CW Complexes

Procedure to Construct CW Complexes

CW Complex

A CW complex is a topological space $X$ constructed inductively from disks of increasing dimension. The construction proceeds as follows:

• Step $0$ ($0$-skeleton): Start with a discrete set of points, $X^0$, called the 0-cells.
• Step $n$ ($n$-skeleton): Assuming the $(n-1)$-skeleton $X^{n-1}$ has been constructed, the n-skeleton $X^n$ is formed by attaching $n$-dimensional disks $D_{\alpha }^n$ to $X^{n-1}$ via attaching maps ${\phi }_{\alpha }:S^{n-1}\to X^{n-1}$, where $S^{n-1}$ is the boundary of $D_{\alpha }^n$. The space $X^n$ is the quotient space: $X^n=X^{n-1}{\coprod }_{\alpha }{D}_{\alpha }^n/(x\sim {\phi }_{\alpha }(x)\text{ for }x\in \partial D_{\alpha }^n)$
• Step $\infty$ (The full space): The CW complex $X$ is the union of all its skeletons, $X={\bigcup }_{n=0}^{\infty }{X}^n$, endowed with the weak topology: a set $A\subset X$ is open (or closed) if and only if its intersection $A\cap X^n$ with every skeleton is open (or closed) in $X^n$.

An n-cell refers to one of the disks $D^n$ attached at step $n$. Specifically, $D_{\alpha }^n$ is a closed n-cell, and its interior, $\text{int}(D_{\alpha }^n)$, is an open n-cell. For each cell $D_{\alpha }^n$, the composition of the quotient map with the inclusion of the disk $D_{\alpha }^n$ gives a characteristic map ${\Phi }_{\alpha }:D_{\alpha }^n\to X$.

Why "CW"?

The C in CW stands for “closure-finite”, and the W for “weak” topology.

e.g.

• The sphere $S^2$ can be given a CW structure with:
  • Two 0-cells: $v$ and $w$. So the 0-skeleton is $(S^2{)}^0=\{v,w\}$.
  • Two 1-cells: $a$ and $b$. The attaching map for $a$ sends the endpoints of $D^1$ to $v$ and $w$, and the map for $b$ sends the endpoints to $w$ and $v$. The 1-skeleton is two arcs joining $v$ and $w$, forming a circle.
  • Two 2-cells: $E$ and $F$. These correspond to the two hemispheres. Their boundaries ($S^1$) are attached to the 1-skeleton. The attaching map for $E$ is ${\phi }_E=a\cdot b$, which traces the circular 1-skeleton. Similarly, the attaching map for $F$ is ${\phi }_F=b\cdot a$, which traces the same circle.
• The closed orientable surface of genus $g$, $M_g$, has a very economical CW structure:
  • One 0-cell: $v$.
  • $2g$ 1-cells: $a_1,b_1,\dots ,a_g,b_g$. The 1-skeleton $(M_g{)}^1$ is a wedge of $2g$ circles.
  • One 2-cell: $E$. This single 2-cell is attached to the 1-skeleton. Its boundary is identified with the path given by the product of commutators: $\phi (\partial E)=[a_1,b_1][a_2,b_2]\cdots [a_g,b_g]=a_1{b}_1{a}_1^{-1}{b}_1^{-1}\cdots a_g{b}_g{a}_g^{-1}{b}_g^{-1}.$

Euler Characteristic

Euler Characteristic

For a finite CW complex $X$ (one with a finite number of cells), the Euler characteristic $\chi (X)$ is defined as the alternating sum of the number of cells in each dimension: $\chi (X)={\sum }_{n\ge 0}(-1{)}^n(\text{number of n-cells in }X)$ This generalizes the familiar formula $V-E+F$ for polyhedra.

e.g.

• For the 2-sphere $S^2$, using the structure with 2 0-cells, 2 1-cells, and 2 2-cells: $\chi (S^2)=2-2+2=2$.
• For the genus $g$ surface $M_g$, using the structure with 1 0-cell, $2g$ 1-cells, and 1 2-cell: $\chi (M_g)=1-2g+1=2-2g$.

The Fundamental Group of a CW Complex

The CW structure is particularly useful for computing the fundamental group.

Proposition

Attaching cells of dimension $n\ge 3$ does not change the fundamental group.

Proof Sketch: Let $Y=X{\cup }_{\phi }{D}^n$ with $n\ge 3$. We apply the van Kampen’s theorem. The space can be divided into two open sets: $A=Y\setminus \{\text{center of }{D}^n\}$ and $B=\text{int}(D^n)$. The intersection $A\cap B$ deformation retracts to $S^{n-1}$. For $n\ge 3$, the sphere $S^{n-1}$ is simply connected, so its fundamental group is trivial. The Seifert-van Kampen theorem then implies that ${\pi }_1(Y)\cong {\pi }_1(A)\ast {\pi }_1(B)\cong {\pi }_1(X)\ast \{e\}\cong {\pi }_1(X)$. $\square$

This means that ${\pi }_1(X)$ depends only on its 2-skeleton, $X^2$. The effect of attaching 2-cells is described by the following theorem.

Theorem

Let $Y$ be the space obtained from a path-connected space $X$ by attaching a 2-cell $D^2$ via an attaching map $\phi :S^1\to X$. Let $x_0$ be a basepoint in $X$. The inclusion map $i:X\hookrightarrow Y$ induces a surjective homomorphism $i_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,x_0)$. The kernel of $i_{\ast }$ is the normal subgroup $N$ generated by the element $[h\cdot \phi \cdot h^{-1}]$, where $h$ is any path from $x_0$ to the loop $\phi (S^1)$. Therefore, ${\pi }_1(Y,x_0)\cong {\pi }_1(X,x_0)/N.$

Proof Let $Y=X{\cup }_{\phi }{D}^2$. We choose two open sets for the theorem:

1. $A=Y\setminus \{p\}$, where $p$ is the center point of the attached disk $D^2$. This set is path-connected and deformation retracts onto $X$, so ${\pi }_1(A,x_0)\cong {\pi }_1(X,x_0)$.
2. $B=\text{int}(D^2)$. This set is an open disk, which is contractible. Therefore, ${\pi }_1(B,x_1)$ is the trivial group for any basepoint $x_1\in B$.

The intersection $A\cap B$ is an open annulus, which deformation retracts to a circle $S^1$. Its fundamental group is infinite cyclic, ${\pi }_1(A\cap B,x_1)\cong \mathbb{Z}$, generated by a loop $[\psi ]$ that goes around the puncture $p$. By the van Kampen’s theorem, ${\pi }_1(Y,x_0)$ is the amalgamated product of ${\pi }_1(A,x_0)$ and ${\pi }_1(B,x_0)$ over ${\pi }_1(A\cap B,x_1)$. The generator $[\psi ]$ is trivial in ${\pi }_1(B)$ and corresponds to the attaching loop $[\phi ]$ in ${\pi }_1(A)$. The theorem states that ${\pi }_1(Y,x_0)$ is the quotient of ${\pi }_1(A,x_0)$ by the normal subgroup generated by the image of the loop from the intersection. To express the loop $[\phi ]$ relative to the basepoint $x_0$, we conjugate it by a path $h$ connecting $x_0$ to the loop. The corresponding element in ${\pi }_1(X,x_0)$ is $[h\cdot \phi \cdot h^{-1}]$. Setting this element to the identity (since it’s trivial in the other part of the amalgam) gives the result: ${\pi }_1(Y,x_0)\cong {\pi }_1(A,x_0)/⟨⟨[h\cdot \phi \cdot h^{-1}]⟩⟩\cong {\pi }_1(X,x_0)/⟨⟨[h\cdot \phi \cdot h^{-1}]⟩⟩$ where $⟨⟨\cdot ⟩⟩$ denotes the normal closure of the subgroup. $\square$

Real Projective Space as a CW Complex

The real projective space $\mathbb{R}{\mathrm{P}}^n$ can be defined in several equivalent ways:

1. As the space of all lines passing through the origin in ${\mathbb{R}}^{n+1}$.
2. As the quotient space $({\mathbb{R}}^{n+1}\setminus \{0\})/\sim$, where $x\sim \lambda x$ for any non-zero real number $\lambda$.
3. As the quotient of the $n$-sphere $S^n$ by identifying antipodal points: $x\sim -x$.
4. As the quotient of the $n$-disk $D^n$ by identifying antipodal points on its boundary: $x\sim -x$ for $x\in \partial D^n$.

Proposition

${\text{RP}}^n$ is a CW complex with exactly one $k$-cell for each dimension $k$ from $0$ to $n$.

Proof Sketch We can show this by induction. We construct ${\text{RP}}^n$ from ${\text{RP}}^{n-1}$ by attaching a single n-cell.

• The attaching map for the n-cell ($D^n$) is the quotient map $p:\partial (D^n)=S^{n-1}\to {\text{RP}}^{n-1}$.
• The resulting space is ${\text{RP}}^{n-1}{\cup }_p{D}^n$. This space is homeomorphic to taking the n-disk $D^n$ and identifying antipodal points $x$ and $-x$ on its boundary $\partial D^n$.
• This construction, $D^n/(x\sim -x\text{ for }x\in \partial D^n)$, is precisely the definition of ${\text{RP}}^n$.
• Since ${\text{RP}}^0$ is a single point (a 0-cell), we can inductively build ${\text{RP}}^n=e^0\cup e^1\cup \cdots \cup e^n$.

$\square$