4888:
4671:
4909:
2134:
4877:
4946:
4919:
4899:
2809:
3208:
1579:, a 1-dimensional CW complex can be constructed in which the 0-cells are the vertices and the 1-cells are the edges of the graph. The endpoints of each edge are identified with the incident vertices to it. This realization of a combinatorial graph as a topological space is sometimes called a
2908:
2624:
3215:
Both of the above examples are particularly simple because the homology is determined by the number of cells—i.e.: the cellular attaching maps have no role in these computations. This is a very special phenomenon and is not indicative of the general case.
3088:
2270:
3556:
1) Adding/removing a generator. Adding a generator, from the perspective of the CW decomposition consists of adding a 1-cell and a 2-cell whose attaching map consists of the new 1-cell and the remainder of the attaching map is in
769:
2976:
1434:
differs from a CW complex in that we allow it to have one extra building block that does not necessarily possess a cellular structure. This extra-block can be treated as a (-1)-dimensional cell in the former definition.
172:
4479:
For example, the opinion "The class of CW complexes (or the class of spaces of the same homotopy type as a CW complex) is the most suitable class of topological spaces in relation to homotopy theory" appears in
432:
1015:
904:
507:
1101:
3781:
3672:
3552:
for group presentations states that there is a sequence of moves we can perform to reduce this group presentation to the trivial presentation of the trivial group. There are two Tietze moves:
2804:{\displaystyle C_{k}={\begin{cases}\mathbb {Z} &k\in \{0,n\}\\0&k\notin \{0,n\}\end{cases}}\quad H_{k}={\begin{cases}\mathbb {Z} &k\in \{0,n\}\\0&k\notin \{0,n\}\end{cases}}}
1159:
3077:
2825:
277:
4124:
This textbook defines CW complexes in the first chapter and uses them throughout; includes an appendix on the topology of CW complexes. A free electronic version is available on the
4145:
1804:
4018:
3847:
3707:
1890:
1833:
587:
940:
826:
3336:
3203:{\displaystyle H^{k}\left(\mathbb {P} ^{n}(\mathbb {C} )\right)={\begin{cases}\mathbb {Z} &0\leqslant k\leqslant 2n,{\text{ even}}\\0&{\text{otherwise}}\end{cases}}}
1649:
1376:
712:
2043:
3891:
3611:
1855:
1746:
653:
617:
3420:
3366:
3270:
2190:
3948:
3296:
1972:
360:
232:
3033:
2579:
1265:
3922:
3812:
3582:
3542:
3515:
3488:
3461:
3003:
2613:
2011:
1939:
1304:
680:
537:
327:
199:
1711:
1682:
3974:
3425:
Another way of stating the above is that a connected CW complex can be replaced by a homotopy-equivalent CW complex whose 0-skeleton consists of a single point.
2283:
has no CW decomposition, because it is not locally contractible at origin. It is also not homotopy equivalent to a CW complex, because it has no good open cover.
1378:
is indeterminate, one takes the direct limit topology, since the diagram is highly suggestive of a direct limit. This turns out to have great technical benefits.
3386:
1328:
1035:
703:
297:
109:
2917:
4024:. i.e.: one can similarly realize elementary matrix operations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.
4949:
2300:
to a CW complex, then it has a good open cover. A good open cover is an open cover, such that every nonempty finite intersection is contractible.
1226:" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). The topology of the CW complex is the topology of the
4449:
4044:
is actually used). Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the
3976:
case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for
4261:
4153:
943:
365:
948:
831:
441:
2181:-skeletons, each of which being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.
2541:
1043:
4547:
3224:
There is a technique, developed by
Whitehead, for replacing a CW complex with a homotopy-equivalent CW complex that has a
2492:
3725:
3616:
3432:
is a simply-connected CW complex whose 0-skeleton consists of a point. Can we, through suitable modifications, replace
4583:
2318:: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups.
2903:{\displaystyle C_{k}={\begin{cases}\mathbb {Z} ^{2}&0\leqslant k\leqslant n\\0&{\text{otherwise}}\end{cases}}}
4937:
4932:
4524:
4370:
4294:
4117:
1113:
3042:
2273:
4927:
237:
4483:
4049:
2510:
1986:-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere"). Inductively, this gives
3231:
Consider, for example, an arbitrary CW complex. Its 1-skeleton can be fairly complicated, being an arbitrary
1414:
is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every
4829:
4539:
4493:
2272:
is homotopy equivalent to a CW complex (the point) but it does not admit a CW decomposition, since it is not
1751:
86:, but still retain a combinatorial nature that allows for computation (often with a much smaller complex).
3979:
3232:
17:
2532:. Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as a
4980:
4970:
4488:
4192:
1341:
can be constructed by repeating the above process countably many times. Since the topology of the union
4837:
4109:
3817:
3677:
2537:
1866:
1809:
771:
The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:
549:
2504:
2433:
1227:
1215:
913:
799:
4456:
3305:
3143:
2847:
2733:
2646:
1610:
1344:
4975:
4636:
4516:
2397:
2016:
1600:
1503:
is the 1-cell. Alternatively, it can be constructed just from a single interval, with no 0-cells.
1415:
3867:
3587:
3242:
in this graph. Since it is a collection of trees, and trees are contractible, consider the space
2978:
This gives the same homology computation above, as the chain complex is exact at all terms except
2265:{\displaystyle \{re^{2\pi i\theta }:0\leq r\leq 1,\theta \in \mathbb {Q} \}\subseteq \mathbb {C} }
1838:
1729:
4922:
4908:
4257:
4067:
3545:
2144:
2086:
625:
596:
4857:
4778:
4655:
4643:
4616:
4576:
3549:
3395:
3341:
3245:
4852:
3718:
2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing
2376:, as expected. In addition, the weak topology on this set often agrees with the more familiar
2108:
has a canonical CW decomposition with only one 0-cell (the compactification point) called the
4699:
4626:
4061:
4045:
3927:
3275:
2460:
2409:
2101:
1944:
332:
204:
4536:
Nonabelian
Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids
3008:
2554:
1244:
4847:
4799:
4773:
4621:
4176:
4071:
4036:
of CW complexes is, in the opinion of some experts, the best if not the only candidate for
3900:
3790:
3560:
3520:
3493:
3466:
3439:
2981:
2591:
1989:
1917:
1277:
658:
515:
305:
177:
59:
4240:
1687:
1658:
1195:
The CW complex construction is a straightforward generalization of the following process:
8:
4694:
3953:
3236:
2525:
2297:
2105:
1403:
4898:
4267:
2820:
Alternatively, if we use the equatorial decomposition with two cells in every dimension
4892:
4862:
4842:
4763:
4753:
4631:
4611:
4431:
4341:
4289:. De Gruyter Studies in Mathematics. Vol. 18. Berlin: Walter de Gruyter & Co.
4141:
3371:
1313:
1268:
1020:
789:
764:{\displaystyle X_{-1}\hookrightarrow X_{0}\hookrightarrow X_{1}\hookrightarrow \cdots }
688:
300:
282:
83:
71:
67:
63:
4426:
4409:
3787:-cell has an attaching map that consists of the new 2-cell and remainder mapping into
2971:{\displaystyle \left({\begin{smallmatrix}1&-1\\1&-1\end{smallmatrix}}\right).}
1914:. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell
1161:
is covered by a finite number of closed cells, each having cell dimension less than k.
4887:
4880:
4746:
4704:
4569:
4543:
4520:
4366:
4333:
4290:
4113:
4033:
4021:
3858:
2529:
2521:
2365:
2315:
2090:
1331:
1223:
51:
4912:
4167:
1391:
is a CW complex whose gluing maps are homeomorphisms. Accordingly, the partition of
4660:
4606:
4421:
4325:
4162:
3463:
consists of a single point? The answer is yes. The first step is to observe that
2377:
2280:
2054:
1591:
4719:
4714:
4172:
2533:
2448:
2328:
The product of two CW complexes can be made into a CW complex. Specifically, if
1859:
1586:
1407:
782:
79:
75:
4902:
2927:
2311:. A compact subspace of a CW complex is always contained in a finite subcomplex.
4809:
4741:
4075:
2322:
2185:
1449:
1204:
1599:
is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from a
4964:
4819:
4729:
4709:
4385:
4358:
4337:
4101:
4041:
2586:
2353:
2308:
2170:
2094:
1219:
1104:
435:
3368:
naturally inherits a CW structure, with cells corresponding to the cells of
4804:
4724:
4670:
4226:
2076:
706:
167:{\displaystyle \emptyset =X_{-1}\subset X_{0}\subset X_{1}\subset \cdots }
4814:
4555:
4405:
2581:
take the cell decomposition with two cells: a single 0-cell and a single
2480:
2304:
2174:
31:
2133:
89:
The C in CW stands for "closure-finite", and the W for "weak" topology.
4689:
4648:
4435:
4393:
4374:
4345:
4125:
2507:
variant of the compact-open topology; the above statements remain true.
2070:
1576:
1171:
106:
is constructed by taking the union of a sequence of topological spaces
4783:
4312:
4329:
4768:
4736:
4685:
4592:
1907:
1651:. This map can be perturbed to be disjoint from the 0-skeleton of
1419:
1307:
427:{\displaystyle g_{\alpha }^{k}:\partial e_{\alpha }^{k}\to X_{k-1}}
35:
4268:"1.3 Introduction to Algebraic Topology. Examples of CW Complexes"
2436:
agrees with the weak topology and therefore defines a CW complex.
2117:
2013:
a CW decomposition with two cells in every dimension k such that
1979:
1974:
to the single 0-cell. An alternative cell decomposition has one (
1411:
1010:{\displaystyle g_{\alpha }^{k}:D^{k}\to {\bar {e}}_{\alpha }^{k}}
899:{\displaystyle {\bar {e}}_{\alpha }^{k}:=cl_{X}(e_{\alpha }^{k})}
58:) of different dimensions in specific ways. It generalizes both
1274:. For each copy, there is a map that "glues" its boundary (the
1835:
has cubical cells that are products of the 0 and 1-cells from
1406:
graph is represented by a regular 1-dimensional CW-complex. A
54:
that is built by gluing together topological balls (so-called
4048:
on the homotopy category have a simple characterisation (the
3298:
if they are contained in a common tree in the maximal forest
1218:
of a 0-dimensional CW complex with one or more copies of the
502:{\displaystyle X_{k}=X_{k-1}\sqcup _{\alpha }e_{\alpha }^{k}}
2062:
The terminology for a generic 2-dimensional CW complex is a
1330:-dimensional complex. The topology of the CW complex is the
4561:
3196:
2896:
2797:
2710:
2293:
CW complexes are locally contractible (Hatcher, prop. A.4).
2516:
4040:
homotopy category (for technical reasons the version for
2479:) is homotopy equivalent to a CW complex by a theorem of
2059:
It admits a CW structure with one cell in each dimension.
1174:
if and only if it meets each closed cell in a closed set.
620:
2177:
and therefore cannot be written as a countable union of
1203:
is just a set of zero or more discrete points (with the
1096:{\displaystyle g_{\alpha }^{k}:B^{k}\to e_{\alpha }^{k}}
1529:. Alternatively, it can be constructed from two points
828:, each with a corresponding closure (or "closed cell")
1941:
is attached by the constant mapping from its boundary
1901:
Some examples of finite-dimensional CW complexes are:
4200:. International Workshop on Combinatorial Algorithms.
3982:
3956:
3930:
3903:
3870:
3820:
3793:
3728:
3680:
3619:
3590:
3563:
3523:
3496:
3469:
3442:
3398:
3374:
3344:
3308:
3278:
3248:
3091:
3045:
3011:
2984:
2920:
2828:
2627:
2594:
2557:
2193:
2019:
1992:
1947:
1920:
1869:
1841:
1812:
1754:
1732:
1690:
1661:
1613:
1347:
1316:
1280:
1247:
1116:
1046:
1023:
951:
916:
834:
802:
715:
691:
661:
628:
599:
552:
518:
444:
368:
335:
308:
285:
240:
207:
180:
112:
4410:"On spaces having the homotopy type of a CW-complex"
4313:"On Spaces Having the Homotopy Type of a CW-Complex"
3428:
Consider climbing up the connectivity ladder—assume
2124:
685:
In the language of category theory, the topology on
2116:and are used in popular computer software, such as
1726:on the real numbers has as 0-skeleton the integers
788:is homeomorphic to a CW complex iff there exists a
4533:
4311:
4215:. Providence, R.I.: American Mathematical Society.
4012:
3968:
3942:
3916:
3885:
3841:
3806:
3776:{\displaystyle {\tilde {X}}=X\cup e^{2}\cup e^{3}}
3775:
3701:
3667:{\displaystyle {\tilde {X}}=X\cup e^{1}\cup e^{2}}
3666:
3605:
3576:
3536:
3509:
3482:
3455:
3414:
3380:
3360:
3330:
3290:
3264:
3202:
3071:
3027:
2997:
2970:
2902:
2803:
2607:
2573:
2400:than the product topology, for example if neither
2264:
2037:
2005:
1966:
1933:
1896:
1884:
1849:
1827:
1798:
1740:
1705:
1676:
1643:
1370:
1322:
1298:
1259:
1153:
1095:
1029:
1009:
934:
898:
820:
763:
697:
674:
647:
611:
581:
531:
501:
426:
354:
321:
291:
271:
226:
193:
166:
4318:Transactions of the American Mathematical Society
2336:are CW complexes, then one can form a CW complex
2112:. Such cell decompositions are frequently called
1460:Some examples of 1-dimensional CW complexes are:
1418:is the 1-skeleton of a regular CW-complex on the
1237:is constructed by taking the disjoint union of a
4962:
4373:. A free electronic version is available on the
3814:. A similar slide gives a homotopy-equivalence
3219:
2424:a CW complex. On the other hand, the product of
4510:
3272:where the equivalence relation is generated by
2914:and the differentials are matrices of the form
1154:{\displaystyle g_{\alpha }^{k}(\partial D^{k})}
1110:(closure-finiteness) The image of the boundary
4066:The notion of CW complex has an adaptation to
3924:consists of a single point. The argument for
3864:one can find a homotopy-equivalent CW complex
3072:{\displaystyle \mathbb {P} ^{n}(\mathbb {C} )}
2396:is finite. However, the weak topology can be
1595:1-dimensional CW complexes. Specifically, if
4577:
4534:Brown, R.; Higgins, P.J.; Sivera, R. (2011).
4154:Bulletin of the American Mathematical Society
4020:(using the presentation matrices coming from
2344:in which each cell is a product of a cell in
1455:
1443:
1190:
2791:
2779:
2761:
2749:
2704:
2692:
2674:
2662:
2251:
2194:
1793:
1755:
1632:
1620:
1509:. It can be constructed from a single point
4287:Quantum invariants of knots and 3-manifolds
4194:The 3-Sphere Regular Cellulation Conjecture
4190:
272:{\displaystyle (e_{\alpha }^{k})_{\alpha }}
4945:
4918:
4584:
4570:
4241:"CW-complex - Encyclopedia of Mathematics"
4140:
4027:
3436:by a homotopy-equivalent CW complex where
3422:is a disjoint union of wedges of circles.
2528:of CW complexes is readily computable via
1806:. Similarly, the standard CW structure on
4519:University Series in Higher Mathematics.
4425:
4210:
4166:
4003:
3147:
3123:
3109:
3062:
3048:
2852:
2737:
2650:
2412:. In this unfavorable case, the product
2258:
2247:
1872:
1843:
1815:
1789:
1734:
1479:(an interval), such that one endpoint of
1467:. It can be constructed from two points (
1425:
4481:
4392:, preliminary version available on the
4265:
4100:
2517:Homology and cohomology of CW complexes
1382:
14:
4963:
4404:
4398:
4309:
4284:
3338:is a homotopy equivalence. Moreover,
2544:is the analogue of cellular homology.
2079:manifolds admit a CW structure called
1799:{\displaystyle \{:n\in \mathbb {Z} \}}
1222:. For each copy, there is a map that "
4565:
4511:Lundell, A. T.; Weingram, S. (1970).
4503:
4365:, Cambridge University Press (2002).
4013:{\displaystyle H_{n}(X;\mathbb {Z} )}
3709:given by sliding the new 2-cell into
3674:then there is a homotopy equivalence
2815:since all the differentials are zero.
2325:of a CW complex is also a CW complex.
4252:
4250:
4211:Davis, James F.; Kirk, Paul (2001).
4136:
4134:
3490:and the attaching maps to construct
2493:compactly generated Hausdorff spaces
2128:
66:and has particular significance for
4213:Lecture Notes in Algebraic Topology
3392:. In particular, the 1-skeleton of
2542:Atiyah–Hirzebruch spectral sequence
2163:
1037:-dimensional closed ball such that
24:
1241:-dimensional CW complex (for some
1135:
387:
113:
25:
4992:
4427:10.1090/s0002-9947-1959-0100267-4
4247:
4131:
3842:{\displaystyle {\tilde {X}}\to X}
3702:{\displaystyle {\tilde {X}}\to X}
2926:
2538:extraordinary (co)homology theory
2125:Infinite-dimensional CW complexes
1499:are the 0-cells; the interior of
1267:) with one or more copies of the
1040:The restriction to the open ball
70:. It was initially introduced by
4944:
4917:
4907:
4897:
4886:
4876:
4875:
4669:
3613:be the corresponding CW complex
2132:
1885:{\displaystyle \mathbb {R} ^{n}}
1828:{\displaystyle \mathbb {R} ^{n}}
582:{\displaystyle X=\cup _{k}X_{k}}
279:, each homeomorphic to the open
4473:
4442:
4379:
4352:
4266:channel, Animated Math (2020).
4168:10.1090/S0002-9904-1949-09175-9
2714:
2114:ideal polyhedral decompositions
1897:Finite-dimensional CW complexes
1452:is a 0-dimensional CW complex.
1339:infinite-dimensional CW complex
935:{\displaystyle e_{\alpha }^{k}}
821:{\displaystyle e_{\alpha }^{k}}
4542:Tracts in Mathematics Vol 15.
4310:Milnor, John (February 1959).
4303:
4278:
4233:
4219:
4204:
4184:
4094:
4074:, which is closely related to
4050:Brown representability theorem
4007:
3993:
3877:
3833:
3827:
3735:
3693:
3687:
3626:
3597:
3331:{\displaystyle X\to X/{\sim }}
3312:
3127:
3119:
3066:
3058:
2511:Cellular approximation theorem
1776:
1758:
1713:are not 0-valence vertices of
1700:
1694:
1671:
1665:
1644:{\displaystyle f:\{0,1\}\to X}
1635:
1475:), and the 1-dimensional ball
1371:{\displaystyle \cup _{k}X_{k}}
1293:
1281:
1148:
1132:
1075:
990:
980:
893:
875:
842:
755:
742:
729:
405:
260:
241:
13:
1:
4540:European Mathematical Society
4082:
3220:Modification of CW structures
2585:-cell. The cellular homology
2287:
2173:is not a CW complex: it is a
2038:{\displaystyle 0\leq k\leq n}
1978:-1)-dimensional sphere (the "
1748:and as 1-cells the intervals
1545:, such that the endpoints of
1408:closed 2-cell graph embedding
1334:defined by these gluing maps.
1230:defined by these gluing maps.
1214:is constructed by taking the
97:
92:
4591:
4513:The topology of CW complexes
4450:"Compactly Generated Spaces"
4191:De Agostino, Sergio (2016).
4146:"Combinatorial homotopy. I."
3886:{\displaystyle {\tilde {X}}}
3606:{\displaystyle {\tilde {X}}}
2467:CW complexes in general. If
2110:Epstein–Penner Decomposition
1850:{\displaystyle \mathbb {R} }
1741:{\displaystyle \mathbb {Z} }
1537:and two 1-dimensional balls
1166:(weak topology) A subset of
655:is open for each k-skeleton
234:by gluing copies of k-cells
7:
4489:Encyclopedia of Mathematics
4390:Vector bundles and K-theory
4055:
2447:be CW complexes. Then the
2420:in the product topology is
2089:, algebraic and projective
1513:and the 1-dimensional ball
1438:
648:{\displaystyle U\cap X_{k}}
434:. The maps are also called
78:. CW complexes have better
10:
4997:
4838:Banach fixed-point theorem
4110:Cambridge University Press
3388:that are not contained in
2615:and homology are given by:
2503:) is often taken with the
2434:compactly generated spaces
2307:. Finite CW complexes are
2102:one-point compactification
2073:is naturally a CW complex.
1487:and the other is glued to
1456:1-dimensional CW complexes
1450:discrete topological space
1444:0-dimensional CW complexes
1191:The construction, in words
612:{\displaystyle U\subset X}
362:by continuous gluing maps
4871:
4828:
4792:
4678:
4667:
4599:
4557:first author's home page]
3415:{\displaystyle X/{\sim }}
3361:{\displaystyle X/{\sim }}
3265:{\displaystyle X/{\sim }}
3235:. Now consider a maximal
2314:CW complexes satisfy the
27:Type of topological space
4087:
2388:, for example if either
2356:. The underlying set of
2169:An infinite-dimensional
2087:Differentiable manifolds
1235:n-dimensional CW complex
4482:Baladze, D.O. (2001) ,
4028:'The' homotopy category
3943:{\displaystyle n\geq 2}
3291:{\displaystyle x\sim y}
1967:{\displaystyle S^{n-1}}
1857:. This is the standard
1557:, and the endpoints of
355:{\displaystyle X_{k-1}}
227:{\displaystyle X_{k-1}}
4893:Mathematics portal
4793:Metrics and properties
4779:Second-countable space
4414:Trans. Amer. Math. Soc
4285:Turaev, V. G. (1994).
4046:representable functors
4014:
3970:
3944:
3918:
3887:
3843:
3808:
3777:
3703:
3668:
3607:
3578:
3538:
3511:
3484:
3457:
3416:
3382:
3362:
3332:
3292:
3266:
3204:
3073:
3029:
3028:{\displaystyle C_{n}.}
2999:
2972:
2904:
2805:
2609:
2575:
2574:{\displaystyle S^{n},}
2540:for a CW complex, the
2266:
2141:This section is empty.
2039:
2007:
1968:
1935:
1886:
1851:
1829:
1800:
1742:
1707:
1678:
1645:
1372:
1324:
1300:
1261:
1260:{\displaystyle k<n}
1212:dimensional CW complex
1201:dimensional CW complex
1155:
1097:
1031:
1011:
936:
900:
822:
765:
699:
676:
649:
613:
583:
533:
503:
428:
356:
323:
293:
273:
228:
195:
168:
34:, and specifically in
4062:Abstract cell complex
4015:
3971:
3945:
3919:
3917:{\displaystyle X^{n}}
3888:
3844:
3809:
3807:{\displaystyle X^{2}}
3778:
3704:
3669:
3608:
3579:
3577:{\displaystyle X^{1}}
3539:
3537:{\displaystyle X^{1}}
3512:
3510:{\displaystyle X^{2}}
3485:
3483:{\displaystyle X^{1}}
3458:
3456:{\displaystyle X^{1}}
3417:
3383:
3363:
3333:
3293:
3267:
3205:
3074:
3030:
3000:
2998:{\displaystyle C_{0}}
2973:
2905:
2806:
2610:
2608:{\displaystyle C_{*}}
2576:
2461:compact-open topology
2267:
2040:
2008:
2006:{\displaystyle S^{n}}
1969:
1936:
1934:{\displaystyle D^{n}}
1887:
1852:
1830:
1801:
1743:
1724:standard CW structure
1708:
1679:
1646:
1589:can be considered as
1426:Relative CW complexes
1373:
1325:
1310:) to elements of the
1301:
1299:{\displaystyle (n-1)}
1262:
1156:
1098:
1032:
1012:
944:continuous surjection
937:
901:
823:
766:
700:
677:
675:{\displaystyle X_{k}}
650:
614:
584:
534:
532:{\displaystyle X_{k}}
504:
429:
357:
324:
322:{\displaystyle B^{k}}
294:
274:
229:
196:
194:{\displaystyle X_{k}}
169:
74:to meet the needs of
4848:Invariance of domain
4800:Euler characteristic
4774:Bundle (mathematics)
4554:More details on the
4227:"CW complex in nLab"
4072:handle decomposition
3980:
3954:
3928:
3901:
3868:
3818:
3791:
3726:
3678:
3617:
3588:
3561:
3521:
3494:
3467:
3440:
3396:
3372:
3342:
3306:
3302:. The quotient map
3276:
3246:
3089:
3043:
3009:
2982:
2918:
2826:
2625:
2592:
2555:
2274:locally contractible
2191:
2017:
1990:
1945:
1918:
1867:
1839:
1810:
1752:
1730:
1706:{\displaystyle f(1)}
1688:
1677:{\displaystyle f(0)}
1659:
1611:
1430:Roughly speaking, a
1420:3-dimensional sphere
1383:Regular CW complexes
1345:
1314:
1278:
1245:
1114:
1044:
1021:
949:
914:
832:
800:
713:
689:
659:
626:
597:
550:
516:
442:
366:
333:
306:
283:
238:
205:
178:
110:
84:simplicial complexes
64:simplicial complexes
4858:Tychonoff's theorem
4853:Poincaré conjecture
4607:General (point-set)
4142:Whitehead, J. H. C.
3969:{\displaystyle n=1}
2505:compactly generated
2483:(1959). Note that
2471:is finite then Hom(
2432:in the category of
2352:, endowed with the
2298:homotopy equivalent
2106:hyperbolic manifold
1911:-dimensional sphere
1432:relative CW complex
1397:regular cellulation
1131:
1092:
1061:
1006:
966:
931:
892:
858:
817:
779: —
498:
404:
383:
258:
4981:Topological spaces
4971:Algebraic topology
4843:De Rham cohomology
4764:Polyhedral complex
4754:Simplicial complex
4504:General references
4363:Algebraic topology
4106:Algebraic topology
4010:
3966:
3950:is similar to the
3940:
3914:
3883:
3839:
3804:
3773:
3699:
3664:
3603:
3574:
3546:group presentation
3534:
3507:
3480:
3453:
3412:
3378:
3358:
3328:
3288:
3262:
3228:CW decomposition.
3200:
3195:
3069:
3025:
2995:
2968:
2959:
2958:
2900:
2895:
2801:
2796:
2709:
2605:
2571:
2262:
2053:-dimensional real
2035:
2003:
1964:
1931:
1882:
1863:cell structure on
1847:
1825:
1796:
1738:
1703:
1674:
1641:
1389:regular CW complex
1368:
1320:
1296:
1257:
1179:This partition of
1151:
1117:
1093:
1078:
1047:
1027:
1007:
983:
952:
932:
917:
896:
878:
835:
818:
803:
796:into "open cells"
777:
761:
695:
672:
645:
609:
579:
529:
499:
484:
424:
390:
369:
352:
319:
289:
269:
244:
224:
191:
164:
72:J. H. C. Whitehead
68:algebraic topology
4958:
4957:
4747:fundamental group
4549:978-3-03719-083-8
4394:author's homepage
4375:author's homepage
4126:author's homepage
4034:homotopy category
4022:cellular homology
3880:
3830:
3738:
3690:
3629:
3600:
3381:{\displaystyle X}
3191:
3177:
2891:
2530:cellular homology
2522:Singular homology
2366:Cartesian product
2316:Whitehead theorem
2303:CW complexes are
2161:
2160:
1581:topological graph
1491:. The two points
1416:2-connected graph
1395:is also called a
1332:quotient topology
1323:{\displaystyle k}
1272:-dimensional ball
1205:discrete topology
1183:is also called a
1030:{\displaystyle k}
993:
942:, there exists a
845:
775:
698:{\displaystyle X}
438:. Thus as a set,
292:{\displaystyle k}
201:is obtained from
52:topological space
16:(Redirected from
4988:
4948:
4947:
4921:
4920:
4911:
4901:
4891:
4890:
4879:
4878:
4673:
4586:
4579:
4572:
4563:
4562:
4553:
4530:
4497:
4496:
4477:
4471:
4470:
4468:
4467:
4461:
4455:. Archived from
4454:
4446:
4440:
4439:
4429:
4402:
4396:
4383:
4377:
4356:
4350:
4349:
4315:
4307:
4301:
4300:
4282:
4276:
4275:
4254:
4245:
4244:
4237:
4231:
4230:
4223:
4217:
4216:
4208:
4202:
4201:
4199:
4188:
4182:
4180:
4170:
4150:
4138:
4129:
4123:
4098:
4068:smooth manifolds
4019:
4017:
4016:
4011:
4006:
3992:
3991:
3975:
3973:
3972:
3967:
3949:
3947:
3946:
3941:
3923:
3921:
3920:
3915:
3913:
3912:
3892:
3890:
3889:
3884:
3882:
3881:
3873:
3853:If a CW complex
3848:
3846:
3845:
3840:
3832:
3831:
3823:
3813:
3811:
3810:
3805:
3803:
3802:
3782:
3780:
3779:
3774:
3772:
3771:
3759:
3758:
3740:
3739:
3731:
3708:
3706:
3705:
3700:
3692:
3691:
3683:
3673:
3671:
3670:
3665:
3663:
3662:
3650:
3649:
3631:
3630:
3622:
3612:
3610:
3609:
3604:
3602:
3601:
3593:
3583:
3581:
3580:
3575:
3573:
3572:
3543:
3541:
3540:
3535:
3533:
3532:
3516:
3514:
3513:
3508:
3506:
3505:
3489:
3487:
3486:
3481:
3479:
3478:
3462:
3460:
3459:
3454:
3452:
3451:
3421:
3419:
3418:
3413:
3411:
3406:
3387:
3385:
3384:
3379:
3367:
3365:
3364:
3359:
3357:
3352:
3337:
3335:
3334:
3329:
3327:
3322:
3297:
3295:
3294:
3289:
3271:
3269:
3268:
3263:
3261:
3256:
3209:
3207:
3206:
3201:
3199:
3198:
3192:
3189:
3178:
3175:
3150:
3134:
3130:
3126:
3118:
3117:
3112:
3101:
3100:
3079:we get similarly
3078:
3076:
3075:
3070:
3065:
3057:
3056:
3051:
3034:
3032:
3031:
3026:
3021:
3020:
3004:
3002:
3001:
2996:
2994:
2993:
2977:
2975:
2974:
2969:
2964:
2960:
2909:
2907:
2906:
2901:
2899:
2898:
2892:
2889:
2861:
2860:
2855:
2838:
2837:
2810:
2808:
2807:
2802:
2800:
2799:
2740:
2724:
2723:
2713:
2712:
2653:
2637:
2636:
2614:
2612:
2611:
2606:
2604:
2603:
2580:
2578:
2577:
2572:
2567:
2566:
2551:For the sphere,
2536:. To compute an
2378:product topology
2281:Hawaiian earring
2271:
2269:
2268:
2263:
2261:
2250:
2218:
2217:
2164:Non CW-complexes
2156:
2153:
2143:You can help by
2136:
2129:
2097:of CW complexes.
2055:projective space
2044:
2042:
2041:
2036:
2012:
2010:
2009:
2004:
2002:
2001:
1973:
1971:
1970:
1965:
1963:
1962:
1940:
1938:
1937:
1932:
1930:
1929:
1891:
1889:
1888:
1883:
1881:
1880:
1875:
1856:
1854:
1853:
1848:
1846:
1834:
1832:
1831:
1826:
1824:
1823:
1818:
1805:
1803:
1802:
1797:
1792:
1747:
1745:
1744:
1739:
1737:
1712:
1710:
1709:
1704:
1683:
1681:
1680:
1675:
1650:
1648:
1647:
1642:
1587:3-regular graphs
1377:
1375:
1374:
1369:
1367:
1366:
1357:
1356:
1329:
1327:
1326:
1321:
1305:
1303:
1302:
1297:
1266:
1264:
1263:
1258:
1160:
1158:
1157:
1152:
1147:
1146:
1130:
1125:
1102:
1100:
1099:
1094:
1091:
1086:
1074:
1073:
1060:
1055:
1036:
1034:
1033:
1028:
1016:
1014:
1013:
1008:
1005:
1000:
995:
994:
986:
979:
978:
965:
960:
941:
939:
938:
933:
930:
925:
906:that satisfies:
905:
903:
902:
897:
891:
886:
874:
873:
857:
852:
847:
846:
838:
827:
825:
824:
819:
816:
811:
780:
770:
768:
767:
762:
754:
753:
741:
740:
728:
727:
704:
702:
701:
696:
681:
679:
678:
673:
671:
670:
654:
652:
651:
646:
644:
643:
618:
616:
615:
610:
588:
586:
585:
580:
578:
577:
568:
567:
546:The topology of
543:of the complex.
538:
536:
535:
530:
528:
527:
508:
506:
505:
500:
497:
492:
483:
482:
473:
472:
454:
453:
433:
431:
430:
425:
423:
422:
403:
398:
382:
377:
361:
359:
358:
353:
351:
350:
328:
326:
325:
320:
318:
317:
298:
296:
295:
290:
278:
276:
275:
270:
268:
267:
257:
252:
233:
231:
230:
225:
223:
222:
200:
198:
197:
192:
190:
189:
173:
171:
170:
165:
157:
156:
144:
143:
131:
130:
82:properties than
44:cellular complex
21:
4996:
4995:
4991:
4990:
4989:
4987:
4986:
4985:
4976:Homotopy theory
4961:
4960:
4959:
4954:
4885:
4867:
4863:Urysohn's lemma
4824:
4788:
4674:
4665:
4637:low-dimensional
4595:
4590:
4560:
4550:
4527:
4506:
4501:
4500:
4478:
4474:
4465:
4463:
4459:
4452:
4448:
4447:
4443:
4403:
4399:
4384:
4380:
4357:
4353:
4330:10.2307/1993204
4308:
4304:
4297:
4283:
4279:
4262:Wayback Machine
4255:
4248:
4239:
4238:
4234:
4225:
4224:
4220:
4209:
4205:
4197:
4189:
4185:
4148:
4139:
4132:
4120:
4099:
4095:
4090:
4085:
4058:
4030:
4002:
3987:
3983:
3981:
3978:
3977:
3955:
3952:
3951:
3929:
3926:
3925:
3908:
3904:
3902:
3899:
3898:
3872:
3871:
3869:
3866:
3865:
3822:
3821:
3819:
3816:
3815:
3798:
3794:
3792:
3789:
3788:
3767:
3763:
3754:
3750:
3730:
3729:
3727:
3724:
3723:
3682:
3681:
3679:
3676:
3675:
3658:
3654:
3645:
3641:
3621:
3620:
3618:
3615:
3614:
3592:
3591:
3589:
3586:
3585:
3568:
3564:
3562:
3559:
3558:
3528:
3524:
3522:
3519:
3518:
3501:
3497:
3495:
3492:
3491:
3474:
3470:
3468:
3465:
3464:
3447:
3443:
3441:
3438:
3437:
3407:
3402:
3397:
3394:
3393:
3373:
3370:
3369:
3353:
3348:
3343:
3340:
3339:
3323:
3318:
3307:
3304:
3303:
3277:
3274:
3273:
3257:
3252:
3247:
3244:
3243:
3222:
3194:
3193:
3188:
3186:
3180:
3179:
3174:
3151:
3146:
3139:
3138:
3122:
3113:
3108:
3107:
3106:
3102:
3096:
3092:
3090:
3087:
3086:
3061:
3052:
3047:
3046:
3044:
3041:
3040:
3016:
3012:
3010:
3007:
3006:
2989:
2985:
2983:
2980:
2979:
2957:
2956:
2948:
2942:
2941:
2933:
2925:
2921:
2919:
2916:
2915:
2894:
2893:
2888:
2886:
2880:
2879:
2862:
2856:
2851:
2850:
2843:
2842:
2833:
2829:
2827:
2824:
2823:
2795:
2794:
2771:
2765:
2764:
2741:
2736:
2729:
2728:
2719:
2715:
2708:
2707:
2684:
2678:
2677:
2654:
2649:
2642:
2641:
2632:
2628:
2626:
2623:
2622:
2599:
2595:
2593:
2590:
2589:
2562:
2558:
2556:
2553:
2552:
2547:Some examples:
2534:homology theory
2519:
2449:function spaces
2410:locally compact
2290:
2257:
2246:
2204:
2200:
2192:
2189:
2188:
2166:
2157:
2151:
2148:
2127:
2018:
2015:
2014:
1997:
1993:
1991:
1988:
1987:
1952:
1948:
1946:
1943:
1942:
1925:
1921:
1919:
1916:
1915:
1899:
1876:
1871:
1870:
1868:
1865:
1864:
1842:
1840:
1837:
1836:
1819:
1814:
1813:
1811:
1808:
1807:
1788:
1753:
1750:
1749:
1733:
1731:
1728:
1727:
1689:
1686:
1685:
1660:
1657:
1656:
1655:if and only if
1612:
1609:
1608:
1601:two-point space
1458:
1446:
1441:
1428:
1385:
1362:
1358:
1352:
1348:
1346:
1343:
1342:
1315:
1312:
1311:
1279:
1276:
1275:
1246:
1243:
1242:
1233:In general, an
1193:
1177:
1142:
1138:
1126:
1121:
1115:
1112:
1111:
1087:
1082:
1069:
1065:
1056:
1051:
1045:
1042:
1041:
1022:
1019:
1018:
1001:
996:
985:
984:
974:
970:
961:
956:
950:
947:
946:
926:
921:
915:
912:
911:
887:
882:
869:
865:
853:
848:
837:
836:
833:
830:
829:
812:
807:
801:
798:
797:
783:Hausdorff space
778:
749:
745:
736:
732:
720:
716:
714:
711:
710:
709:of the diagram
690:
687:
686:
666:
662:
660:
657:
656:
639:
635:
627:
624:
623:
598:
595:
594:
573:
569:
563:
559:
551:
548:
547:
523:
519:
517:
514:
513:
493:
488:
478:
474:
462:
458:
449:
445:
443:
440:
439:
412:
408:
399:
394:
378:
373:
367:
364:
363:
340:
336:
334:
331:
330:
313:
309:
307:
304:
303:
284:
281:
280:
263:
259:
253:
248:
239:
236:
235:
212:
208:
206:
203:
202:
185:
181:
179:
176:
175:
174:such that each
152:
148:
139:
135:
123:
119:
111:
108:
107:
100:
95:
76:homotopy theory
28:
23:
22:
15:
12:
11:
5:
4994:
4984:
4983:
4978:
4973:
4956:
4955:
4953:
4952:
4942:
4941:
4940:
4935:
4930:
4915:
4905:
4895:
4883:
4872:
4869:
4868:
4866:
4865:
4860:
4855:
4850:
4845:
4840:
4834:
4832:
4826:
4825:
4823:
4822:
4817:
4812:
4810:Winding number
4807:
4802:
4796:
4794:
4790:
4789:
4787:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4750:
4749:
4744:
4742:homotopy group
4734:
4733:
4732:
4727:
4722:
4717:
4712:
4702:
4697:
4692:
4682:
4680:
4676:
4675:
4668:
4666:
4664:
4663:
4658:
4653:
4652:
4651:
4641:
4640:
4639:
4629:
4624:
4619:
4614:
4609:
4603:
4601:
4597:
4596:
4589:
4588:
4581:
4574:
4566:
4559:
4558:
4548:
4531:
4525:
4507:
4505:
4502:
4499:
4498:
4472:
4441:
4420:(2): 272–280.
4397:
4386:Hatcher, Allen
4378:
4359:Hatcher, Allen
4351:
4324:(2): 272–280.
4302:
4295:
4277:
4246:
4232:
4218:
4203:
4183:
4161:(5): 213–245.
4130:
4118:
4102:Hatcher, Allen
4092:
4091:
4089:
4086:
4084:
4081:
4080:
4079:
4076:surgery theory
4064:
4057:
4054:
4042:pointed spaces
4029:
4026:
4009:
4005:
4001:
3998:
3995:
3990:
3986:
3965:
3962:
3959:
3939:
3936:
3933:
3911:
3907:
3879:
3876:
3851:
3850:
3838:
3835:
3829:
3826:
3801:
3797:
3783:where the new
3770:
3766:
3762:
3757:
3753:
3749:
3746:
3743:
3737:
3734:
3715:
3714:
3698:
3695:
3689:
3686:
3661:
3657:
3653:
3648:
3644:
3640:
3637:
3634:
3628:
3625:
3599:
3596:
3571:
3567:
3550:Tietze theorem
3531:
3527:
3504:
3500:
3477:
3473:
3450:
3446:
3410:
3405:
3401:
3377:
3356:
3351:
3347:
3326:
3321:
3317:
3314:
3311:
3287:
3284:
3281:
3260:
3255:
3251:
3221:
3218:
3213:
3212:
3211:
3210:
3197:
3187:
3185:
3182:
3181:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3145:
3144:
3142:
3137:
3133:
3129:
3125:
3121:
3116:
3111:
3105:
3099:
3095:
3081:
3080:
3068:
3064:
3060:
3055:
3050:
3036:
3035:
3024:
3019:
3015:
2992:
2988:
2967:
2963:
2955:
2952:
2949:
2947:
2944:
2943:
2940:
2937:
2934:
2932:
2929:
2928:
2924:
2912:
2911:
2910:
2897:
2887:
2885:
2882:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2859:
2854:
2849:
2848:
2846:
2841:
2836:
2832:
2817:
2816:
2813:
2812:
2811:
2798:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2770:
2767:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2735:
2734:
2732:
2727:
2722:
2718:
2711:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2683:
2680:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2648:
2647:
2645:
2640:
2635:
2631:
2617:
2616:
2602:
2598:
2570:
2565:
2561:
2518:
2515:
2514:
2513:
2508:
2437:
2348:and a cell in
2326:
2323:covering space
2319:
2312:
2301:
2296:If a space is
2294:
2289:
2286:
2285:
2284:
2277:
2260:
2256:
2253:
2249:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2216:
2213:
2210:
2207:
2203:
2199:
2196:
2186:hedgehog space
2182:
2165:
2162:
2159:
2158:
2152:September 2024
2139:
2137:
2126:
2123:
2122:
2121:
2098:
2084:
2081:Schubert cells
2074:
2067:
2060:
2046:
2034:
2031:
2028:
2025:
2022:
2000:
1996:
1961:
1958:
1955:
1951:
1928:
1924:
1898:
1895:
1894:
1893:
1879:
1874:
1845:
1822:
1817:
1795:
1791:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1736:
1720:
1719:
1718:
1702:
1699:
1696:
1693:
1673:
1670:
1667:
1664:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1570:
1504:
1457:
1454:
1445:
1442:
1440:
1437:
1427:
1424:
1384:
1381:
1380:
1379:
1365:
1361:
1355:
1351:
1335:
1319:
1295:
1292:
1289:
1286:
1283:
1256:
1253:
1250:
1231:
1228:quotient space
1216:disjoint union
1208:
1192:
1189:
1176:
1175:
1164:
1163:
1162:
1150:
1145:
1141:
1137:
1134:
1129:
1124:
1120:
1108:
1090:
1085:
1081:
1077:
1072:
1068:
1064:
1059:
1054:
1050:
1026:
1004:
999:
992:
989:
982:
977:
973:
969:
964:
959:
955:
929:
924:
920:
895:
890:
885:
881:
877:
872:
868:
864:
861:
856:
851:
844:
841:
815:
810:
806:
773:
760:
757:
752:
748:
744:
739:
735:
731:
726:
723:
719:
694:
669:
665:
642:
638:
634:
631:
608:
605:
602:
576:
572:
566:
562:
558:
555:
539:is called the
526:
522:
496:
491:
487:
481:
477:
471:
468:
465:
461:
457:
452:
448:
436:attaching maps
421:
418:
415:
411:
407:
402:
397:
393:
389:
386:
381:
376:
372:
349:
346:
343:
339:
316:
312:
288:
266:
262:
256:
251:
247:
243:
221:
218:
215:
211:
188:
184:
163:
160:
155:
151:
147:
142:
138:
134:
129:
126:
122:
118:
115:
99:
96:
94:
91:
26:
9:
6:
4:
3:
2:
4993:
4982:
4979:
4977:
4974:
4972:
4969:
4968:
4966:
4951:
4943:
4939:
4936:
4934:
4931:
4929:
4926:
4925:
4924:
4916:
4914:
4910:
4906:
4904:
4900:
4896:
4894:
4889:
4884:
4882:
4874:
4873:
4870:
4864:
4861:
4859:
4856:
4854:
4851:
4849:
4846:
4844:
4841:
4839:
4836:
4835:
4833:
4831:
4827:
4821:
4820:Orientability
4818:
4816:
4813:
4811:
4808:
4806:
4803:
4801:
4798:
4797:
4795:
4791:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4748:
4745:
4743:
4740:
4739:
4738:
4735:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4707:
4706:
4703:
4701:
4698:
4696:
4693:
4691:
4687:
4684:
4683:
4681:
4677:
4672:
4662:
4659:
4657:
4656:Set-theoretic
4654:
4650:
4647:
4646:
4645:
4642:
4638:
4635:
4634:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4617:Combinatorial
4615:
4613:
4610:
4608:
4605:
4604:
4602:
4598:
4594:
4587:
4582:
4580:
4575:
4573:
4568:
4567:
4564:
4556:
4551:
4545:
4541:
4537:
4532:
4528:
4526:0-442-04910-2
4522:
4518:
4514:
4509:
4508:
4495:
4491:
4490:
4485:
4476:
4462:on 2016-03-03
4458:
4451:
4445:
4437:
4433:
4428:
4423:
4419:
4415:
4411:
4407:
4401:
4395:
4391:
4387:
4382:
4376:
4372:
4371:0-521-79540-0
4368:
4364:
4360:
4355:
4347:
4343:
4339:
4335:
4331:
4327:
4323:
4319:
4314:
4306:
4298:
4296:9783110435221
4292:
4288:
4281:
4273:
4269:
4263:
4259:
4253:
4251:
4242:
4236:
4228:
4222:
4214:
4207:
4196:
4195:
4187:
4181:(open access)
4178:
4174:
4169:
4164:
4160:
4156:
4155:
4147:
4143:
4137:
4135:
4127:
4121:
4119:0-521-79540-0
4115:
4111:
4107:
4103:
4097:
4093:
4077:
4073:
4069:
4065:
4063:
4060:
4059:
4053:
4051:
4047:
4043:
4039:
4035:
4025:
4023:
3999:
3996:
3988:
3984:
3963:
3960:
3957:
3937:
3934:
3931:
3909:
3905:
3896:
3874:
3863:
3861:
3856:
3836:
3824:
3799:
3795:
3786:
3768:
3764:
3760:
3755:
3751:
3747:
3744:
3741:
3732:
3721:
3717:
3716:
3712:
3696:
3684:
3659:
3655:
3651:
3646:
3642:
3638:
3635:
3632:
3623:
3594:
3584:. If we let
3569:
3565:
3555:
3554:
3553:
3551:
3547:
3529:
3525:
3502:
3498:
3475:
3471:
3448:
3444:
3435:
3431:
3426:
3423:
3408:
3403:
3399:
3391:
3375:
3354:
3349:
3345:
3324:
3319:
3315:
3309:
3301:
3285:
3282:
3279:
3258:
3253:
3249:
3241:
3238:
3234:
3229:
3227:
3217:
3183:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3140:
3135:
3131:
3114:
3103:
3097:
3093:
3085:
3084:
3083:
3082:
3053:
3038:
3037:
3022:
3017:
3013:
2990:
2986:
2965:
2961:
2953:
2950:
2945:
2938:
2935:
2930:
2922:
2913:
2883:
2876:
2873:
2870:
2867:
2864:
2857:
2844:
2839:
2834:
2830:
2822:
2821:
2819:
2818:
2814:
2788:
2785:
2782:
2776:
2773:
2768:
2758:
2755:
2752:
2746:
2743:
2730:
2725:
2720:
2716:
2701:
2698:
2695:
2689:
2686:
2681:
2671:
2668:
2665:
2659:
2656:
2643:
2638:
2633:
2629:
2621:
2620:
2619:
2618:
2600:
2596:
2588:
2587:chain complex
2584:
2568:
2563:
2559:
2550:
2549:
2548:
2545:
2543:
2539:
2535:
2531:
2527:
2523:
2512:
2509:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2438:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2391:
2387:
2383:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2354:weak topology
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2324:
2320:
2317:
2313:
2310:
2306:
2302:
2299:
2295:
2292:
2291:
2282:
2278:
2275:
2254:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2214:
2211:
2208:
2205:
2201:
2197:
2187:
2183:
2180:
2176:
2172:
2171:Hilbert space
2168:
2167:
2155:
2146:
2142:
2138:
2135:
2131:
2130:
2119:
2115:
2111:
2107:
2103:
2099:
2096:
2095:homotopy type
2092:
2088:
2085:
2082:
2078:
2075:
2072:
2068:
2065:
2061:
2058:
2056:
2052:
2047:
2032:
2029:
2026:
2023:
2020:
1998:
1994:
1985:
1981:
1977:
1959:
1956:
1953:
1949:
1926:
1922:
1913:
1912:
1910:
1904:
1903:
1902:
1877:
1862:
1861:
1860:cubic lattice
1820:
1785:
1782:
1779:
1773:
1770:
1767:
1764:
1761:
1725:
1721:
1716:
1697:
1691:
1668:
1662:
1654:
1638:
1629:
1626:
1623:
1617:
1614:
1606:
1602:
1598:
1594:
1593:
1588:
1585:
1584:
1582:
1578:
1574:
1571:
1568:
1564:
1561:are glued to
1560:
1556:
1552:
1549:are glued to
1548:
1544:
1540:
1536:
1532:
1528:
1525:are glued to
1524:
1521:endpoints of
1520:
1516:
1512:
1508:
1505:
1502:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1463:
1462:
1461:
1453:
1451:
1436:
1433:
1423:
1421:
1417:
1413:
1409:
1405:
1400:
1398:
1394:
1390:
1363:
1359:
1353:
1349:
1340:
1336:
1333:
1317:
1309:
1306:-dimensional
1290:
1287:
1284:
1273:
1271:
1254:
1251:
1248:
1240:
1236:
1232:
1229:
1225:
1221:
1220:unit interval
1217:
1213:
1209:
1206:
1202:
1198:
1197:
1196:
1188:
1186:
1182:
1173:
1169:
1165:
1143:
1139:
1127:
1122:
1118:
1109:
1106:
1105:homeomorphism
1088:
1083:
1079:
1070:
1066:
1062:
1057:
1052:
1048:
1039:
1038:
1024:
1002:
997:
987:
975:
971:
967:
962:
957:
953:
945:
927:
922:
918:
909:
908:
907:
888:
883:
879:
870:
866:
862:
859:
854:
849:
839:
813:
808:
804:
795:
791:
787:
784:
772:
758:
750:
746:
737:
733:
724:
721:
717:
708:
692:
683:
667:
663:
640:
636:
632:
629:
622:
606:
603:
600:
592:
591:weak topology
574:
570:
564:
560:
556:
553:
544:
542:
524:
520:
510:
494:
489:
485:
479:
475:
469:
466:
463:
459:
455:
450:
446:
437:
419:
416:
413:
409:
400:
395:
391:
384:
379:
374:
370:
347:
344:
341:
337:
314:
310:
302:
286:
264:
254:
249:
245:
219:
216:
213:
209:
186:
182:
161:
158:
153:
149:
145:
140:
136:
132:
127:
124:
120:
116:
105:
90:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4950:Publications
4815:Chern number
4805:Betti number
4758:
4688: /
4679:Key concepts
4627:Differential
4535:
4517:Van Nostrand
4512:
4487:
4484:"CW-complex"
4475:
4464:. Retrieved
4457:the original
4444:
4417:
4413:
4406:Milnor, John
4400:
4389:
4381:
4362:
4354:
4321:
4317:
4305:
4286:
4280:
4271:
4258:Ghostarchive
4256:Archived at
4235:
4221:
4212:
4206:
4193:
4186:
4158:
4152:
4105:
4096:
4037:
4031:
3894:
3859:
3854:
3852:
3784:
3719:
3710:
3433:
3429:
3427:
3424:
3389:
3299:
3239:
3230:
3225:
3223:
3214:
2582:
2546:
2520:
2500:
2496:
2488:
2484:
2476:
2472:
2468:
2464:
2459:) (with the
2456:
2452:
2444:
2440:
2429:
2425:
2421:
2417:
2413:
2405:
2401:
2393:
2389:
2385:
2381:
2373:
2369:
2364:is then the
2361:
2357:
2349:
2345:
2341:
2337:
2333:
2329:
2178:
2149:
2145:adding to it
2140:
2113:
2109:
2104:of a cusped
2080:
2077:Grassmannian
2063:
2050:
2048:
1983:
1975:
1908:
1905:
1900:
1858:
1723:
1714:
1652:
1604:
1596:
1590:
1580:
1572:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1530:
1526:
1522:
1518:
1517:, such that
1514:
1510:
1506:
1500:
1496:
1492:
1488:
1484:
1483:is glued to
1480:
1476:
1472:
1468:
1464:
1459:
1447:
1431:
1429:
1401:
1396:
1392:
1388:
1386:
1338:
1269:
1238:
1234:
1211:
1200:
1194:
1184:
1180:
1178:
1167:
793:
785:
774:
707:direct limit
684:
590:
545:
540:
511:
103:
101:
88:
55:
48:cell complex
47:
43:
39:
29:
18:CW-complexes
4913:Wikiversity
4830:Key results
2481:John Milnor
2305:paracompact
2175:Baire space
1982:") and two
1465:An interval
1185:cellulation
593:: a subset
80:categorical
32:mathematics
4965:Categories
4759:CW complex
4700:Continuity
4690:Closed set
4649:cohomology
4466:2012-08-26
4083:References
3897:-skeleton
3862:-connected
3176: even
2526:cohomology
2288:Properties
2071:polyhedron
541:k-skeleton
104:CW complex
98:CW complex
93:Definition
40:CW complex
4938:geometric
4933:algebraic
4784:Cobordism
4720:Hausdorff
4715:connected
4632:Geometric
4622:Continuum
4612:Algebraic
4494:EMS Press
4338:0002-9947
4144:(1949a).
4070:called a
3935:≥
3878:~
3834:→
3828:~
3761:∪
3748:∪
3736:~
3694:→
3688:~
3652:∪
3639:∪
3627:~
3598:~
3409:∼
3355:∼
3325:∼
3313:→
3283:∼
3259:∼
3190:otherwise
3163:⩽
3157:⩽
2951:−
2936:−
2890:otherwise
2874:⩽
2868:⩽
2777:∉
2747:∈
2690:∉
2660:∈
2601:∗
2495:, so Hom(
2255:⊆
2244:∈
2241:θ
2232:≤
2226:≤
2215:θ
2209:π
2093:have the
2091:varieties
2030:≤
2024:≤
1957:−
1786:∈
1636:→
1350:∪
1288:−
1136:∂
1123:α
1084:α
1076:→
1053:α
1017:from the
998:α
991:¯
981:→
958:α
923:α
910:For each
884:α
850:α
843:¯
809:α
790:partition
759:⋯
756:↪
743:↪
730:↪
722:−
633:∩
604:⊂
561:∪
490:α
480:α
476:⊔
467:−
417:−
406:→
396:α
388:∂
375:α
345:−
265:α
250:α
217:−
162:⋯
159:⊂
146:⊂
133:⊂
125:−
114:∅
60:manifolds
4903:Wikibook
4881:Category
4769:Manifold
4737:Homotopy
4695:Interior
4686:Open set
4644:Homology
4593:Topology
4408:(1959).
4260:and the
4104:(2002).
4056:See also
1575:Given a
1573:A graph.
1507:A circle
1439:Examples
1404:loopless
619:is open
36:topology
4928:general
4730:uniform
4710:compact
4661:Digital
4436:1993204
4346:1993204
4272:Youtube
4177:0030759
3544:form a
3226:simpler
2309:compact
2118:SnapPea
1980:equator
1592:generic
1412:surface
776:Theorem
705:is the
50:) is a
4923:Topics
4725:metric
4600:Fields
4546:
4523:
4434:
4369:
4344:
4336:
4293:
4175:
4116:
3893:whose
3548:. The
3237:forest
2463:) are
2064:shadow
1448:Every
1308:sphere
1172:closed
42:(also
4705:Space
4460:(PDF)
4453:(PDF)
4432:JSTOR
4342:JSTOR
4198:(PDF)
4149:(PDF)
4088:Notes
3517:from
3233:graph
2398:finer
1577:graph
1410:on a
1224:glues
1103:is a
512:Each
329:, to
56:cells
4544:ISBN
4521:ISBN
4367:ISBN
4334:ISSN
4291:ISBN
4114:ISBN
4032:The
3039:For
3005:and
2524:and
2491:are
2487:and
2451:Hom(
2443:and
2439:Let
2428:and
2404:nor
2372:and
2332:and
2279:The
2184:The
2100:The
2049:The
1722:The
1684:and
1569:too.
1565:and
1553:and
1541:and
1533:and
1519:both
1495:and
1471:and
1252:<
1210:A 1-
1199:A 0-
301:ball
62:and
38:, a
4422:doi
4326:doi
4163:doi
4052:).
4038:the
3857:is
3722:by
2465:not
2422:not
2408:is
2392:or
2380:on
2368:of
2147:.
1906:An
1603:to
1337:An
1170:is
792:of
621:iff
589:is
46:or
30:In
4967::
4538:.
4515:.
4492:,
4486:,
4430:.
4418:90
4416:.
4412:.
4388:,
4361:,
4340:.
4332:.
4322:90
4320:.
4316:.
4270:.
4264::
4249:^
4173:MR
4171:.
4159:55
4157:.
4151:.
4133:^
4112:.
4108:.
2416:Ă—
2384:Ă—
2360:Ă—
2340:Ă—
2321:A
2069:A
1607:,
1583:.
1422:.
1402:A
1399:.
1387:A
1207:).
1187:.
860::=
781:A
682:.
509:.
102:A
4585:e
4578:t
4571:v
4552:.
4529:.
4469:.
4438:.
4424::
4348:.
4328::
4299:.
4274:.
4243:.
4229:.
4179:.
4165::
4128:.
4122:.
4078:.
4008:)
4004:Z
4000:;
3997:X
3994:(
3989:n
3985:H
3964:1
3961:=
3958:n
3938:2
3932:n
3910:n
3906:X
3895:n
3875:X
3860:n
3855:X
3849:.
3837:X
3825:X
3800:2
3796:X
3785:3
3769:3
3765:e
3756:2
3752:e
3745:X
3742:=
3733:X
3720:X
3713:.
3711:X
3697:X
3685:X
3660:2
3656:e
3647:1
3643:e
3636:X
3633:=
3624:X
3595:X
3570:1
3566:X
3530:1
3526:X
3503:2
3499:X
3476:1
3472:X
3449:1
3445:X
3434:X
3430:X
3404:/
3400:X
3390:F
3376:X
3350:/
3346:X
3320:/
3316:X
3310:X
3300:F
3286:y
3280:x
3254:/
3250:X
3240:F
3184:0
3172:,
3169:n
3166:2
3160:k
3154:0
3148:Z
3141:{
3136:=
3132:)
3128:)
3124:C
3120:(
3115:n
3110:P
3104:(
3098:k
3094:H
3067:)
3063:C
3059:(
3054:n
3049:P
3023:.
3018:n
3014:C
2991:0
2987:C
2966:.
2962:)
2954:1
2946:1
2939:1
2931:1
2923:(
2884:0
2877:n
2871:k
2865:0
2858:2
2853:Z
2845:{
2840:=
2835:k
2831:C
2792:}
2789:n
2786:,
2783:0
2780:{
2774:k
2769:0
2762:}
2759:n
2756:,
2753:0
2750:{
2744:k
2738:Z
2731:{
2726:=
2721:k
2717:H
2705:}
2702:n
2699:,
2696:0
2693:{
2687:k
2682:0
2675:}
2672:n
2669:,
2666:0
2663:{
2657:k
2651:Z
2644:{
2639:=
2634:k
2630:C
2597:C
2583:n
2569:,
2564:n
2560:S
2501:Y
2499:,
2497:X
2489:Y
2485:X
2477:Y
2475:,
2473:X
2469:X
2457:Y
2455:,
2453:X
2445:Y
2441:X
2430:Y
2426:X
2418:Y
2414:X
2406:Y
2402:X
2394:Y
2390:X
2386:Y
2382:X
2374:Y
2370:X
2362:Y
2358:X
2350:Y
2346:X
2342:Y
2338:X
2334:Y
2330:X
2276:.
2259:C
2252:}
2248:Q
2238:,
2235:1
2229:r
2223:0
2220::
2212:i
2206:2
2202:e
2198:r
2195:{
2179:n
2154:)
2150:(
2120:.
2083:.
2066:.
2057:.
2051:n
2045:.
2033:n
2027:k
2021:0
1999:n
1995:S
1984:n
1976:n
1960:1
1954:n
1950:S
1927:n
1923:D
1909:n
1892:.
1878:n
1873:R
1844:R
1821:n
1816:R
1794:}
1790:Z
1783:n
1780::
1777:]
1774:1
1771:+
1768:n
1765:,
1762:n
1759:[
1756:{
1735:Z
1717:.
1715:X
1701:)
1698:1
1695:(
1692:f
1672:)
1669:0
1666:(
1663:f
1653:X
1639:X
1633:}
1630:1
1627:,
1624:0
1621:{
1618::
1615:f
1605:X
1597:X
1567:y
1563:x
1559:B
1555:y
1551:x
1547:A
1543:B
1539:A
1535:y
1531:x
1527:x
1523:B
1515:B
1511:x
1501:B
1497:y
1493:x
1489:y
1485:x
1481:B
1477:B
1473:y
1469:x
1393:X
1364:k
1360:X
1354:k
1318:k
1294:)
1291:1
1285:n
1282:(
1270:n
1255:n
1249:k
1239:k
1181:X
1168:X
1149:)
1144:k
1140:D
1133:(
1128:k
1119:g
1107:.
1089:k
1080:e
1071:k
1067:B
1063::
1058:k
1049:g
1025:k
1003:k
988:e
976:k
972:D
968::
963:k
954:g
928:k
919:e
894:)
889:k
880:e
876:(
871:X
867:l
863:c
855:k
840:e
814:k
805:e
794:X
786:X
751:1
747:X
738:0
734:X
725:1
718:X
693:X
668:k
664:X
641:k
637:X
630:U
607:X
601:U
575:k
571:X
565:k
557:=
554:X
525:k
521:X
495:k
486:e
470:1
464:k
460:X
456:=
451:k
447:X
420:1
414:k
410:X
401:k
392:e
385::
380:k
371:g
348:1
342:k
338:X
315:k
311:B
299:-
287:k
261:)
255:k
246:e
242:(
220:1
214:k
210:X
187:k
183:X
154:1
150:X
141:0
137:X
128:1
121:X
117:=
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.