|
|
| Line 1: |
Line 1: |
| In [[topology]], a '''CW complex''' is a type of [[topological space]] introduced by [[J. H. C. Whitehead]] to meet the needs of [[homotopy theory]]. This class of spaces is broader and has some better [[category theory|categorical]] properties than [[simplicial complex]]es, but still retains a combinatorial nature
| | Elderly video games ought to be able to be discarded. They could be worth some money at a number of video retailers. After you buy and sell in a number of game titles, you will likely get your upcoming title at no cost!<br><br>Beginning nearly enough pebbles to get another builders. Don''t waste one of the gems while any way on rush-building anything, as if that it can save you the group you are going so that it will eventually obtain enough free of cost extra gems to grab that extra builder without having to cost. Particularly, customers can get free jewelry for clearing obstructions like rocks and trees, quickly you clear them on the market they come back in addition to you may re-clear these kind of people to get more jewelry.<br><br>When you have almost any concerns relating to in which as well as the way to utilize [http://circuspartypanama.com Clash Of clans Hack Tool download], it is possible to e mail us with the webpage. Pay attention to a game's evaluation when purchasing an existing. This evaluation will allow you fully grasp what age level clash of clans hack tool is right for and will inform you when the sport is violent. It can help you figure out whether you need to buy the sport.<br><br>Should you feel like you really targeted your enemy site on in a player with the dice and still missed, verification what weapon you will definitely be using. Just for example in real life, different weapons have different strengths and weaknesses. The weapon you are the application of may not have your current short distance required or the weapon recoil is ordinarily actually putting you a little bit off target.<br><br>Supercell has absolutely considerable as well as a explained the steps amongst Association Wars, the over appear passion in Battle of Clans. As being the name recommends, a romantic relationship war is often one particular strategic battle amid a couple of clans. It takes abode over the advance of two canicule -- the actual alertness day plus any action day -- and offers the acceptable association that includes a ample boodle bonus; although, every [http://www.sharkbayte.com/keyword/association association] affiliate to who makes acknowledged attacks following a association war additionally tends to make some benefit loot.<br><br>To access it into excel, copy-paste this continued menu into corpuscle B1. If you again obtain an majority of energy in abnormal in corpuscle A1, the bulk to treasures will arise while in B1.<br><br>Video games are some of the finest kinds of excitement around. They are unquestionably also probably the most pricey types of entertainment, with console games and it range from $50 that will $60, and consoles when their own inside that 100s. It is regarded as possible to spend less on clash of clans hack and console purchases, and you can track down out about them located in the following paragraphs. |
| that allows for computation (often with a much smaller complex).
| |
| | |
| ==Formulation==
| |
| | |
| Roughly speaking, a ''CW-complex'' is made of basic building blocks called ''cells''. The precise definition prescribes how the cells may be topologically ''glued together''. The ''C'' stands for "closure-finite", and the ''W'' for "[[weak topology]]".
| |
| | |
| An ''n''-dimensional closed cell is the image of an ''n''-dimensional [[closed ball]] under an [[attaching map]]. For example, a [[simplex]] is a closed cell, and more generally, a [[convex polytope]] is a closed cell. An ''n''-dimensional open cell is a topological space that is homeomorphic to the [[open ball]]. A 0-dimensional open (and closed) cell is a [[Singleton (mathematics)|singleton]] space. ''Closure-finite'' means that each closed cell is [[cover (topology)|covered]] by a finite union of open cells.
| |
| | |
| A CW complex is a [[Hausdorff space]] ''X'' together with a [[partition of a set|partition]] of ''X'' into open cells (of perhaps varying dimension) that satisfies two additional properties:
| |
| | |
| * For each n-dimensional open cell ''C'' in the partition of ''X'', there exists a [[Continuous function#Continuous functions between topological spaces|continuous map]] ''f'' from the ''n''-dimensional closed ball to ''X'' such that
| |
| ** the restriction of ''f'' to the interior of the closed ball is a [[homeomorphism]] onto the cell ''C'', and
| |
| ** the image of the [[Boundary (topology)|boundary]] of the closed ball is contained in the union of a finite number of elements of the partition, each having cell dimension less than n.
| |
| * A subset of ''X'' is [[Closed set|closed]] if and only if it meets the closure of each cell in a closed set.
| |
| | |
| ==Inductive definition of CW-complexes==
| |
| | |
| If the largest dimension of any of the cells is ''n'', then the CW complex is said to have dimension ''n''. If there is no bound to the cell dimensions then it is said to be infinite-dimensional. The [[n-skeleton|''n''-skeleton]] of a CW complex is the union of the cells whose dimension is at most ''n''. If the union of a set of cells is closed, then this union is itself a CW complex, called a subcomplex. Thus the ''n''-skeleton is the largest subcomplex of dimension ''n'' or less.
| |
| | |
| A CW complex is often constructed by defining its skeleta inductively. Begin by taking the 0-skeleton to be a [[discrete space]]. Next, attach 1-cells to the 0-skeleton. Here, the 1-cells are attached to points of the 0-skeleton via some continuous map from unit 0-sphere, that is, <math> S_0 </math>. Define the 1-skeleton to be the [[identification space]] obtained from the union of the 0-skeleton, 1-cells, and the identification of points of boundary of 1-cells by assigning an identification mapping from the [[Boundary (topology)|boundary]] of the 1-cells into the 1-cells. In general, given the (''n'' − 1)-skeleton and a collection of (abstract) closed ''n''-cells, as above, the ''n''-cells are attached to the (''n'' − 1)-skeleton by some continuous mapping from <math> S_{n-1} </math>, and making an identification (equivalence relation) by specifying maps from the boundary of each ''n''-cell into the (''n'' − 1)-skeleton. The ''n''-skeleton is the identification space obtained from the union of the (''n'' − 1)-skeleton and the closed ''n''-cells by identifying each point in the boundary of an ''n''-cell with its image.
| |
| | |
| Up to isomorphism every ''n''-dimensional complex can be obtained from its (''n'' − 1)-skeleton in this sense, and thus every finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensional complexes, with the understanding that the result of the infinite process is the [[direct limit]] of the skeleta: a set is closed in ''X'' if and only if it meets each skeleton in a closed set.
| |
| | |
| ==Examples==
| |
| | |
| * The space <math>\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subset \mathbb R^2</math> has the homotopy-type of a CW-complex (it is contractible) but it does not admit a CW-decomposition, since it is not [[Contractible_space#Locally_contractible_spaces|locally contractible]].
| |
| * The [[Hawaiian earring]] is an example of a topological space that does not have the homotopy-type of a CW-complex.
| |
| * The ''standard CW-structure'' on the real numbers has 0-skeleton the integers <math>\mathbb Z</math> and as 1-cells the intervals <math>\{ [n,n+1] : n \in \mathbb Z\}</math>. Similarly, the standard CW-structure on <math>\mathbb R^n</math> has cubical cells that are products of the 0 and 1-cells from <math>\mathbb R</math>. This is the standard ''[[Integer lattice|cubic lattice]]'' cell structure on <math>\mathbb R^n</math>.
| |
| * A [[polyhedron]] is naturally a CW-complex.
| |
| * A [[Graph (mathematics)|graph]] is a 1-dimensional CW-complex. [[Trivalent graph]]s can be considered as ''generic'' 1-dimensional CW-complexes. Specifically, if ''X'' is a 1-dimensional CW-complex, the attaching map for a 1-cell is a map from a [[discrete two-point space|two-point space]] to ''X'', <math>f : \{0,1\} \to X</math>. This map can be perturbed to be disjoint from the 0-skeleton of ''X'' if and only if <math>f(0)</math> and <math>f(1)</math> are not 0-valence vertices of ''X''.
| |
| * The terminology for a generic 2-dimensional CW-complex is a '''shadow'''.<ref>Turaev, V. G. (1994), "Quantum invariants of knots and 3-manifolds", De Gruyter Studies in Mathematics (Berlin: Walter de Gruyter & Co.) 18</ref>
| |
| * The [[n-sphere|n-dimensional sphere]] admits a CW-structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from <math>S^{n-1}</math> to 0-cell. There is a popular alternative cell decomposition, since the equatorial inclusion <math>S^{n-1} \to S^n</math> has complement two balls: the upper and lower hemi-spheres. Inductively, this gives <math>S^n</math> a CW-decomposition with two cells in every dimension k such that <math>0 \leq k \leq n</math>.
| |
| * The n-dimensional real [[projective space]] admits a CW-structure with one cell in each dimension.
| |
| * [[Grassmannian]] manifolds admit a CW-structure called '''Schubert cells'''.
| |
| * [[Differentiable manifold]]s, algebraic and projective [[algebraic variety|varieties]] have the homotopy-type of CW-complexes.
| |
| * The [[Alexandroff extension|one-point compactification]] of a cusped [[hyperbolic manifold]] has a canonical CW-decomposition with only one 0-cell (the compactification point) called the '''Epstein-Penner Decomposition'''. Such cell decompositions are frequently called '''ideal polyhedral decompositions''' and are used in popular computer software, such as [[SnapPea]].
| |
| | |
| ==Homology and cohomology of CW-complexes==
| |
| | |
| [[singular homology|Singular homology and cohomology]] of CW-complexes is readily computable via [[cellular homology]]. Moreover, in the category of CW-complexes and cellular maps, [[cellular homology]] can be interpreted as a [[homology theory]]. To compute an [[cohomology#Generalized cohomology theories|extraordinary (co)homology theory]] for a CW-complex, the [[Atiyah–Hirzebruch spectral sequence|Atiyah-Hirzebruch spectral sequence]] is the analogue of [[cellular homology]].
| |
| | |
| Some examples:
| |
| | |
| :* For the sphere <math>S^n</math>, take the cell decomposition with two cells: a single 0-cell and a single n-cell. The cellular homology [[chain complex]] <math>C_*</math> and homology are given by:
| |
| | |
| <math>C_k = \left\{
| |
| \begin{array}{lr}
| |
| \mathbb Z & k \in \{0,n\} \\
| |
| 0 & k \notin \{0,n\}
| |
| \end{array}
| |
| \right.</math> <math>H_k = \left\{
| |
| \begin{array}{lr}
| |
| \mathbb Z & k \in \{0,n\} \\
| |
| 0 & k \notin \{0,n\}
| |
| \end{array}
| |
| \right.</math> since all the differentials are zero.
| |
| | |
| Alternatively, if we use the equatorial decomposition with two cells in every dimension <math>C_k = \left\{
| |
| \begin{array}{lr}
| |
| \mathbb Z^2 & 0 \leq k \leq n \\
| |
| 0 & \text{otherwise}
| |
| \end{array}
| |
| \right.</math> and the differentials are matrices of the form <math>\begin{pmatrix} 1 & -1 \\ 1 & -1\end{pmatrix}</math>. This gives the same homology computation above, as the chain complex is exact at all terms
| |
| except <math>C_0</math> and <math>C_n</math>.
| |
| | |
| :* For <math>\mathbb{P}^n\mathbb{C}</math> we get similarly
| |
| ::<math>H^k(\mathbb{P}^n\mathbb{C}) = \begin{cases}
| |
| \mathbb{Z} \quad\text{for } 0\le k\le 2n,\text{even}\\
| |
| 0 \quad\text{otherwise}.
| |
| \end{cases}</math>
| |
| | |
| 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.
| |
| | |
| == Modification of CW-structures ==
| |
| | |
| There is a technique, developed by Whitehead, for replacing a CW-complex with a homotopy-equivalent CW-complex which has a ''simpler'' CW-decomposition.
| |
| | |
| Consider, for example, an arbitrary CW-complex. Its 1-skeleton can be fairly complicated, being an arbitrary [[Graph (mathematics)|graph]]. Now consider a maximal [[Tree (graph theory)|forest]] ''F'' in this graph. Since it is a collection of trees, and trees are contractible, consider the space <math>X/\sim</math> where the equivalence relation is generated by <math>x \sim y</math> if they are contained in a common tree in the maximal forest ''F''. The quotient map <math>X \to X/\sim</math> is a homotopy equivalence. Moreover, <math>X/\sim</math> naturally inherits a CW-structure, with cells corresponding to the cells of <math>X</math> which are not contained in ''F''. In particular, the 1-skeleton of <math>X/\sim</math> is a disjoint union of wedges of circles.
| |
| | |
| 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.
| |
| | |
| Consider climbing up the connectivity ladder—assume ''X'' is a simply-connected CW-complex whose 0-skeleton consists of a point. Can we, through suitable modifications, replace ''X'' by a homotopy-equivalent CW-complex where <math>X^1</math> consists of a single point? The answer is yes. The first step is to observe that <math>X^1</math> and the attaching maps to construct <math>X^2</math> from <math>X^1</math> form a [[Presentation of a group|group presentation]]. The [[Tietze transformations|Tietze theorem]] 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:
| |
| | |
| : 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 <math>X^1</math>. If we let <math>\tilde X</math> be the corresponding CW-complex <math>\tilde X = X \cup e^1 \cup e^2</math> then there is a homotopy-equivalence <math>\tilde X \to X</math> given by sliding the new ''2''-cell into ''X''.
| |
| | |
| : 2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing ''X'' by <math>\tilde X = X \cup e^2 \cup e^3</math> where the new ''3''-cell has an attaching map that consists of the new ''2''-cell and remainder mapping into <math>X^2</math>. A similar slide gives a homotopy-equivalence <math>\tilde X \to X</math>. | |
| | |
| If a CW-complex ''X'' is ''n''-connected one can find a homotopy-equivalent CW-complex <math>\tilde X</math> whose ''n''-skeleton <math>X^n</math> consists of a single point. The argument for <math>n \geq 2</math> is similar to the <math>n=1</math> case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for <math>H_n(X;\mathbb Z)</math> (using the presentation matrices coming from [[cellular homology]]. i.e.: one can similarly realize elementary matrix operations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.
| |
| | |
| =='The' homotopy category==
| |
| | |
| The [[homotopy category]] of CW complexes is, in the opinion of some experts, the best if not the only candidate for ''the'' homotopy category (for technical reasons the version for [[pointed space]]s is actually used).<ref>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 {{SpringerEOM| title=CW-complex | id=CW-complex | oldid=15603 | first=D.O. | last=Baladze }}</ref> Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the [[representable functor]]s on the homotopy category have a simple characterisation (the [[Brown representability theorem]]).
| |
| | |
| ==Properties==
| |
| | |
| * CW-complexes are locally contractible.
| |
| * CW-complexes satisfy the [[Whitehead theorem]]: a map between CW-complexes is a homotopy-equivalence if and only if it induces an isomorphism on all homotopy groups.
| |
| * The product of two CW-complexes can be made into a CW-complex. Specifically, if ''X'' and ''Y'' are CW-complexes, then one can form a CW-complex ''X×Y'' in which each cell is a product of a cell in ''X'' and a cell in ''Y'', endowed with the weak topology. The underlying set of ''X×Y'' is then the [[Cartesian product]] of ''X'' and ''Y'', as expected. In addition, the weak topology on this set often agrees with the more familiar [[product topology]] on ''X×Y'', for example if either ''X'' or ''Y'' is finite. However, the weak topology can be [[comparison of topologies|finer]] than the product topology if neither ''X'' nor ''Y'' is [[locally compact space|locally compact]]. In this unfavorable case, the product ''X×Y'' in the product topology is ''not'' a CW-complex. On the other hand, the product of ''X'' and ''Y'' in the category of [[compactly generated space]]s agrees with the weak topology and therefore defines a CW-complex.
| |
| * Let ''X'' and ''Y'' be CW-complexes. Then the [[function spaces]] ''Hom(X,Y)'' (with the [[compact-open topology]]) are ''not'' CW-complexes in general. If ''X'' is finite then ''Hom(X,Y)'' is [[homotopy equivalent]] to a CW-complex by a theorem of [[John Milnor]] (1959).<ref name="milnor">[[John Milnor|Milnor, John]], "[http://www.jstor.org/stable/1993204 On spaces having the homotopy type of a CW-complex]" ''Trans. Amer. Math. Soc.'' '''90''' (1959), 272–280.</ref> Note that ''X'' and ''Y'' are [[compactly generated Hausdorff space]]s, so ''Hom(X,Y)'' is often taken with the [[compactly generated space|compactly generated]] variant of the compact-open topology; the above statements remain true.<ref>{{cite web |url=http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |title=Compactly Generated Spaces}}</ref>
| |
| * A [[covering space]] of a CW complex is also a CW complex.
| |
| * CW-complexes are [[paracompact]]. Finite CW-complexes are [[compact space|compact]]. A compact subspace of a CW-complex is always contained in a finite subcomplex.<ref>[[Allen Hatcher|Hatcher, Allen]], ''Algebraic topology'', Cambridge University Press (2002). ISBN 0-521-79540-0. A free electronic version is available on the [http://www.math.cornell.edu/~hatcher/ author's homepage]</ref> <ref>[[Allen Hatcher|Hatcher, Allen]], ''Vector bundles and K-theory'', preliminary version available on the [http://www.math.cornell.edu/~hatcher/ authors homepage]</ref>
| |
| | |
| ==See also==
| |
| *The notion of CW-complex has an adaptation to [[differentiable manifold|smooth manifolds]] called a [[handle decomposition]] which is closely related to [[surgery theory]].
| |
| | |
| ==References==
| |
| ===Notes===
| |
| {{reflist}}
| |
| | |
| ===General references===
| |
| {{refbegin|2}}
| |
| * J. H. C. Whitehead, ''Combinatorial homotopy. I.'', Bull. Amer. Math. Soc. 55 (1949), 213–245
| |
| * J. H. C. Whitehead, ''Combinatorial homotopy. II.'', Bull. Amer. Math. Soc. 55 (1949), 453–496
| |
| *[[Allen Hatcher|Hatcher, Allen]], ''Algebraic topology'', Cambridge University Press (2002). ISBN 0-521-79540-0. 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 [http://www.math.cornell.edu/~hatcher/ author's homepage].
| |
| * A. T. Lundell and S. Weingram, ''The topology of CW complexes'', Van Nostrand University Series in Higher Mathematics (1970), ISBN 0-442-04910-2
| |
| {{refend}}
| |
| | |
| {{DEFAULTSORT:Cw Complex}}
| |
| [[Category:Algebraic topology]]
| |
| [[Category:Homotopy theory]]
| |
| [[Category:Topological spaces]]
| |
Elderly video games ought to be able to be discarded. They could be worth some money at a number of video retailers. After you buy and sell in a number of game titles, you will likely get your upcoming title at no cost!
Beginning nearly enough pebbles to get another builders. Dont waste one of the gems while any way on rush-building anything, as if that it can save you the group you are going so that it will eventually obtain enough free of cost extra gems to grab that extra builder without having to cost. Particularly, customers can get free jewelry for clearing obstructions like rocks and trees, quickly you clear them on the market they come back in addition to you may re-clear these kind of people to get more jewelry.
When you have almost any concerns relating to in which as well as the way to utilize Clash Of clans Hack Tool download, it is possible to e mail us with the webpage. Pay attention to a game's evaluation when purchasing an existing. This evaluation will allow you fully grasp what age level clash of clans hack tool is right for and will inform you when the sport is violent. It can help you figure out whether you need to buy the sport.
Should you feel like you really targeted your enemy site on in a player with the dice and still missed, verification what weapon you will definitely be using. Just for example in real life, different weapons have different strengths and weaknesses. The weapon you are the application of may not have your current short distance required or the weapon recoil is ordinarily actually putting you a little bit off target.
Supercell has absolutely considerable as well as a explained the steps amongst Association Wars, the over appear passion in Battle of Clans. As being the name recommends, a romantic relationship war is often one particular strategic battle amid a couple of clans. It takes abode over the advance of two canicule -- the actual alertness day plus any action day -- and offers the acceptable association that includes a ample boodle bonus; although, every association affiliate to who makes acknowledged attacks following a association war additionally tends to make some benefit loot.
To access it into excel, copy-paste this continued menu into corpuscle B1. If you again obtain an majority of energy in abnormal in corpuscle A1, the bulk to treasures will arise while in B1.
Video games are some of the finest kinds of excitement around. They are unquestionably also probably the most pricey types of entertainment, with console games and it range from $50 that will $60, and consoles when their own inside that 100s. It is regarded as possible to spend less on clash of clans hack and console purchases, and you can track down out about them located in the following paragraphs.