4887:
4670:
4908:
2134:
4876:
4945:
4918:
4898:
2808:
3207:
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
2907:
2623:
3214:
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.
3087:
2270:
3555:
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:
2975:
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:
4478:
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:
3780:
3671:
3551:
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:
2803:{\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:
3076:
2824:
277:
4123:
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
4144:
1804:
4017:
3846:
3706:
1890:
1833:
587:
940:
826:
3335:
3202:{\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:
3890:
3610:
1855:
1746:
653:
617:
3419:
3365:
3269:
2190:
3947:
3295:
1972:
360:
232:
3032:
2578:
1265:
3921:
3811:
3581:
3541:
3514:
3487:
3460:
3002:
2612:
2011:
1939:
1304:
680:
537:
327:
199:
1711:
1682:
3973:
3424:
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.
4948:
4152:
3385:
1328:
1035:
703:
297:
109:
2916:
4023:. i.e.: one can similarly realize elementary matrix operations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.
2536:
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
4448:
4043:
is actually used). Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the
3975:
case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for
4546:
4260:
365:
2492:
948:
831:
4582:
4936:
4931:
4523:
4369:
4293:
4116:
441:
2181:-skeletons, each of which being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.
4926:
1043:
4048:
4828:
4538:
4492:
3223:
There is a technique, developed by
Whitehead, for replacing a CW complex with a homotopy-equivalent CW complex that has a
3724:
3615:
3431:
is a simply-connected CW complex whose 0-skeleton consists of a point. Can we, through suitable modifications, replace
3231:
2318:: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups.
4979:
4969:
4487:
2902:{\displaystyle C_{k}={\begin{cases}\mathbb {Z} ^{2}&0\leqslant k\leqslant n\\0&{\text{otherwise}}\end{cases}}}
1113:
3041:
4836:
4108:
2504:
2433:
1227:
1215:
237:
4482:
4974:
4907:
4635:
4515:
2397:
1986:-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere"). Inductively, this gives
1600:
1415:
3230:
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
2272:
is homotopy equivalent to a CW complex (the point) but it does not admit a CW decomposition, since it is not
4921:
4066:
3544:
2086:
1751:
86:, but still retain a combinatorial nature that allows for computation (often with a much smaller complex).
3978:
4856:
4851:
4777:
4654:
4642:
4615:
4575:
3548:
4191:
4698:
4625:
4060:
4044:
2460:
2409:
2101:
1341:
can be constructed by repeating the above process countably many times. Since the topology of the union
4886:
3816:
3676:
1866:
1809:
771:
The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:
549:
4846:
4798:
4772:
4620:
4070:
913:
799:
59:
4455:
3304:
3142:
2846:
2732:
2645:
1610:
1344:
4897:
4693:
3235:
2297:
2105:
2016:
1503:
is the 1-cell. Alternatively, it can be constructed just from a single interval, with no 0-cells.
1403:
943:
3866:
3586:
3241:
in this graph. Since it is a collection of trees, and trees are contractible, consider the space
2977:
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:
4891:
4841:
4762:
4752:
4630:
4610:
4430:
4340:
4256:
4140:
2273:
1268:
789:
625:
300:
83:
71:
67:
63:
4861:
596:
3394:
3340:
3244:
4879:
4745:
4703:
4568:
4542:
4519:
4365:
4332:
4289:
4112:
4032:
4020:
3857:
3717:
2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing
2540:
2524:
2520:
2516:
2376:, as expected. In addition, the weak topology on this set often agrees with the more familiar
2365:
2315:
2108:
has a canonical CW decomposition with only one 0-cell (the compactification point) called the
2090:
1331:
1223:
51:
3926:
3274:
1944:
332:
204:
4659:
4605:
4535:
Nonabelian
Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids
4420:
4324:
4161:
3007:
2553:
2377:
2280:
2054:
1591:
1244:
4175:
4035:
of CW complexes is, in the opinion of some experts, the best if not the only candidate for
3899:
3789:
3559:
3519:
3492:
3465:
3438:
2980:
2590:
1989:
1917:
1277:
658:
515:
305:
177:
4718:
4713:
4239:
4171:
2528:
2448:
1859:
1687:
1658:
1586:
1407:
1195:
The CW complex construction is a straightforward generalization of the following process:
782:
79:
75:
3952:
4266:
2819:
Alternatively, if we use the equatorial decomposition with two cells in every dimension
4808:
4740:
4288:. De Gruyter Studies in Mathematics. Vol. 18. Berlin: Walter de Gruyter & Co.
4074:
3370:
2322:
2185:
1449:
1313:
1204:
1020:
764:{\displaystyle X_{-1}\hookrightarrow X_{0}\hookrightarrow X_{1}\hookrightarrow \cdots }
688:
282:
4425:
4408:
3786:-cell has an attaching map that consists of the new 2-cell and remainder mapping into
2970:{\displaystyle \left({\begin{smallmatrix}1&-1\\1&-1\end{smallmatrix}}\right).}
2133:
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.
4963:
4818:
4728:
4708:
4384:
4357:
4100:
4040:
2585:
2353:
2308:
2170:
2094:
1219:
1104:
435:
4911:
4166:
1391:
is a CW complex whose gluing maps are homeomorphisms. Accordingly, the partition of
4803:
4723:
4669:
3462:
consists of a single point? The answer is yes. The first step is to observe that
2076:
706:
2328:
The product of two CW complexes can be made into a CW complex. Specifically, if
4901:
4813:
4404:
2926:
2480:
2311:. A compact subspace of a CW complex is always contained in a finite subcomplex.
2304:
2174:
31:
17:
4688:
4647:
2532:
2070:
1599:
is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from a
1576:
1171:
4336:
4782:
3367:
naturally inherits a CW structure, with cells corresponding to the cells of
4225:
167:{\displaystyle \emptyset =X_{-1}\subset X_{0}\subset X_{1}\subset \cdots }
4767:
4735:
4684:
4591:
4554:
2580:
take the cell decomposition with two cells: a single 0-cell and a single
1907:
1419:
1307:
35:
89:
The C in CW stands for "closure-finite", and the W for "weak" topology.
4434:
4392:
4373:
4344:
4124:
2507:
variant of the compact-open topology; the above statements remain true.
2117:
1979:
1411:
106:
is constructed by taking the union of a sequence of topological spaces
4311:
4328:
1651:. This map can be perturbed to be disjoint from the 0-skeleton of
427:{\displaystyle g_{\alpha }^{k}:\partial e_{\alpha }^{k}\to X_{k-1}}
4267:"1.3 Introduction to Algebraic Topology. Examples of CW Complexes"
2436:
agrees with the weak topology and therefore defines a CW complex.
2013:
a CW decomposition with two cells in every dimension k such that
1974:
to the single 0-cell. An alternative cell decomposition has one (
1010:{\displaystyle g_{\alpha }^{k}:D^{k}\to {\bar {e}}_{\alpha }^{k}}
899:{\displaystyle {\bar {e}}_{\alpha }^{k}:=cl_{X}(e_{\alpha }^{k})}
2523:. Moreover, in the category of CW complexes and cellular maps,
58:) of different dimensions in specific ways. It generalizes both
4564:
1274:. For each copy, there is a map that "glues" its boundary (the
2128:
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
620:
54:
that is built by gluing together topological balls (so-called
4047:
on the homotopy category have a simple characterisation (the
3297:
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
4560:
3195:
2895:
2796:
2709:
2293:
CW complexes are locally contractible (Hatcher, prop. A.4).
4039:
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.
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}}
2144:
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:
4199:. International Workshop on Combinatorial Algorithms.
3981:
3955:
3929:
3902:
3869:
3819:
3792:
3727:
3679:
3618:
3589:
3562:
3522:
3495:
3468:
3441:
3397:
3373:
3343:
3307:
3277:
3247:
3090:
3044:
3010:
2983:
2919:
2827:
2626:
2593:
2556:
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:
4409:"On spaces having the homotopy type of a CW-complex"
4312:"On Spaces Having the Homotopy Type of a CW-Complex"
3427:
Consider climbing up the connectivity ladder—assume
685:
In the language of category theory, the topology on
4827:
4791:
4677:
4598:
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
4310:
4214:. Providence, R.I.: American Mathematical Society.
4011:
3967:
3941:
3915:
3884:
3840:
3805:
3775:{\displaystyle {\tilde {X}}=X\cup e^{2}\cup e^{3}}
3774:
3700:
3666:{\displaystyle {\tilde {X}}=X\cup e^{1}\cup e^{2}}
3665:
3604:
3575:
3535:
3508:
3481:
3454:
3413:
3379:
3359:
3329:
3289:
3263:
3201:
3070:
3026:
2996:
2969:
2901:
2802:
2606:
2572:
2400:than the product topology, for example if neither
2264:
2037:
2005:
1966:
1933:
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:
4317: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
4372:. A free electronic version is available on the
3813:. A similar slide gives a homotopy-equivalence
2424:a CW complex. On the other hand, the product of
3271:where the equivalence relation is generated by
2913:and the differentials are matrices of the form
1154:{\displaystyle g_{\alpha }^{k}(\partial D^{k})}
1110:(closure-finiteness) The image of the boundary
4065:The notion of CW complex has an adaptation to
3923:consists of a single point. The argument for
3863:one can find a homotopy-equivalent CW complex
3071:{\displaystyle \mathbb {P} ^{n}(\mathbb {C} )}
2396:is finite. However, the weak topology can be
1595:1-dimensional CW complexes. Specifically, if
4576:
4533:Brown, R.; Higgins, P.J.; Sivera, R. (2011).
4153:Bulletin of the American Mathematical Society
4019:(using the presentation matrices coming from
2344:in which each cell is a product of a cell in
8:
2790:
2778:
2760:
2748:
2703:
2691:
2673:
2661:
2251:
2194:
1793:
1755:
1632:
1620:
1509:. It can be constructed from a single point
4286:Quantum invariants of knots and 3-manifolds
4193:The 3-Sphere Regular Cellulation Conjecture
775:
272:{\displaystyle (e_{\alpha }^{k})_{\alpha }}
4944:
4917:
4583:
4569:
4561:
4240:"CW-complex - Encyclopedia of Mathematics"
3435:by a homotopy-equivalent CW complex where
3421:is a disjoint union of wedges of circles.
2519:of CW complexes is readily computable via
1806:. Similarly, the standard CW structure on
4518:University Series in Higher Mathematics.
4424:
4165:
4002:
4001:
3986:
3980:
3954:
3928:
3907:
3901:
3871:
3870:
3868:
3821:
3820:
3818:
3797:
3791:
3766:
3753:
3729:
3728:
3726:
3681:
3680:
3678:
3657:
3644:
3620:
3619:
3617:
3591:
3590:
3588:
3567:
3561:
3527:
3521:
3500:
3494:
3473:
3467:
3446:
3440:
3406:
3401:
3396:
3372:
3352:
3347:
3342:
3322:
3317:
3306:
3276:
3256:
3251:
3246:
3187:
3173:
3146:
3145:
3137:
3122:
3121:
3112:
3108:
3107:
3095:
3089:
3061:
3060:
3051:
3047:
3046:
3043:
3015:
3009:
2988:
2982:
2924:
2918:
2887:
2855:
2851:
2850:
2841:
2832:
2826:
2736:
2735:
2727:
2718:
2649:
2648:
2640:
2631:
2625:
2598:
2592:
2561:
2555:
2412:. In this unfavorable case, the product
2258:
2257:
2247:
2246:
2204:
2192:
2018:
1997:
1991:
1952:
1946:
1925:
1919:
1876:
1872:
1871:
1868:
1843:
1842:
1840:
1819:
1815:
1814:
1811:
1789:
1788:
1753:
1734:
1733:
1731:
1689:
1660:
1612:
1479:(an interval), such that one endpoint of
1467:. It can be constructed from two points (
1362:
1352:
1346:
1315:
1279:
1246:
1142:
1126:
1121:
1115:
1087:
1082:
1069:
1056:
1051:
1045:
1022:
1001:
996:
985:
984:
974:
961:
956:
950:
926:
921:
915:
887:
882:
869:
853:
848:
837:
836:
833:
812:
807:
801:
749:
736:
720:
714:
690:
666:
660:
639:
627:
598:
573:
563:
551:
523:
517:
493:
488:
478:
462:
449:
443:
412:
399:
394:
378:
373:
367:
340:
334:
313:
307:
284:
263:
253:
248:
239:
212:
206:
185:
179:
152:
139:
123:
111:
4391:, preliminary version available on the
4092:
2512:Homology and cohomology of CW complexes
3337:is a homotopy equivalence. Moreover,
2079:manifolds admit a CW structure called
1799:{\displaystyle \{:n\in \mathbb {Z} \}}
1222:. For each copy, there is a map that "
4510:Lundell, A. T.; Weingram, S. (1970).
4364:, Cambridge University Press (2002).
4012:{\displaystyle H_{n}(X;\mathbb {Z} )}
3708:given by sliding the new 2-cell into
3673:then there is a homotopy equivalence
2814:since all the differentials are zero.
2325:of a CW complex is also a CW complex.
7:
4251:
4249:
4210:Davis, James F.; Kirk, Paul (2001).
4135:
4133:
4026:
3489:and the attaching maps to construct
2493:compactly generated Hausdorff spaces
66:and has particular significance for
4212:Lecture Notes in Algebraic Topology
3391:. In particular, the 1-skeleton of
2537:Atiyah–Hirzebruch spectral sequence
1037:-dimensional closed ball such that
1241:-dimensional CW complex (for some
1135:
387:
113:
25:
4426:10.1090/s0002-9947-1959-0100267-4
3841:{\displaystyle {\tilde {X}}\to X}
3701:{\displaystyle {\tilde {X}}\to X}
2925:
2533: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
4943:
4916:
4906:
4896:
4885:
4875:
4874:
4668:
3612:be the corresponding CW complex
2517:Singular homology and cohomology
2132:
1885:{\displaystyle \mathbb {R} ^{n}}
1828:{\displaystyle \mathbb {R} ^{n}}
582:{\displaystyle X=\cup _{k}X_{k}}
279:, each homeomorphic to the open
4265:channel, Animated Math (2020).
4167:10.1090/S0002-9904-1949-09175-9
2713:
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}}
4541:Tracts in Mathematics Vol 15.
4309:Milnor, John (February 1959).
4073:, which is closely related to
4049:Brown representability theorem
4006:
3992:
3876:
3832:
3826:
3734:
3692:
3686:
3625:
3596:
3330:{\displaystyle X\to X/{\sim }}
3311:
3126:
3118:
3065:
3057:
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:
1:
4539:European Mathematical Society
3219:Modification of CW structures
2584:-cell. The cellular homology
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
4512:The topology of CW complexes
4449:"Compactly Generated Spaces"
4190:De Agostino, Sergio (2016).
4145:"Combinatorial homotopy. I."
3885:{\displaystyle {\tilde {X}}}
3605:{\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
4488:Encyclopedia of Mathematics
4389:Vector bundles and K-theory
2447:be CW complexes. Then the
2420:in the product topology is
2089:, algebraic and projective
1513:and the 1-dimensional ball
648:{\displaystyle U\cap X_{k}}
434:. The maps are also called
78:. CW complexes have better
4996:
4837:Banach fixed-point theorem
4109:Cambridge University Press
3387:that are not contained in
2614: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
4870:
4666:
4556:first author's home page]
3414:{\displaystyle X/{\sim }}
3360:{\displaystyle X/{\sim }}
3264:{\displaystyle X/{\sim }}
3234:. Now consider a maximal
2314:CW complexes satisfy the
27:Type of topological space
2527:can be interpreted as a
2388:, for example if either
2356:. The underlying set of
2169:An infinite-dimensional
2087:Differentiable manifolds
1235:n-dimensional CW complex
4481:Baladze, D.O. (2001) ,
4027:'The' homotopy category
3942:{\displaystyle n\geq 2}
3290:{\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}}
4892:Mathematics portal
4792:Metrics and properties
4778:Second-countable space
4413:Trans. Amer. Math. Soc
4284:Turaev, V. G. (1994).
4045:representable functors
4013:
3969:
3943:
3917:
3886:
3842:
3807:
3776:
3702:
3667:
3606:
3577:
3537:
3510:
3483:
3456:
3415:
3381:
3361:
3331:
3291:
3265:
3203:
3072:
3028:
3027:{\displaystyle C_{n}.}
2998:
2971:
2903:
2804:
2608:
2574:
2573:{\displaystyle S^{n},}
2535: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
4061:Abstract cell complex
4014:
3970:
3944:
3918:
3916:{\displaystyle X^{n}}
3887:
3843:
3808:
3806:{\displaystyle X^{2}}
3777:
3703:
3668:
3607:
3578:
3576:{\displaystyle X^{1}}
3538:
3536:{\displaystyle X^{1}}
3511:
3509:{\displaystyle X^{2}}
3484:
3482:{\displaystyle X^{1}}
3457:
3455:{\displaystyle X^{1}}
3416:
3382:
3362:
3332:
3292:
3266:
3204:
3073:
3029:
2999:
2997:{\displaystyle C_{0}}
2972:
2904:
2805:
2609:
2607:{\displaystyle C_{*}}
2575:
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
4847:Invariance of domain
4799:Euler characteristic
4773:Bundle (mathematics)
4553:More details on the
4226:"CW complex in nLab"
4071:handle decomposition
3979:
3953:
3927:
3900:
3867:
3817:
3790:
3725:
3677:
3616:
3587:
3560:
3520:
3493:
3466:
3439:
3395:
3371:
3341:
3305:
3301:. The quotient map
3275:
3245:
3088:
3042:
3008:
2981:
2917:
2825:
2624:
2591:
2554:
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
4857:Tychonoff's theorem
4852:Poincaré conjecture
4606:General (point-set)
4141:Whitehead, J. H. C.
3968:{\displaystyle n=1}
2539:is the analogue of
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:
4980:Topological spaces
4970:Algebraic topology
4842:De Rham cohomology
4763:Polyhedral complex
4753:Simplicial complex
4503:General references
4362:Algebraic topology
4105:Algebraic topology
4009:
3965:
3949:is similar to the
3939:
3913:
3882:
3838:
3803:
3772:
3698:
3663:
3602:
3573:
3545:group presentation
3533:
3506:
3479:
3452:
3411:
3377:
3357:
3327:
3287:
3261:
3227:CW decomposition.
3199:
3194:
3068:
3024:
2994:
2967:
2958:
2957:
2899:
2894:
2800:
2795:
2708:
2604:
2570:
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
4957:
4956:
4746:fundamental group
4548:978-3-03719-083-8
4393:author's homepage
4374:author's homepage
4125:author's homepage
4033:homotopy category
4021:cellular homology
3879:
3829:
3737:
3689:
3628:
3599:
3380:{\displaystyle X}
3190:
3176:
2890:
2541:cellular homology
2525:cellular homology
2521:cellular 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:
698:{\displaystyle X}
438:. Thus as a set,
292:{\displaystyle k}
201:is obtained from
52:topological space
16:(Redirected from
4987:
4947:
4946:
4920:
4919:
4910:
4900:
4890:
4889:
4878:
4877:
4672:
4585:
4578:
4571:
4562:
4552:
4529:
4496:
4495:
4476:
4470:
4469:
4467:
4466:
4460:
4454:. Archived from
4453:
4445:
4439:
4438:
4428:
4401:
4395:
4382:
4376:
4355:
4349:
4348:
4314:
4306:
4300:
4299:
4281:
4275:
4274:
4253:
4244:
4243:
4236:
4230:
4229:
4222:
4216:
4215:
4207:
4201:
4200:
4198:
4187:
4181:
4179:
4169:
4149:
4137:
4128:
4122:
4097:
4067:smooth manifolds
4018:
4016:
4015:
4010:
4005:
3991:
3990:
3974:
3972:
3971:
3966:
3948:
3946:
3945:
3940:
3922:
3920:
3919:
3914:
3912:
3911:
3891:
3889:
3888:
3883:
3881:
3880:
3872:
3852:If a CW complex
3847:
3845:
3844:
3839:
3831:
3830:
3822:
3812:
3810:
3809:
3804:
3802:
3801:
3781:
3779:
3778:
3773:
3771:
3770:
3758:
3757:
3739:
3738:
3730:
3707:
3705:
3704:
3699:
3691:
3690:
3682:
3672:
3670:
3669:
3664:
3662:
3661:
3649:
3648:
3630:
3629:
3621:
3611:
3609:
3608:
3603:
3601:
3600:
3592:
3582:
3580:
3579:
3574:
3572:
3571:
3542:
3540:
3539:
3534:
3532:
3531:
3515:
3513:
3512:
3507:
3505:
3504:
3488:
3486:
3485:
3480:
3478:
3477:
3461:
3459:
3458:
3453:
3451:
3450:
3420:
3418:
3417:
3412:
3410:
3405:
3386:
3384:
3383:
3378:
3366:
3364:
3363:
3358:
3356:
3351:
3336:
3334:
3333:
3328:
3326:
3321:
3296:
3294:
3293:
3288:
3270:
3268:
3267:
3262:
3260:
3255:
3208:
3206:
3205:
3200:
3198:
3197:
3191:
3188:
3177:
3174:
3149:
3133:
3129:
3125:
3117:
3116:
3111:
3100:
3099:
3078:we get similarly
3077:
3075:
3074:
3069:
3064:
3056:
3055:
3050:
3033:
3031:
3030:
3025:
3020:
3019:
3003:
3001:
3000:
2995:
2993:
2992:
2976:
2974:
2973:
2968:
2963:
2959:
2908:
2906:
2905:
2900:
2898:
2897:
2891:
2888:
2860:
2859:
2854:
2837:
2836:
2809:
2807:
2806:
2801:
2799:
2798:
2739:
2723:
2722:
2712:
2711:
2652:
2636:
2635:
2613:
2611:
2610:
2605:
2603:
2602:
2579:
2577:
2576:
2571:
2566:
2565:
2550:For the sphere,
2531:. 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:
18:Cellular complex
4995:
4994:
4990:
4989:
4988:
4986:
4985:
4984:
4975:Homotopy theory
4960:
4959:
4958:
4953:
4884:
4866:
4862:Urysohn's lemma
4823:
4787:
4673:
4664:
4636:low-dimensional
4594:
4589:
4559:
4549:
4532:
4526:
4509:
4505:
4500:
4499:
4480:
4477:
4473:
4464:
4462:
4458:
4451:
4447:
4446:
4442:
4403:
4402:
4398:
4383:
4379:
4356:
4352:
4329:10.2307/1993204
4308:
4307:
4303:
4296:
4283:
4282:
4278:
4264:
4261:Wayback Machine
4254:
4247:
4238:
4237:
4233:
4224:
4223:
4219:
4209:
4208:
4204:
4196:
4189:
4188:
4184:
4147:
4139:
4138:
4131:
4119:
4099:
4098:
4094:
4089:
4084:
4057:
4029:
3982:
3977:
3976:
3951:
3950:
3925:
3924:
3903:
3898:
3897:
3865:
3864:
3815:
3814:
3793:
3788:
3787:
3762:
3749:
3723:
3722:
3675:
3674:
3653:
3640:
3614:
3613:
3585:
3584:
3563:
3558:
3557:
3523:
3518:
3517:
3496:
3491:
3490:
3469:
3464:
3463:
3442:
3437:
3436:
3393:
3392:
3369:
3368:
3339:
3338:
3303:
3302:
3273:
3272:
3243:
3242:
3221:
3193:
3192:
3185:
3179:
3178:
3150:
3138:
3106:
3105:
3101:
3091:
3086:
3085:
3045:
3040:
3039:
3011:
3006:
3005:
2984:
2979:
2978:
2956:
2955:
2947:
2941:
2940:
2932:
2920:
2915:
2914:
2893:
2892:
2885:
2879:
2878:
2861:
2849:
2842:
2828:
2823:
2822:
2794:
2793:
2770:
2764:
2763:
2740:
2728:
2714:
2707:
2706:
2683:
2677:
2676:
2653:
2641:
2627:
2622:
2621:
2594:
2589:
2588:
2557:
2552:
2551:
2546:Some examples:
2529:homology theory
2514:
2449:function spaces
2410:locally compact
2290:
2200:
2189:
2188:
2166:
2157:
2151:
2148:
2127:
2015:
2014:
1993:
1988:
1987:
1948:
1943:
1942:
1921:
1916:
1915:
1899:
1870:
1865:
1864:
1837:
1836:
1813:
1808:
1807:
1750:
1749:
1728:
1727:
1686:
1685:
1657:
1656:
1655:if and only if
1609:
1608:
1601:two-point space
1458:
1446:
1441:
1428:
1385:
1358:
1348:
1343:
1342:
1312:
1311:
1276:
1275:
1243:
1242:
1233:In general, an
1193:
1177:
1138:
1112:
1111:
1065:
1042:
1041:
1019:
1018:
970:
947:
946:
912:
911:
865:
830:
829:
798:
797:
783:Hausdorff space
778:
745:
732:
716:
711:
710:
709:of the diagram
687:
686:
662:
657:
656:
635:
624:
623:
595:
594:
569:
559:
548:
547:
519:
514:
513:
474:
458:
445:
440:
439:
408:
364:
363:
336:
331:
330:
309:
304:
303:
281:
280:
259:
236:
235:
208:
203:
202:
181:
176:
175:
174:such that each
148:
135:
119:
108:
107:
100:
95:
76:homotopy theory
28:
23:
22:
15:
12:
11:
5:
4993:
4991:
4983:
4982:
4977:
4972:
4962:
4961:
4955:
4954:
4952:
4951:
4941:
4940:
4939:
4934:
4929:
4914:
4904:
4894:
4882:
4871:
4868:
4867:
4865:
4864:
4859:
4854:
4849:
4844:
4839:
4833:
4831:
4825:
4824:
4822:
4821:
4816:
4811:
4809:Winding number
4806:
4801:
4795:
4793:
4789:
4788:
4786:
4785:
4780:
4775:
4770:
4765:
4760:
4755:
4750:
4749:
4748:
4743:
4741:homotopy group
4733:
4732:
4731:
4726:
4721:
4716:
4711:
4701:
4696:
4691:
4681:
4679:
4675:
4674:
4667:
4665:
4663:
4662:
4657:
4652:
4651:
4650:
4640:
4639:
4638:
4628:
4623:
4618:
4613:
4608:
4602:
4600:
4596:
4595:
4590:
4588:
4587:
4580:
4573:
4565:
4558:
4557:
4547:
4530:
4524:
4506:
4504:
4501:
4498:
4497:
4471:
4440:
4419:(2): 272–280.
4396:
4385:Hatcher, Allen
4377:
4358:Hatcher, Allen
4350:
4323:(2): 272–280.
4301:
4294:
4276:
4245:
4231:
4217:
4202:
4182:
4160:(5): 213–245.
4129:
4117:
4101:Hatcher, Allen
4091:
4090:
4088:
4085:
4083:
4080:
4079:
4078:
4075:surgery theory
4063:
4056:
4053:
4041:pointed spaces
4028:
4025:
4008:
4004:
4000:
3997:
3994:
3989:
3985:
3964:
3961:
3958:
3938:
3935:
3932:
3910:
3906:
3878:
3875:
3850:
3849:
3837:
3834:
3828:
3825:
3800:
3796:
3782:where the new
3769:
3765:
3761:
3756:
3752:
3748:
3745:
3742:
3736:
3733:
3714:
3713:
3697:
3694:
3688:
3685:
3660:
3656:
3652:
3647:
3643:
3639:
3636:
3633:
3627:
3624:
3598:
3595:
3570:
3566:
3549:Tietze theorem
3530:
3526:
3503:
3499:
3476:
3472:
3449:
3445:
3409:
3404:
3400:
3376:
3355:
3350:
3346:
3325:
3320:
3316:
3313:
3310:
3286:
3283:
3280:
3259:
3254:
3250:
3220:
3217:
3212:
3211:
3210:
3209:
3196:
3186:
3184:
3181:
3180:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3144:
3143:
3141:
3136:
3132:
3128:
3124:
3120:
3115:
3110:
3104:
3098:
3094:
3080:
3079:
3067:
3063:
3059:
3054:
3049:
3035:
3034:
3023:
3018:
3014:
2991:
2987:
2966:
2962:
2954:
2951:
2948:
2946:
2943:
2942:
2939:
2936:
2933:
2931:
2928:
2927:
2923:
2911:
2910:
2909:
2896:
2886:
2884:
2881:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2858:
2853:
2848:
2847:
2845:
2840:
2835:
2831:
2816:
2815:
2812:
2811:
2810:
2797:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2769:
2766:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2734:
2733:
2731:
2726:
2721:
2717:
2710:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2682:
2679:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2647:
2646:
2644:
2639:
2634:
2630:
2616:
2615:
2601:
2597:
2569:
2564:
2560:
2513:
2510:
2509:
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4992:
4981:
4978:
4976:
4973:
4971:
4968:
4967:
4965:
4950:
4942:
4938:
4935:
4933:
4930:
4928:
4925:
4924:
4923:
4915:
4913:
4909:
4905:
4903:
4899:
4895:
4893:
4888:
4883:
4881:
4873:
4872:
4869:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4843:
4840:
4838:
4835:
4834:
4832:
4830:
4826:
4820:
4819:Orientability
4817:
4815:
4812:
4810:
4807:
4805:
4802:
4800:
4797:
4796:
4794:
4790:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4759:
4756:
4754:
4751:
4747:
4744:
4742:
4739:
4738:
4737:
4734:
4730:
4727:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4706:
4705:
4702:
4700:
4697:
4695:
4692:
4690:
4686:
4683:
4682:
4680:
4676:
4671:
4661:
4658:
4656:
4655:Set-theoretic
4653:
4649:
4646:
4645:
4644:
4641:
4637:
4634:
4633:
4632:
4629:
4627:
4624:
4622:
4619:
4617:
4616:Combinatorial
4614:
4612:
4609:
4607:
4604:
4603:
4601:
4597:
4593:
4586:
4581:
4579:
4574:
4572:
4567:
4566:
4563:
4555:
4550:
4544:
4540:
4536:
4531:
4527:
4525:0-442-04910-2
4521:
4517:
4513:
4508:
4507:
4502:
4494:
4490:
4489:
4484:
4475:
4472:
4461:on 2016-03-03
4457:
4450:
4444:
4441:
4436:
4432:
4427:
4422:
4418:
4414:
4410:
4406:
4400:
4397:
4394:
4390:
4386:
4381:
4378:
4375:
4371:
4370:0-521-79540-0
4367:
4363:
4359:
4354:
4351:
4346:
4342:
4338:
4334:
4330:
4326:
4322:
4318:
4313:
4305:
4302:
4297:
4295:9783110435221
4291:
4287:
4280:
4277:
4272:
4268:
4262:
4258:
4252:
4250:
4246:
4241:
4235:
4232:
4227:
4221:
4218:
4213:
4206:
4203:
4195:
4194:
4186:
4183:
4180:(open access)
4177:
4173:
4168:
4163:
4159:
4155:
4154:
4146:
4142:
4136:
4134:
4130:
4126:
4120:
4118:0-521-79540-0
4114:
4110:
4106:
4102:
4096:
4093:
4086:
4081:
4076:
4072:
4068:
4064:
4062:
4059:
4058:
4054:
4052:
4050:
4046:
4042:
4038:
4034:
4024:
4022:
3998:
3995:
3987:
3983:
3962:
3959:
3956:
3936:
3933:
3930:
3908:
3904:
3895:
3873:
3862:
3860:
3855:
3835:
3823:
3798:
3794:
3785:
3767:
3763:
3759:
3754:
3750:
3746:
3743:
3740:
3731:
3720:
3716:
3715:
3711:
3695:
3683:
3658:
3654:
3650:
3645:
3641:
3637:
3634:
3631:
3622:
3593:
3583:. If we let
3568:
3564:
3554:
3553:
3552:
3550:
3546:
3528:
3524:
3501:
3497:
3474:
3470:
3447:
3443:
3434:
3430:
3425:
3422:
3407:
3402:
3398:
3390:
3374:
3353:
3348:
3344:
3323:
3318:
3314:
3308:
3300:
3284:
3281:
3278:
3257:
3252:
3248:
3240:
3237:
3233:
3228:
3226:
3218:
3216:
3182:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3139:
3134:
3130:
3113:
3102:
3096:
3092:
3084:
3083:
3082:
3081:
3052:
3037:
3036:
3021:
3016:
3012:
2989:
2985:
2964:
2960:
2952:
2949:
2944:
2937:
2934:
2929:
2921:
2912:
2882:
2875:
2872:
2869:
2866:
2863:
2856:
2843:
2838:
2833:
2829:
2821:
2820:
2818:
2817:
2813:
2787:
2784:
2781:
2775:
2772:
2767:
2757:
2754:
2751:
2745:
2742:
2729:
2724:
2719:
2715:
2700:
2697:
2694:
2688:
2685:
2680:
2670:
2667:
2664:
2658:
2655:
2642:
2637:
2632:
2628:
2620:
2619:
2618:
2617:
2599:
2595:
2587:
2586:chain complex
2583:
2567:
2562:
2558:
2549:
2548:
2547:
2544:
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2511:
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:
2287:
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:
2163:
2155:
2146:
2142:
2138:
2135:
2131:
2130:
2124:
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:
1896:
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:
1455:
1453:
1451:
1443:
1438:
1436:
1433:
1425:
1423:
1421:
1417:
1413:
1409:
1405:
1400:
1398:
1394:
1390:
1382:
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:
1190:
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:
97:
92:
90:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4949:Publications
4814:Chern number
4804:Betti number
4757:
4687: /
4678:Key concepts
4626:Differential
4534:
4516:Van Nostrand
4511:
4486:
4483:"CW-complex"
4474:
4463:. Retrieved
4456:the original
4443:
4416:
4412:
4405:Milnor, John
4399:
4388:
4380:
4361:
4353:
4320:
4316:
4304:
4285:
4279:
4270:
4257:Ghostarchive
4255:Archived at
4234:
4220:
4211:
4205:
4192:
4185:
4157:
4151:
4104:
4095:
4036:
4030:
3893:
3858:
3853:
3851:
3783:
3718:
3709:
3432:
3428:
3426:
3423:
3388:
3298:
3238:
3229:
3224:
3222:
3213:
2581:
2545:
2515:
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:
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1269:
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1234:
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1200:
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1178:
1167:
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785:
774:
707:direct limit
684:
590:
545:
540:
511:
103:
101:
88:
55:
48:cell complex
47:
43:
39:
29:
4912:Wikiversity
4829: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
4964:Categories
4758:CW complex
4699:Continuity
4689:Closed set
4648:cohomology
4465:2012-08-26
4082:References
3896:-skeleton
3861:-connected
3175: even
2288:Properties
2071:polyhedron
541:k-skeleton
104:CW complex
98:CW complex
93:Definition
40:CW complex
4937:geometric
4932:algebraic
4783:Cobordism
4719:Hausdorff
4714:connected
4631:Geometric
4621:Continuum
4611:Algebraic
4493:EMS Press
4337:0002-9947
4143:(1949a).
4069:called a
3934:≥
3877:~
3833:→
3827:~
3760:∪
3747:∪
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3693:→
3687:~
3651:∪
3638:∪
3626:~
3597:~
3408:∼
3354:∼
3324:∼
3312:→
3282:∼
3258:∼
3189:otherwise
3162:⩽
3156:⩽
2950:−
2935:−
2889:otherwise
2873:⩽
2867:⩽
2776:∉
2746:∈
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2659:∈
2600:∗
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2255:⊆
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2209:π
2093:have the
2091:varieties
2030:≤
2024:≤
1957:−
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1636:→
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1017:from the
998:α
991:¯
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958:α
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910:For each
884:α
850:α
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809:α
790:partition
759:⋯
756:↪
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730:↪
722:−
633:∩
604:⊂
561:∪
490:α
480:α
476:⊔
467:−
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406:→
396:α
388:∂
375:α
345:−
265:α
250:α
217:−
162:⋯
159:⊂
146:⊂
133:⊂
125:−
114:∅
60:manifolds
4902:Wikibook
4880:Category
4768:Manifold
4736:Homotopy
4694:Interior
4685:Open set
4643:Homology
4592:Topology
4407:(1959).
4259:and the
4103:(2002).
4055:See also
1575:Given a
1573:A graph.
1507:A circle
1439:Examples
1404:loopless
619:is open
36:topology
4927:general
4729:uniform
4709:compact
4660:Digital
4435:1993204
4345:1993204
4271:Youtube
4176:0030759
3543:form a
3225:simpler
2309:compact
2118:SnapPea
1980:equator
1592:generic
1412:surface
776:Theorem
705:is the
50:) is a
4922:Topics
4724:metric
4599:Fields
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3892:whose
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2463:) are
2064:shadow
1448:Every
1308:sphere
1172:closed
42:(also
4704:Space
4459:(PDF)
4452:(PDF)
4431:JSTOR
4341:JSTOR
4197:(PDF)
4148:(PDF)
4087:Notes
3516:from
3232:graph
2398:finer
1577:graph
1410:on a
1224:glues
1103:is a
512:Each
329:, to
56:cells
4543:ISBN
4520:ISBN
4366:ISBN
4333:ISSN
4290:ISBN
4113:ISBN
4031:The
3038:For
3004:and
2491:are
2487:and
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2443:and
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2428:and
2404:nor
2372:and
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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
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4421:doi
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