:
2635:
2633:
2632:
2627:
2616:
2615:
2602:
2600:
2599:
2594:
2583:
2582:
2566:
2564:
2563:
2558:
2547:
2546:
2533:
2531:
2530:
2525:
2523:
2522:
2510:
2509:
2490:
2488:
2487:
2482:
2465:
2464:
2444:
2442:
2441:
2436:
2434:
2433:
2418:
2417:
2401:
2399:
2398:
2393:
2391:
2390:
2375:
2374:
2358:
2356:
2355:
2350:
2303:
2302:
2290:
2289:
2246:
2244:
2243:
2238:
2214:
2212:
2211:
2206:
2198:
2197:
2185:
2184:
2165:
2163:
2162:
2157:
2121:
2119:
2118:
2113:
2095:
2093:
2092:
2087:
2085:
2084:
2066:
2065:
2049:
2047:
2046:
2041:
2039:
2038:
2026:
2025:
2003:
2001:
2000:
1995:
1971:
1969:
1968:
1963:
1955:
1954:
1942:
1941:
1922:
1920:
1919:
1914:
1902:
1900:
1899:
1894:
1882:
1880:
1879:
1874:
1859:
1857:
1856:
1851:
1840:
1839:
1810:
1808:
1807:
1802:
1784:
1782:
1781:
1776:
1765:
1764:
1740:
1739:
1719:
1717:
1716:
1711:
1703:
1702:
1686:
1684:
1683:
1678:
1666:
1664:
1663:
1658:
1646:
1644:
1643:
1638:
1630:
1629:
1597:
1595:
1594:
1589:
1581:
1580:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1516:
1515:
1483:
1481:
1480:
1475:
1455:
1453:
1452:
1447:
1433:
1432:
1410:
1408:
1407:
1402:
1390:
1388:
1387:
1382:
1371:
1370:
1349:
1348:
1327:
1326:
1310:
1308:
1307:
1302:
1288:
1287:
1266:
1265:
1253:
1252:
1219:
1218:
1199:
1197:
1196:
1191:
1179:
1177:
1176:
1171:
1155:
1153:
1152:
1147:
1119:
1117:
1116:
1111:
1095:
1093:
1092:
1087:
1069:
1067:
1066:
1061:
1049:
1047:
1046:
1041:
1029:
1027:
1026:
1021:
966:
964:
963:
958:
947:
946:
907:
906:
890:
888:
887:
882:
870:
868:
867:
862:
850:
848:
847:
842:
827:
825:
824:
819:
807:
805:
804:
799:
787:
785:
784:
779:
777:
776:
764:
763:
744:
742:
741:
736:
724:
722:
721:
716:
701:
699:
698:
693:
681:
679:
678:
673:
661:
659:
658:
653:
625:
623:
622:
617:
605:
603:
602:
597:
584:Poincaré duality
577:
575:
574:
569:
553:
551:
550:
545:
534:
533:
494:
493:
477:
475:
474:
469:
457:
455:
454:
449:
437:
435:
434:
429:
410:
408:
407:
402:
390:
388:
387:
382:
370:
368:
367:
362:
360:
359:
336:
334:
333:
328:
305:
303:
302:
297:
265:
263:
262:
257:
244:Poincaré duality
174:
172:
171:
166:
118:Poincaré complex
104:is a concept in
93:
86:
82:
79:
73:
68:this article by
59:inline citations
38:
37:
30:
21:
3957:
3956:
3952:
3951:
3950:
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3918:
3899:10.1.1.309.8451
3883:
3870:
3851:10.1.1.309.8886
3837:
3830:
3823:
3801:
3788:Springer-Verlag
3781:
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3298:
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3262:
3197:
3196:
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3163:
3127:
3126:
3072:
3071:
3026:
3021:
3020:
2999:
2994:
2993:
2957:
2952:
2951:
2897:
2896:
2871:
2870:
2833:
2832:
2810:
2809:
2787:
2786:
2741:
2740:
2721:
2720:
2687:
2686:
2675:
2673:Homotopy theory
2638:
2637:
2605:
2604:
2572:
2571:
2570:Hence studying
2536:
2535:
2514:
2501:
2496:
2495:
2454:
2453:
2425:
2409:
2404:
2403:
2382:
2366:
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2360:
2294:
2281:
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2248:
2217:
2216:
2189:
2176:
2171:
2170:
2124:
2123:
2104:
2103:
2076:
2057:
2052:
2051:
2030:
2017:
2006:
2005:
1974:
1973:
1946:
1933:
1928:
1927:
1905:
1904:
1885:
1884:
1865:
1864:
1829:
1828:
1825:
1787:
1786:
1756:
1731:
1726:
1725:
1694:
1689:
1688:
1669:
1668:
1649:
1648:
1621:
1604:
1603:
1572:
1561:
1560:
1538:
1537:
1501:
1490:
1489:
1466:
1465:
1424:
1413:
1412:
1393:
1392:
1362:
1340:
1318:
1313:
1312:
1279:
1257:
1244:
1210:
1205:
1204:
1182:
1181:
1162:
1161:
1126:
1125:
1102:
1101:
1072:
1071:
1052:
1051:
1032:
1031:
1006:
1005:
982:
977:
938:
898:
893:
892:
873:
872:
853:
852:
833:
832:
810:
809:
790:
789:
768:
755:
750:
749:
727:
726:
707:
706:
684:
683:
664:
663:
632:
631:
608:
607:
588:
587:
560:
559:
525:
485:
480:
479:
460:
459:
440:
439:
420:
419:
393:
392:
373:
372:
351:
346:
345:
319:
318:
276:
275:
248:
247:
228:
187:actually being
157:
156:
110:William Browder
94:
83:
77:
74:
63:
49:related reading
39:
35:
28:
23:
22:
15:
12:
11:
5:
3955:
3953:
3945:
3944:
3942:Surgery theory
3934:
3933:
3930:
3929:
3916:
3881:
3868:
3835:
3821:
3810:(3): 257–266,
3799:
3777:
3774:
3761:
3758:
3755:
3735:
3732:
3729:
3726:
3723:
3720:
3700:
3697:
3694:
3690:
3686:
3666:
3663:
3659:
3655:
3641:
3640:
3629:
3626:
3623:
3619:
3615:
3595:
3592:
3588:
3584:
3564:
3560:
3556:
3542:
3541:
3530:
3527:
3524:
3521:
3518:
3514:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3484:
3481:
3478:
3472:
3449:
3446:
3443:
3439:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3409:
3406:
3400:
3377:
3374:
3370:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3340:
3314:
3310:
3306:
3286:
3283:
3280:
3275:
3261:
3258:
3246:
3243:
3240:
3236:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3206:
3171:
3151:
3148:
3144:
3140:
3137:
3134:
3111:
3108:
3104:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3059:
3056:
3053:
3050:
3047:
3042:
3035:
3032:
3006:
3002:
2981:
2978:
2975:
2972:
2969:
2964:
2960:
2939:
2936:
2932:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2884:
2881:
2878:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2820:
2817:
2797:
2794:
2767:
2764:
2761:
2758:
2755:
2750:
2728:
2707:
2704:
2701:
2696:
2674:
2671:
2658:
2655:
2652:
2647:
2625:
2622:
2619:
2614:
2592:
2589:
2586:
2581:
2556:
2553:
2550:
2545:
2521:
2517:
2513:
2508:
2504:
2480:
2477:
2474:
2471:
2468:
2463:
2432:
2428:
2424:
2421:
2416:
2412:
2389:
2385:
2381:
2378:
2373:
2369:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2301:
2297:
2293:
2288:
2284:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2236:
2233:
2230:
2227:
2224:
2204:
2201:
2196:
2192:
2188:
2183:
2179:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2111:
2083:
2079:
2075:
2072:
2069:
2064:
2060:
2037:
2033:
2029:
2024:
2020:
2016:
2013:
1993:
1990:
1987:
1984:
1981:
1961:
1958:
1953:
1949:
1945:
1940:
1936:
1912:
1892:
1872:
1849:
1846:
1843:
1838:
1824:
1821:
1813:
1812:
1800:
1797:
1794:
1774:
1771:
1768:
1763:
1759:
1755:
1752:
1749:
1746:
1743:
1738:
1734:
1709:
1706:
1701:
1697:
1676:
1656:
1636:
1633:
1628:
1624:
1620:
1617:
1614:
1611:
1587:
1584:
1579:
1575:
1571:
1568:
1545:
1525:
1522:
1519:
1514:
1511:
1508:
1504:
1500:
1497:
1473:
1458:
1457:
1445:
1442:
1439:
1436:
1431:
1427:
1423:
1420:
1400:
1380:
1377:
1374:
1369:
1365:
1361:
1358:
1355:
1352:
1347:
1343:
1339:
1336:
1333:
1330:
1325:
1321:
1300:
1297:
1294:
1291:
1286:
1282:
1278:
1275:
1272:
1269:
1264:
1260:
1256:
1251:
1247:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1217:
1213:
1202:surgery kernel
1189:
1169:
1145:
1142:
1139:
1136:
1133:
1109:
1085:
1082:
1079:
1059:
1039:
1019:
1016:
1013:
1002:
1001:
981:
978:
976:
973:
969:
968:
956:
953:
950:
945:
941:
937:
934:
931:
928:
925:
922:
919:
916:
913:
910:
905:
901:
880:
860:
840:
829:
817:
797:
775:
771:
767:
762:
758:
747:tangent bundle
734:
714:
703:
691:
671:
651:
648:
645:
642:
639:
615:
595:
567:
556:
555:
543:
540:
537:
532:
528:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
492:
488:
467:
447:
427:
412:
400:
380:
358:
354:
326:
315:
295:
292:
289:
286:
283:
255:
227:
224:
193:Sergei Novikov
164:
125:Poincaré space
114:surgery theory
96:
95:
53:external links
42:
40:
33:
26:
24:
14:
13:
10:
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6:
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181:C. T. C. Wall
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2681:be a finite
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64:Please help
56:
3019:has a lift
206:are called
70:introducing
3776:References
2359:such that
2050:such that
1464:, the map
1311:, that is
1000:-manifold?
580:CW-complex
416:degree one
240:CW-complex
226:Definition
116:. Given a
102:normal map
3894:CiteSeerX
3846:CiteSeerX
3497:≅
3422:≅
3353:≅
3219:≅
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1957:→
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1600:immersion
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317:a bundle
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200:cobordism
163:≥
78:June 2020
3936:Category
3804:Topology
3188:Stasheff
2677:1.' Let
2215:, where
1972:, where
578:(i.e. a
238:(i.e. a
3926:1687388
3878:2061749
3796:0358813
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2452:1.' Is
1722:Whitney
215:surgery
179:due to
108:due to
66:improve
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3876:
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3794:
3746:, and
3460:, and
3388:, and
3184:Gitler
630:a map
274:a map
139:and a
3831:(PDF)
828:, and
725:over
411:, and
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51:, or
3912:ISBN
3864:ISBN
3677:and
3606:and
3186:and
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198:The
141:Thom
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1460:By
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