CW complex

A CW-complex is a type of topological space with an additional structure (cell division) introduced by Whitehead to satisfy the needs of homotopy theory . The literature in Russian also uses the names cell space , cell decomposition and cell complex . The class of cell complexes is wider than the class of simplicial complexes , but at the same time it preserves the combinatorial nature, which allows efficient calculations.

Definitions

An open n -dimensional cell is a topological space homeomorphic to an open n -dimensional ball (in particular, a zero-dimensional cell is a singleton space). A CW complex is a Hausdorff topological space X represented as a union of open cells in such a way that for every open n- dimensional cell there is a continuous mapping f from a closed n- dimensional ball to X , the restriction of which to the interior of the ball is a homeomorphism onto this cell ( characteristic mapping ). In this case, two properties are assumed to be fulfilled:

(C) The boundary of each cell is contained in the union of a finite number of cells of smaller dimensions;
(W) A subset of the space X is closed if and only if its intersection with the closure of each cell is closed.

The designations C and W come from the English words closure-finiteness and weak topology . ^[1] ^[2]

The dimension of the cell complex is defined as the upper bound on the dimensions of its cells. The nth skeleton of the cell complex is the union of all its cells whose dimension does not exceed n , the standard notation for the nth skeleton of the cell complex X is X ⁿ or sk _n X. A subset of a cell complex is called a subcomplex if it is closed and consists of whole cells; In particular, any skeleton of a complex is its subcomplex.

Any CW-complex can be constructed inductively, using the following procedure: ^[3]

start with a discrete set $X^{0}$ ${\ displaystyle X ^ {0}}$ $X^{0}$ whose points are considered zero-dimensional cells;
by induction we form the nth skeleton from the ( n - 1) th, gluing n- dimensional cells to it by means of arbitrary continuous mappings $\varphi _{\alpha }:S^{n-1}\to X^{n-1}.$ ${\ displaystyle \ varphi _ {\ alpha}: S ^ {n-1} \ to X ^ {n-1}.}$ $\varphi _{\alpha }:S^{{n-1}}\to X^{{n-1}}.$ In other words, space $X^{n}$ ${\ displaystyle X ^ {n}}$ $X^n$ Is a factor space of a disconnected union $X^{n-1}$ ${\ displaystyle X ^ {n-1}}$ $X^{{n-1}}$ and set of balls $D_{\alpha }$ ${\ displaystyle D _ {\ alpha}}$ $D_{\alpha }$ in relation to equivalence $x\sim \varphi _{\alpha }(x),$ ${\ displaystyle x \ sim \ varphi _ {\ alpha} (x),}$ $x\sim \varphi _{\alpha }(x),$ if a $x\in \partial D_{\alpha }.$ ${\ displaystyle x \ in \ partial D _ {\ alpha}.}$
You can complete the inductive process at the final stage by setting $X=X^{n},$ ${\ displaystyle X = X ^ {n},}$ either continue it endlessly by putting $X=\varinjlim X_{i}$ ${\ displaystyle X = \ varinjlim X_ {i}}$ ^[4] . The direct limit topology coincides with the weak topology: a subset $\varinjlim X_{i}$ ${\ displaystyle \ varinjlim X_ {i}}$ is closed if and only if its intersection with each $X_{i}.$ ${\ displaystyle X_ {i}.}$

Examples

Space $\{re^{2\pi i\theta }:0\leq r\leq 1,\theta \in \mathbb {Q} \}\subset \mathbb {C}$ ${\ displaystyle \ {re ^ {2 \ pi i \ theta}: 0 \ leq r \ leq 1, \ theta \ in \ mathbb {Q} \} \ subset \ mathbb {C}}$ is homotopy equivalent to a CW-complex (since it is contractible ), but it is impossible to introduce the structure of a CW-complex on it (since all CW-complexes are locally contractible ).
A Hawaiian earring is an example of a topological space that is not homotopy equivalent to any CW-complex.
Any polyhedron is naturally endowed with the structure of a CW complex, and a graph with a one-dimensional CW complex.
An n -dimensional sphere admits a cell structure with one zero-dimensional cell and one n- dimensional cell (since the n- dimensional sphere is homeomorphic to the quotient space of an n- dimensional ball along its boundary). Another cellular partition uses the fact that the embedding of the "equator" $S^{n-1}\to S^{n}$ ${\ displaystyle S ^ {n-1} \ to S ^ {n}}$ divides the sphere into two n- dimensional cells (upper and lower hemispheres). By induction, this allows one to obtain a cell decomposition of an n- dimensional sphere with two cells in each dimension from 0 to n , and the use of the direct limit construction allows one to obtain a cell decomposition of a sphere $S^{\infty }$ ${\ displaystyle S ^ {\ infty}}$ .
$\mathbb {RP} ^{n}$ ${\ displaystyle \ mathbb {RP} ^ {n}}$ allows a cell structure with one cell in each dimension, and $\mathbb {CP} ^{n}$ ${\ displaystyle \ mathbb {CP} ^ {n}}$ - with one cell in each even dimension.
Grassmannian can be divided into cells called Schubert cells .
For any compact smooth manifold, we can construct a CW-complex homotopy equivalent to it (for example, using the Morse function ).

Cell homology

The singular homology of the CW complex can be calculated using cell homology , i.e. homology of the cell chain complex

\cdots \to {H_{n+1}}(X^{n+1},X^{n})\to {H_{n}}(X^{n},X^{n-1})\to {H_{n-1}}(X^{n-1},X^{n-2})\to \cdots ,

{\ displaystyle \ cdots \ to {H_ {n + 1}} (X ^ {n + 1}, X ^ {n}) \ to {H_ {n}} (X ^ {n}, X ^ {n- 1}) \ to {H_ {n-1}} (X ^ {n-1}, X ^ {n-2}) \ to \ cdots,}

Where $X^{-1}$ ${\ displaystyle X ^ {- 1}}$ defined as an empty set.

Group ${H_{n}}(X^{n},X^{n-1})$ ${\ displaystyle {H_ {n}} (X ^ {n}, X ^ {n-1})}$ is a free abelian group whose generators can be identified with oriented n- dimensional cells of the CW complex. Boundary mappings are constructed as follows. Let be $e_{n}^{\alpha }$ ${\ displaystyle e_ {n} ^ {\ alpha}}$ Is an arbitrary n- dimensional cell $X,$ ${\ displaystyle X,}$ $\chi _{n}^{\alpha }:\partial e_{n}^{\alpha }\cong S^{n-1}\to X^{n-1}$ ${\ displaystyle \ chi _ {n} ^ {\ alpha}: \ partial e_ {n} ^ {\ alpha} \ cong S ^ {n-1} \ to X ^ {n-1}}$ Is the restriction of its characteristic mapping to the boundary, and $e_{n-1}^{\beta }$ ${\ displaystyle e_ {n-1} ^ {\ beta}}$ Is an arbitrary ( n - 1) -dimensional cell. Consider the composition

\chi _{n}^{\alpha \beta }:S^{n-1}\,{\stackrel {\cong }{\longrightarrow }}\,\partial e_{n}^{\alpha }\,{\stackrel {\chi _{n}^{\alpha }}{\longrightarrow }}\,X^{n-1}\,{\stackrel {q}{\longrightarrow }}\,X^{n-1}/\left(X^{n-1}\setminus e_{n-1}^{\beta }\right)\,{\stackrel {\cong }{\longrightarrow }}\,S^{n-1},

{\ displaystyle \ chi _ {n} ^ {\ alpha \ beta}: S ^ {n-1} \, {\ stackrel {\ cong} {\ longrightarrow}} \, \ partial e_ {n} ^ {\ alpha } \, {\ stackrel {\ chi _ {n} ^ {\ alpha}} {\ longrightarrow}} \, X ^ {n-1} \, {\ stackrel {q} {\ longrightarrow}} \, X ^ {n-1} / \ left (X ^ {n-1} \ setminus e_ {n-1} ^ {\ beta} \ right) \, {\ stackrel {\ cong} {\ longrightarrow}} \, S ^ {n-1},}

where the first mapping identifies $S^{n-1}$ ${\ displaystyle S ^ {n-1}}$ with $\partial e_{n}^{\alpha },$ ${\ displaystyle \ partial e_ {n} ^ {\ alpha},}$ display $q$ ${\ displaystyle q}$ - factorization, and the last mapping identifies $X^{n-1}/\left(X^{n-1}\setminus e_{n-1}^{\beta }\right)$ {\ displaystyle X ^ {n-1} / \ left (X ^ {n-1} \ setminus e_ {n-1} ^ {\ beta} \ right)} with $S^{n-1}$ ${\ displaystyle S ^ {n-1}}$ using characteristic cell imaging $e_{n-1}^{\beta }$ ${\ displaystyle e_ {n-1} ^ {\ beta}}$ . Then the boundary mapping

d_{n}:{H_{n}}(X_{n},X_{n-1})\to {H_{n-1}}(X_{n-1},X_{n-2})

{\ displaystyle d_ {n}: {H_ {n}} (X_ {n}, X_ {n-1}) \ to {H_ {n-1}} (X_ {n-1}, X_ {n-2 })}

is given by the formula

{d_{n}}(e_{n}^{\alpha })=\sum _{\beta }\deg \left(\chi _{n}^{\alpha \beta }\right)e_{n-1}^{\beta },

{\ displaystyle {d_ {n}} (e_ {n} ^ {\ alpha}) = \ sum _ {\ beta} \ deg \ left (\ chi _ {n} ^ {\ alpha \ beta} \ right) e_ {n-1} ^ {\ beta},}

Where $\deg \left(\chi _{n}^{\alpha \beta }\right)$ ${\ displaystyle \ deg \ left (\ chi _ {n} ^ {\ alpha \ beta} \ right)}$ - degree of display $\chi _{n}^{\alpha \beta }$ ${\ displaystyle \ chi _ {n} ^ {\ alpha \ beta}}$ and the sum is taken over all ( n - 1) -dimensional cells $X$ ${\ displaystyle X}$ .

In particular, if there are no two cells in the cell complex whose dimensions differ by one, then all the boundary mappings vanish and the homology groups are free. For example, ${H_{n}}(\mathbb {CP} ^{n},\mathbb {Z} )=\mathbb {Z}$ ${\ displaystyle {H_ {n}} (\ mathbb {CP} ^ {n}, \ mathbb {Z}) = \ mathbb {Z}}$ for even $n$ ${\ displaystyle n}$ and zero for odd ones.

Properties

The homotopy category of CW-complexes, according to some experts, is the best option for constructing a theory of homotopy. ^[5] One of the “good” properties of CW-complexes is ( weak homotopy equivalence between CW-complexes is homotopy equivalence). For any topological space, there exists a weakly homotopy equivalent CW-complex to it. ^[6] Another useful result is that representable functors in the homotopy category of CW-complexes have a simple characterization in categorical terms ( ). The cylinder, cone, and superstructure above the CW complex have a natural cellular structure.

On the other hand, the product of CW-complexes with a natural decomposition into cells is not always a CW-complex - the product topology may not coincide with the weak topology if both complexes are not locally compact. However, the product topology in the category of compactly generated spaces coincides with the weak topology and always defines a CW-complex ^[7] . The space of functions Hom ( X , Y ) with a compact-open topology , generally speaking, is not a CW-complex, however, according to John Milnor's theorem ^[8] , it is homotopy equivalent to a CW-complex under the condition that X is compact.

The covering of the CW complex X can be endowed with the structure of the CW complex in such a way that its cells map homeomorphically onto the cells X.

The final CW complexes (complexes with a finite number of cells) are compact. Any compact subset of a CW complex is contained in a finite subcomplex.

Notes

↑ Whitehead, 1949 , p. 214.
↑ Fomenko, Fuchs, 1989 , p. 35.
↑ Hatcher, 2011 , p. 14.
↑ See article direct limit .
↑ For example, see D.O. Baladze . Cell division - article from the Mathematical Encyclopedia.
↑ Hatcher, 2011 , p. 445-446.
↑ Martin Arkowitz. Introduction to Homotopy Theory. - Springer, 2011 .-- S. 302 . - ISBN 9781441973290 .
↑ Milnor, John. On spaces having the homotopy type of a CW-complex // Trans. Amer. Math. Soc .. - 1959.- T. 90 . - S. 272–280 .

Literature

JHC Whitehead. Combinatorial homotopy. I. // Bull. Amer. Math. Soc .. - 1949. - Vol. 55. - P. 213–245.
JHC Whitehead. Combinatorial homotopy. II. // Bull. Amer. Math. Soc .. - 1949. - Vol. 55. - P. 453–496.
Hatcher, A. Algebraic Topology / Per. from English V.V. Prasolova, ed. T.E. Panova. - M .: ICMMO, 2011 .-- 688 p. - ISBN 978-5-94057-748-5 .
A. T. Fomenko, D. B. Fuchs. Homotopy topology course. - M .: Nauka, 1989 .-- 528 p.

[_d23c5fcbad0b897f-1] Whitehead, 1949 , p. 214.

[_8aa406956b89c474-2] Fomenko, Fuchs, 1989 , p. 35.

[_f5d1cbb67686d7b8-3] Hatcher, 2011 , p. 14.

[4] See article direct limit .

[5] For example, see D.O. Baladze . Cell division - article from the Mathematical Encyclopedia.

[_70b146e1c8fc4fb1-6] Hatcher, 2011 , p. 445-446.

[7] Martin Arkowitz. Introduction to Homotopy Theory. - Springer, 2011 .-- S. 302 . - ISBN 9781441973290 .

[8] Milnor, John. On spaces having the homotopy type of a CW-complex // Trans. Amer. Math. Soc .. - 1959.- T. 90 . - S. 272–280 .