CW complex

A CW complex is built inductively by attaching discs along maps from their boundary spheres. Its $n$-skeleton contains all cells of dimension at most $n$.

A compact subset of a CW complex is contained in a finite CW complex that is a subcomplex of the original complex. Otherwise choose one point in each of infinitely many cells met by the subset. The closure-finiteness axiom makes each closed cell meet the chosen set in finitely many points. The weak topology axiom then makes the chosen set, and each of its subsets, closed. It is consequently an infinite closed discrete subspace of the compact subset, a contradiction. Only finitely many cells are met; their closures form a finite subcomplex containing the subset.

The $q$-skeleton of a CW complex is the subcomplex consisting of its topological cells of dimension at most $q$. A cellular map preserves these skeleta. The cellular approximation theorem lets continuous maps respect this filtration up to homotopy.

An open $q$-cell of a CW complex is a copy of an open $q$-dimensional ball. Its characteristic map is a map from a closed ball whose interior parametrizes that cell and whose boundary lands in the preceding skeleton. The characteristic map need not be injective on the boundary. Each oriented cell gives one generator of the cellular chain complex.

A CW pair consists of a CW complex $X$ and a subcomplex $A$. Its inclusion is a cofibration, so homotopies on $A$ extend across $X$. The relative lifting property of a Serre fibration for such pairs allows specified boundary lifts to be retained while lifting a homotopy.

Attaching a $q$-cell along $\varphi :S^{q-1}\to X$ forms the quotient of $X\bigsqcup D^q$ identifying each boundary point with its image under $\varphi$. A constant attaching map gives $X\vee S^q$. An attaching map can affect the cohomology ring even when it does not change the additive cohomology groups.

Attaching cells of dimension $i+1$ to a connected CW complex, for $i\ge 2$, preserves all homotopy groups below degree $i$. In degree $i$ it takes the quotient by the subgroup generated by the attaching maps and their translates under the fundamental group action. This follows from the relative homotopy sequence and the generation of the first nonzero relative group by the new cells. In particular, a single attaching map representing a generator of ${\pi }_i\cong \mathbb{Z}$ kills that group even when the fundamental group is nontrivial. Higher attachments can create higher homotopy groups, which may then be killed at later stages.

The weak topology of a CW complex is determined by its closed cells: a subset is closed if and only if its inverse image under every characteristic map from a closed cell is closed. A map out of the complex is continuous when its restrictions along those characteristic maps are continuous. This is distinct from the weak topology of a vector space. Taking the product with a locally finite CW complex, such as a closed interval with its finite cell structure, retains the corresponding CW weak topology.

A topological graph is a CW complex with only zero-cells and one-cells, called vertices and edges. Edge endpoints attach to vertices; loops, multiple edges and infinitely many edges are allowed. A subgraph in this setting is a CW subcomplex. A connected acyclic graph is a tree and is a contractible space.

A CW subcomplex is a union of cells of a CW complex containing the attaching boundary of every included cell. In a combinatorial 2-complex, including a two-cell therefore includes every edge in its boundary. A connected CW subcomplex has a connected 1-skeleton, whose intrinsic path metric may differ from the ambient graph metric.

The one-skeleton $X^{(1)}$ of a CW complex is the union of its zero-cells and one-cells. It is a graph. The unit-edge path metric measures the infimum of lengths of paths in this graph; it is different from metrics that allow paths through two-cell interiors.

A two-dimensional CW complex has cells only in dimensions zero, one and two. In a combinatorial two-complex the two-cells are polygons attached by combinatorial edge paths. Its 1-skeleton carries the path metric assigning length one to each edge. Compact combinatorial complexes have finitely many cells, so their cell-boundary lengths have a finite maximum.

A disc diagram over a combinatorial 2-complex is a finite contractible planar combinatorial complex together with a combinatorial map to that complex. Its exterior boundary circuit records a null-homotopic path. Tree portions and cut vertices are allowed; the diagram need not be an embedded disk or map injectively. Its area is its number of two-cells.

A ladder is a disc diagram whose blocks form a linear chain, each block a two-cell or a connecting edge, with cells attached successively along interior arcs. Its boundary consists of two opposite side paths between its ends. Every cell meets both side paths. If cell perimeters are bounded by $L$, each point of one side is within $L/2$ of the other by traveling around a cell boundary; connecting edge paths coincide on the two sides.

A spur is an exposed degree-one vertex and its incident edge in a disc diagram. The exterior boundary runs out along that edge and immediately back. Thus a spur in the interior of a boundary side contradicts that side being a reduced combinatorial path or a metric geodesic. A diagram with no two-cells may be a tree and have spurs rather than shells.

A shell is an exposed two-cell in a disc diagram, with perimeter split as an exterior boundary arc $Q$ and an interior path $P$. An $i$-shell has $P$ consisting of $i$ interior pieces. Under $C^{\prime }(1/6)$, an $i$-shell with $i\le 3$ has $|P|<|\partial R|/2<|Q|$, so removal replaces a long exterior path by a strictly shorter path.

A closed combinatorial path in a 2-complex is null-homotopic exactly when it is the boundary path of a disc diagram over that complex. A finite cellular null-homotopy can be arranged into such a planar diagram; conversely the contractible diagram supplies a null-homotopy. In a group presentation, this is the geometric form of expressing a trivial word as a product of conjugates of relators.

A disc diagram is reduced if it has no adjacent pair of two-cells that can be folded together across their common boundary and removed while preserving the outside boundary. A minimum-area diagram is reduced. Among diagrams of minimum area, minimizing the number of edges removes unnecessary tree portions. Internal common arcs in a reduced diagram satisfy the applicable small cancellation theory piece bounds.

A cellular map $f:X\to Y$ between CW complexes satisfies $f(X^q)\subseteq Y^q$ for every skeleton. It induces a chain map $f_{\#}:C_{\ast }^{\mathrm{cell}}(X)\to C_{\ast }^{\mathrm{cell}}(Y)$.

A continuous map between CW complexes is homotopic to a cellular map. If the map is already cellular on a subcomplex, the homotopy can be taken relative to that subcomplex. In particular, a map from a $d$-dimensional CW complex can be deformed into the $d$-skeleton of the target. This permits finite cellular chain complexes to compute induced homology maps, and turns maps to Complex projective space into maps to an appropriate finite-dimensional projective skeleton.

A map from the $n$-sphere to a CW complex can be homotoped into its $n$-skeleton. By compact subsets of CW complexes lie in finite subcomplexes, its image lies in a finite subcomplex. For a top-dimensional cell of dimension $d>n$, use smooth approximation of maps into Euclidean space in its interior, with a cutoff supported away from the cell boundary. The Sard theorem supplies a point in a smaller interior ball omitted by the smoothed map; the cutoff is chosen so the transition region also avoids that point. Radial retraction from the omitted point pushes the image into the cell boundary and fixes all other cells. Repeating over the finitely many higher-dimensional cells proves the assertion, without assuming the initial continuous image has measure zero.

A finite CW complex has finitely many cells. Its cellular chain complex consists of finitely generated free abelian groups and vanishes outside finitely many degrees, so its integral homology groups are finitely generated.

Suppose a compact skeleton $Y$ embeds as $e(Y)\subset {\mathbb{R}}^m$. Attach a disk by $a:S^{k-1}\to Y$ and extend its boundary coordinate map radially by $G(ru)=re(a(u))$. Map the disk to $(G(x),(1-|x|)x,1-|x|)$ and $Y$ to $(e(y),0,0)$. The boundary identifications agree; a positive last coordinate recovers each interior point. This defines a continuous injection of the compact quotient into Euclidean space, hence an embedding. Repeating proves Hausdorffness and Euclidean embeddability for every finite CW complex, without assuming triangulability of arbitrary attaching maps.

A CW approximation of a space $X$ is a CW complex $X_{\mathrm{cw}}$ with a weak homotopy equivalence $X_{\mathrm{cw}}\to X$. The realization of the singular simplicial set of $X$ supplies one functorially.

For simply connected $X$ with finitely generated integral homology in every degree, the Hurewicz theorem modulo a Serre class gives finitely generated homotopy groups. Realize finite generating sets of successive relative homotopy groups by attaching finitely many cells at each stage. The resulting weak equivalence has finitely many cells in each dimension.

Start with a finite $n$-dimensional $n$-connected approximation. Its degree-$n$ homology kernel is finite free. Realize a basis via the Relative Hurewicz theorem and attach $(n+1)$-cells; their boundary is injective, so they kill precisely that kernel without adding top homology. The homological Whitehead theorem gives a finite weak equivalence of dimension at most $n+1$.

A CW filtration builds a CW complex by its skeletawhere each quotient $X^n/X^{n-1}$ is a wedge of $n$-spheres, one for each $n$-cell.

For an abelian group $G$ and an integer $m\ge 1$, a Moore space $M(G,m)$ is a CW complex whose reduced homology is $G$ in degree $m$ and zero in every other degree.

Attach an $(i+1)$-cell to $S^i$ by a degree-$n$ map, with $i\ge 1$ and $n\ge 2$. The integral cellular differential is multiplication by $n$, while its modulo-$n$ reduction is zero. Lifting the degree-$i$ generator modulo $n^2$ and applying the differential gives $n$, so the Bockstein homomorphism sends that generator to the degree-$(i+1)$ generator. This supplies a nonzero example in every positive degree.

Attaching a two-cell to $S^1$ by a map of degree $n$ gives the Moore space $M(\mathbb{Z}/n,1)$. Its positive-dimensional cellular chain complex isso $H_1\cong \mathbb{Z}/n$ and all its other reduced integral homology groups vanish.