\section{CW-complexes} \begin{cislo}\label{CWconstruction}{\bf Constructive definition of CW-complexes.} \emph{CW-complexes}\index{CW-complex} are all the spaces which can be obtained by the following construction: \begin{enumerate} \item We start with a discrete space $X^0$. Single points of $X^0$ are called 0-dimensional cells. \item Suppose that we have already constructed $X^{n-1}$. For every element $\alpha$ of an index set $J_n$ take a map $f_{\alpha}:S^{n-1}=\partial D^n_{\alpha}\to X^{n-1}$ and put $$X^n=\bigcup_{\alpha}\left(X^{n-1}\cup_{f_{\alpha}} D_{\alpha}^n\right ).$$ Interiors of discs $D_{\alpha}^n$ are called $n$-dimensional cells and denoted by $e_{\alpha}^n$. \item We can stop our construction for some $n$ and put $X=X^n$ or we can proceed with $n$ to infinity and put $$X=\bigcup_{n=0}^{\infty}X^n.$$ In the latter case $X$ is equipped with inductive topology which means that $A\subseteq X$ is closed (open) iff $A\cap X^n$ is closed (open) in $X^n$ for every $n$. \end{enumerate} \begin{example} The sphere $S^n$ is a CW-complex with one cell $e^0$ in dimension $0$, one cell $e^n$ in dimension $n$ and the constant attaching map $f:S^{n-1}\to e^0$. \end{example} \begin{example} The real projective space $\mathbb{RP}^n$ is the space of $1$-dimensional linear subspaces in $\mathbb R^{n+1}$. It is homeomorhic to $$S^n/(v\simeq -v)\cong D^n/(w\simeq-w),\quad \text{ for }w\in \partial D^n=S^{n-1}.$$ However, $S^{n-1}/(w\simeq-w)\cong\mathbb{RP}^{n-1}$. So $\mathbb{RP}^n$ arises from $\mathbb{RP}^{n-1}$ by attaching one $n$-dimensional cell using the projection $f:S^{n-1}\to \mathbb{RP}^{n-1}$. Hence $\mathbb{RP}^{n}$ is a CW-complex with one cell in every dimension from $0$ to $n$. We define $\mathbb{RP^{\infty}}=\bigcup_{n=1}^{\infty}\mathbb{RP}^n$. It is again a CW-complex. \end{example} \begin{example} The complex projective space $\mathbb{CP}^n$ is the space of complex $1$-dimensional linear subspaces in $\mathbb C^{n+1}$. It is homeomorhic to \begin{align*} S^{2n+1}/(v\simeq \lambda v) &\cong\{(w,\sqrt{1-\vert w\vert^2})\in\mathbb C^{n+1};\ \Vert w\Vert\le 1\}/((w,0)\simeq\lambda (w,0),\ \Vert w\Vert =1)\\ &\cong D^{2n}/(w\simeq\lambda w;\ w\in\partial D^{2n}) \end{align*} for all $\lambda\in \mathbb C$, $\vert \lambda\vert=1$. However, $\partial D^{2n}/(w\simeq\lambda w)\cong\mathbb{CP}^{n-1}$. So $\mathbb{CP}^n$ arises from $\mathbb{CP}^{n-1}$ by attaching one $2n$-dimensional cell using the projection $f:S^{2n-1}=\partial D^{2n}\to \mathbb{CP}^{n-1}$. Hence $\mathbb{CP}^{n}$ is a CW-complex with one cell in every even dimension from $0$ to $2n$. Define $\mathbb{CP^{\infty}}=\bigcup_{n=1}^{\infty}\mathbb{CP}^n$. It is again a CW-complex. \end{example} \end{cislo} \begin{cislo}\label{CWdef}{\bf Another definition of CW-complexes.} Sometimes it is advantageous to be able to describe CW-complexes by their properties. We carry it out in this paragraph. Then we show that the both definitions of CW-complexes are equivalent. \begin{defin*}\label{CW} A \emph{cell complex}\index{cell complex} is a Hausdorff topological space $X$ such that \begin{enumerate} \item $X$ as a set is a disjoint union of cells $e_{\alpha}$ $$X=\bigcup_{\alpha\in J}e_{\alpha}.$$ \item For every cell $e_{\alpha}$ there is a number, called dimension. $$X^n=\bigcup_{\dim e_{\alpha}\le n}e_{\alpha}$$ is the $n$-skeleton of $X$. \item Cells of dimension 0 are points. For every cell of dimension $\ge 1$ there is a characteristic map $$\varphi_{\alpha}:(D^n,S^{n-1})\to (X,X^{n-1})$$ which is a homeomorphism of $\inte D^n$ onto $e_{\alpha}$. \end{enumerate} The \emph{cell subcomplex}\index{cell subcomplex} $Y$ of a cell complex $X$ is a union $Y=\bigcup_{\alpha\in K}e_{\alpha}$ , $K\subseteq J$, which is a cell complex with the same characterictic maps as the complex $X$. A \emph{CW-complex}\index{CW-complex} is a cell complex satisfying the following conditions: \begin{enumerate} \item[(C)] Closure finite property. The closure of every cell belongs to a finite subcomplex, i.~e. subcomplex consisting only from a finite number of cells. \item[(W)] Weak topology property. $F$ is closed in $X$ if and only if $F\cap \bar e_{\alpha}$ is closed for every $\alpha$. \end{enumerate} \end{defin*} \end{cislo} \begin{example*} Examples of cell complexes which are not CW-complexes: \begin{enumerate} \item $S^2$ where every point is 0-cell. It does not satisfy property (W). \item $D^3$ with cells $e^3=\inte B^3$, $e_{x}^0=\{x\}$ for all $x\in S^2$. It does not satisfy (C). \item $X=\{1/n;\ n\ge 1\}\cup\{0\}\subset \mathbb R$. It does not satisfy (W). \item $X=\bigcup_{n=1}^{\infty}\{x\in\mathbb R^2;\ \Vert x-(1/n,0)\Vert=1/n\}\subset\mathbb R^2$. If it were a CW-complex, the set $\{(1/n,0)\in\mathbb R^2;\ n\ge 1\}$ would be closed in $X$, and consequently in $\mathbb R^2$. \end{enumerate} \end{example*} \begin{cislo}\label{CWeq}{\bf Equivalence of definitions.} \begin{prop*} The definitions \ref{CWconstruction} and \ref{CW} of CW-complexes are equivalent. \end{prop*} \begin{proof} We will show that a space $X$ constructed according to \ref{CWconstruction} satisfies definition \ref{CW}. The proof in the opposite direction is left as an exercise to the reader. The cells of dimension 0 are points of $X^0$. The cells of dimension $n$ are interiors of discs $D_{\alpha}^n$ attached to $X^{n-1}$ with charakteristic maps $$\varphi_{\alpha}:(D_{\alpha}^n,S^{n-1}_{\alpha})\to (X^{n-1}\cup_{f_{\alpha}}D_{\alpha}^n,X^{n-1})$$ induced by identity on $D_{\alpha}^n$. So $X$ is a cell complex. From the construction \ref{CWconstruction} it follows that $X$ satisfies property (W). It remains to prove property (C). We will carry it out by induction. Let $n=0$. Then $\overline{e_{\alpha}^0}=e_{\alpha}^0$. Let (C) holds for all cells of dimension $\le n-1$. $\overline{e_{\alpha}^n}$ is a compact set (since it is an image of $D_{\alpha}^n$). Its boundary $\partial e_{\alpha}^n$ is compact in $X^{n-1}$. Consider the set of indices $$K=\{\beta\in J;\ \partial e_{\alpha}^n\cap e_{\beta}\ne\emptyset\}.$$ If we show that $K$ is finite, from the inductive assumption we get that $\bar e_{\alpha}^n$ lies in a finite subcomplex which is a union of finite subcomplexes for $\bar e_{\beta}$, $\beta\in K$. Choosing one point from every intersection $\partial e_{\alpha}^n\cap e_{\beta}$, $\beta\in K$ we form a set $A$. $A$ is closed since any intersection with a cell is empty or a onepoint set. Simultaneously, it is open, since every its element $a$ forms an open subset (for $A-\{a\}$ is closed). So $A$ is a discrete subset in the compact set $\partial e_{\alpha}^n$, consequently, it is finite. \end{proof} \end{cislo} \begin{cislo}\label{CWcompact} {\bf Compact sets in CW complexes.} \begin{lemma*} Let $X$ be a CW-complex. Then any compact set $A\subseteq X$ lies in a finite subcomplex, particularly, there is $n$ such that $A\subseteq X^{n}$. \end{lemma*} \begin{proof} Consider the set of indices $$K=\{\beta\in J;\ A\cap e_{\beta}\ne\emptyset\}.$$ Similarly as in \ref{CWeq} we will show that $K$ is a finite set. Then $A\subseteq\bigcup_{\beta\in K}\bar e_{\beta}$ and every $\bar e_{\beta}$ lies in a finite subcomplexes. Hence $A$ itself is a subset of a finite subcomplex. \end{proof} \end{cislo} \begin{cislo}\label{CWmaps}{\bf Cellular maps.} Let $X$ and $Y$ be CW-complexes. A map $f:X\to Y$ is called a \emph{cellular map}\index{cellular map} if $f(X^n)\subseteq Y^n$ for all $n$. In Section 5 %\ref{HCWca} we will prove that every map $g:X\to Y$ is homotopic to a cellular map $f:X\to Y$. If moreover, $g$ restricted to a subcomplex $A\subset X$ is already cellular, $f$ can be chosen in such a way that $f=g$ on $A$. \end{cislo} \begin{cislo}{\bf Spaces homotopy equivalent to CW-complexes.} One can show that every open subset of $\mathbb R^n$ is a CW-complex. In [Hatcher], %\cite{Ha}, Theorem A.11, it is proved that every retract of a CW-complex is homotopy equivalent to a CW-complex. These two facts imply that every compact manifold with or without boundary is homotopy equivalent to a CW-complex. (See [Hatcher], %\cite{Ha}, Corollary A.12.) \end{cislo} \begin{cislo}\label{CWhep}{\bf CW complexes and HEP.} The most important result of this section is the following theorem: \begin{thm*} Let $A$ be a subcomplex of a CW-complex $X$. Then the pair $(X,A)$ has the homotopy extension property. \end{thm*} \begin{proof} According to the last theorem in Section 1 it is sufficient to prove that $X\times\{0\}\cup A\times I$ is a retract of $X\times I$. We will prove that it is even a deformation retract. There is a retraction $r_n:D^n\times I\to D^n\times\{0\}\cup S^{n-1}\times I$. (See Section 1.) Then $h_n:D^n\times I\times I\to D^n\times I$ defined by $$h_n(x,s,t)=(1-t)(x,s)+tr_n(x,s)$$ is a deformation retraction, i.e. a homotopy between $\id$ and $r_n$. Put $Y^{-1}=A$, $Y^{n}=X^{n}\cup A$. Using $h_n$ we can define a deformation retraction $H_n:Y^{n}\times I\times I \to Y^n\times I$ for the retract $Y^n\times \{0\}\cup Y^{n-1}\times I$ of $Y^n\times I$. Now define the deformation retraction $H: X\times I\times I\to X\times I$ for the retract $X\times\{0\}\cup A\times I$ succesively on the subspaces $X\times \{0\}\times I\cup Y^n\times I\times I$ with values in $X\times \{0\}\cup Y^n\times I$. For $n=0$ put \begin{align*} H(x,s,t)&=(x,s) \qquad \text{for }(x,s)\in X\times\{0\} \text{ or } t\in [0,1/2],\\ H(x,s,t)&=H_0(x,s,2(t-1/2)) \quad \text{for }x\in Y^0 \text{ and } t\in [1/2,1]. \end{align*} Suppose that we have already defined $H$ on $X\times \{0\}\cup Y^{n-1}\times I$. On $X\times \{0\}\cup Y^n\times I$ we put \begin{align*} H(x,s,t)&=(x,s) \qquad \text{for }(x,s)\in X\times\{0\} \text{ or } t\in [0,1/2^{n+1}],\\ H(x,s,t)&=H_n(x,s,2^{n+1}(t-1/2^{n+1})) \quad \text{for }x\in Y^n \text{ and } t\in [1/2^{n+1},1/2^n],\\ H(x,s,t)&=H(H(x,s,1/2^n),t) \quad \text{ for }x\in Y^n \text{ and } t\in [1/2^{n},1]. \end{align*} $H:X\times I\times I\to X\times I$ is continuous since so are its restrictions on $X\times \{0\}\times I\cup Y^n\times I\times I$ and the space $X\times I\times I$ is a direct limit of the subspaces $X\times \{0\}\times I\cup Y^n\times I\times I$. %vlozeno 20130304 \begin{figure}[htb] \centering \def\svgwidth{400pt} \input{img/02_fig-5.pdf_tex} \caption{Image of H depending on t} \end{figure} %vlozeno 20130304 \end{proof} \end{cislo} \begin{cislo}\label{CW1crit}{\bf First criterion for homotopy equivalence.} \begin{prop*}Suppose that a pair $(X,A)$ has the homotopy extension property and that $A$ is contractible (in $A$). Then the canonical projection $q:X\to X/A$ is a homotopy equivalence. \end{prop*} \begin{proof} Since $A$ is contractible, there is a homotopy $h:A\times I\to A$ between $\id_A$ and constant map. This homotopy together with $\id_X:X\to X$ can be extended to a homotopy $f:X\times I\to X$. Since $f(A,t)\subseteq A$ for all $t\in I$, there is a homotopy $\tilde f:X/A\times I\to X/A$ such that the diagram $$ \xymatrix{ X\times I \ar[r]^f \ar[d]_q& X \ar[d]^q \\ X/A\times I \ar[r]_{\tilde f} & X/A } $$ commutes. Define $g:X/A\to X$ by $g([x])=f(x,1)$. Then $\id_X\sim g\circ q$ via the homotopy $f$ and $\id_{X/A}\sim q\circ g$ via the homotopy $\tilde f$. Hence $X$ is homotopy equivalent to $X/A$. \end{proof} \begin{ex} Using the previous criterion show that $S^2/S^0\sim S^2\lor S^1$. \end{ex} \begin{ex} Using the previous criterion show that the suspension and the reduced suspension of a CW-complex are homotopy equivalent. \end{ex} \end{cislo} \begin{cislo}\label{CW2crit} {\bf Second criterion for homotopy equivalence.} \begin{prop*} Let $(X,A)$ be a pair of CW-complexes and let $Y$ be a space. Suppose that $f,g:A\to Y$ are homotopic maps. Then $X\cup_{f}Y$ and $X\cup_{g}Y$ are homotopy equivalent. \end{prop*} \begin{proof} Let $F:A\times I\to Y$ be a homotopy between $f$ and $g$. We will show that $X\cup_{f}Y$ and $X\cup_{g}Y$ are both deformation retracts of $(X\times I)\cup_{F}Y$. Consequently, they have to be homotopy equivalent. We construct a deformation retraction in two steps. \begin{enumerate} \item $(X\times\{0\})\cup_f Y$ is a deformation retract of $(X\times\{0\}\cup A\times I)\cup_F Y$. \item $(X\times\{0\}\cup A\times I)\cup_F Y$ is a deformation retract of $(X\times I)\cup_F Y$. \end{enumerate} \end{proof} \begin{ex*} Let $(X,A)$ be a pair of CW-complexes. Suppose that $A$ is a contractible in $X$, i.~e. there is a homotopy $F:A\to X$ between $\id_X$ and $\operatorname{const}$. Using the first criterion show that $X/A\cong X\cup CA/CA\sim X\cup CA$. Using the second criterion prove that $X\cup CA\sim X\lor SA$. Then $$X/A\sim X\lor SA.$$ Apply it to compute $S^n/S^i$, $i