Homotopy Theory

Arun Ram
Department of Mathematics
University of Wisconsin
Madison, WI 53706 USA
ram@math.wisc.edu

Maps

If $X$ and $Y$ are topological spaces

$\mathrm{Map}(X,Y)=\{f:X\to Y\mid f\text{is continuous}\}.$

A based space is a topological space $X$ with a distinguished point$x_0$, the basepoint of $X$. If $X$ and$Y$ are based spaces

${\mathrm{Map}}_{*}(X,Y)=\{f\in \mathrm{Map}(X,Y)\mid f(x_0)=y_0\}\text{,}$

where $x_0$ is the basepoint of $X$ and $y_0$ is the basepoint of $Y$.

The wedge of $X$ and $Y$ is the subspace of $X\times Y$ given by

$X\vee Y=(X\times y_0)\cup (x_0\times Y)\text{and}X\wedge Y=\frac{X\times Y}{langle;(x,y_0)=(x_0,y_0)=(x_0,y)rangle;}$

is the smash of $X$ and $Y$. Then

$\mathrm{Map}(X\vee Y,Z)=\mathrm{Map}(X,\mathrm{Map}(Y,Z\left)\text{and}{\mathrm{Map}}_{*}\right(X\wedge Y,Z)={\mathrm{Map}}_{*}(X,{\mathrm{Map}}_{*}(Y,Z))$.

Homotopy is the equivalence relation on $\mathrm{Map}(X,Y)$ given by

$f_1\simeq f_2\text{if there exists}F:X\times \left[0.1\right]⟶Y\text{such that}F(x,0)=f_1\left(x\right)\text{ and }F(x,1)=f_2\left(x\right)$,

for $x\in X$. Let

$[X,Y]=\mathrm{Map}(X,Y)/\text{homotopy}$.

Fundamental group, loop space and suspension

The $n$-sphere is

${\displaystyle S^n=\frac{[\mathrm{0,1}{]}^n}{langle;(s_1,\dots ,s_n)=(0,\dots ,0)\mathrm{if\; some}{s}_i=0\text{or}1rangle;}\text{so that}{S}^1=\frac{\left[\mathrm{0,1}\right]}{?0=1?}\text{and}{S}^n=\underset{\underset{n\text{factors}}{︸}}{S^1\wedge \cdots \wedge S^1}}$.

The suspension of $X$ is

$\mathrm{SX}=S^1\wedge X\text{and}\Omega X={\mathrm{Map}}_{*}(S^1,X)\text{with basepoint}\begin{array}{cccc}*:& S^1& \to & X\\ & s& \mapsto & x_0\end{array}$

is the loop space of $X$. Then

${\mathrm{Map}}_{*}(\mathrm{SX},Y)\simeq {\mathrm{Map}}_{*}(X,\Omega Y),{\Omega }^{n}X={\mathrm{Map}}_{*}(S^n,X),S^{n}X=S^n\wedge X,\mathrm{and}[S^{n}X,Y]=[X,{\Omega }^{n}Y]$.

The fundamental group of $X$ is ${\pi }_1(X,x_0)$, where

${\pi }_n(X,x_0)=\left[\right(S^n,s_0),(X,x_0\left)\right]\text{with product}(f_1*f_2)(s_1,\dots ,s_n)=\{\begin{array}{cc}f(2s_1,\dots s_n),& 0\le s_1\le \frac{1}{2},\\ g(2s_1–1,s_2,\dots ,s_n),& \frac{1}{2}\le s_1\le 1,\end{array}$

is the $n^{\mathrm{th}}$ homotopy group of $X$. Let $G$ be a group. The Eilenberg-Maclane space is a space $K(G,n)$ that has the homotopy type of a CW-complex,

${\pi }_n\left(K\right(G,n),x_0)=G\text{and}{\pi }_i\left(K\right(G,n),x_0)=0,\text{for }i\ne n.$

These are important because

$H_n\left(G\right)\simeq H_n\left(K\right(G,1\left)\right)\text{and}{H}^n\left(G\right)\simeq H^n\left(K\right(G,1\left)\right)\text{.}$

Fibrations

A map $E\stackrel{p}{\to }X$ has the homotopy lifting property with respect to $Y$ if a lift of the $0$ end of a homotopy $Y\times \left[\mathrm{0,1}\right]\stackrel{h}{\to }X$ extends to a lift of the entire homotopy, i.e. given

$\begin{array}{ccc}Y\times 0& \stackrel{f}{⟶}& E\\ \downarrow & & \downarrow p\\ Y\times \left[\mathrm{0,1}\right]& \stackrel{h}{⟶}& X\end{array}$ there exists $H:Y\times \left[\mathrm{0,1}\right]\to E$ making the diagram commute.

A Hurewicz fibration is $E\stackrel{p}{\to }X$ such that the homotopy lifting property holds for all spaces $Y$. A Serre fibration is $E\stackrel{p}{\to }X$ such that the homotopy lifting property holds for all simplicial complexes $Y$.

Let $f:Y\to X$ be a map of based spaces. The homotopy fiber of $f$ is

the fibre $X/Y$ of $P⟶X$ where $P=\left\{\right(y,\omega )\in Y\times X^{\left[0.1\right]}\mid f(y)=\omega (1\left)\right\}$

is the push out of of $f$,

$\begin{array}{ccc}P& \stackrel{{\mathrm{pr}}_2}{⟶}& X^{\left[\mathrm{0,1}\right]}\\ \downarrow {\mathrm{pr}}_1& & \downarrow \\ Y& \stackrel{f}{⟶}& X\end{array}$ $\begin{array}{ccc}(y,\omega )& \mapsto & \omega \\ \downarrow & & \downarrow \\ y& \mapsto & \omega \left(1\right)=f\left(y\right)\end{array}$

Explicitly, $X/Y=\left\{\right(y,\omega )\in Y\times X^{\left[\mathrm{0,1}\right]}\mid \omega (0)=x_0,\omega (1)=f(y\left)\right\}$ and

$\begin{array}{ccc}X/Y& \stackrel{{\mathrm{pr}}_2}{⟶}& \mathrm{PX}\\ \downarrow & & \downarrow \\ Y& \stackrel{f}{⟶}& X\end{array}$.

Let $\varphi :X\times X$. A tricky way to view the fixed points of $\varphi$,

$\begin{array}{l}{X}^{\varphi }=\{x\in X\mid \varphi (x)=x\}\\ \phantom{X^{\mathrm{phiv}}}=\left\{\right(x,x)\in X\times X\mid \varphi (x)=x\}\\ \phantom{X^{\mathrm{phiv}}}=\left\{\right(x_1,x_2)\in X\times X\mid (x_1,\varphi \left(x_1\right))=(x_2,x_2\left)\right\}\end{array}$

is as the push out

$\begin{array}{ccc}{X}^{\varphi }& ⟶& X\\ \downarrow & & \downarrow \Delta \\ X& \stackrel{(\mathrm{id},\varphi )}{⟶}& X\times X\end{array}\text{,}$ where $\Delta \left(x\right)=(x,x)$.

The map $\stackrel{\sim }{\Delta }:X^{\left[\mathrm{0,1}\right]}?X\times X$ given by

$\stackrel{\sim }{\Delta }\left(\omega \right)=\left(\omega \right(0),\omega (1\left)\right),\text{for}\omega :\left[\mathrm{0,1}\right]\to X$

is homotopic to $\Delta$ and the homotopy fixed points of $\varphi$,

$X^{\varphi }=\left\{\right(x,\omega )\in X\times X^{\left[\mathrm{0,1}\right]}\mid \omega (0)=x,\omega (1)=\varphi (x\left)\right\}$,

is the pushout

$\begin{array}{ccc}{X}^{h\varphi }& \stackrel{{\mathrm{pr}}_2}{⟶}& X^{\left[\mathrm{0,1}\right]}\\ \downarrow {\mathrm{pr}}_1& & \downarrow \stackrel{\sim }{\Delta }\\ X& \stackrel{(\mathrm{id},\varphi )}{⟶}& X\times X\end{array}$.

Fibre bundles and classifying spaces

A fibre bundle with fibre $F$ is a surjective map

$\begin{array}{cc}E& \text{total space}\\ \downarrow p& \\ X& \text{base space}\end{array}$ such that      if $x\in X$ then $p^{–1}\left(x\right)\simeq F$,

and there is an open covering $\left\{U_{\alpha }\right\}$ of $X$ and homeomorphisms ${\varphi }_{\alpha }$ with

$\begin{array}{ccc}{U}_{\alpha }\times F& \stackrel{{\varphi }_{\alpha }}{⟶}& p^{–1}\left(U_{\alpha }\right)\\ \downarrow {\mathrm{pr}}_1& & \downarrow p\\ U_{\alpha }& =& U_{\alpha }\end{array}$

If $f:Y\to X$ and $E\stackrel{p}{\to }X$ is a fibre bundle the pullback $f^{*}\left(E\right)$ is

$f^{*}\left(E\right)\stackrel{f^{*}\left(p\right)}{?}Y$ given by $\begin{array}{ccc}{f}^{*}\left(E\right)& \stackrel{{\mathrm{pr}}_2}{⟶}& E\\ \downarrow {\mathrm{pr}}_1& & \downarrow p\\ Y& \stackrel{f}{⟶}& X\end{array}$

so that $f^{*}\left(E\right)=\left\{\right(y,e)\mid f(y)=p(e\left)\right\}$.

A covering space of $X$ is a fibre bundle with discrete fiber. The universal cover of a path connected space is a covering space $E$ of $X$ which is path connected and has ${\pi }_1(E,e_0)=0$ (is simply connected?).

Example. Picture of Mobius band.

Let $G$ be a group. A principal $G$-bundle is a fibre bundle $E\to X$ with fiber $G$ and a right action $E\times G\to G$. A universal $G$-bundle is a principal $G$-bundle

$\begin{array}{cc}EG& \\ \downarrow & \\ BG& \text{classifying space}\end{array}$ such that $\begin{array}{ccc}[X,BG]& ⟶& \left\{\text{principal}G\text{-bundles on}X\right\}\\ f& \mapsto & f^{*}\left(EG\right)\end{array}$ is a bijection.

Then

• If $G$ is discrete $BG=K(G,1)$ and $EG$ is the universal cover of $BG$.
• $EG$ is the unique, up to homotopy, contractible space on which $G$ acts freely.
• $\Omega BG\simeq G$.

Let $X$ and $Y$ be topological spaces. The join of $X$ and $Y$ is

$X*Y=\frac{X\times \left[0.1\right]\times Y}{langle;(x,0,y)=(x^{\prime },0,y),(x,1,y)=(x,1,y^{\prime })rangle;}$

The Milnor construction of the classifying space of $G$ is by letting $G$ act on

$EG=G*G*\cdots ={\displaystyle \frac{\left\{\right(t_1{g}_1,t_2{g}_2,\dots )\mid t_i\in [\mathrm{0,1}],\sum t_i=1,\text{most }{t}_i=0\}}{\{(t_1{g}_1,t_2{g}_2,\dots )=(t_1{g}_1^{\prime },t_2{g}_1^{\prime },\dots )\text{ if }{g}_i=g_i^{\prime }\text{ for }{t}_i\ne 0\}}}$

by $(t_1{g}_1,t_2{g}_2,\dots )g=(t_1{g}_{1}g,t_2{g}_{2}g,\dots )$ and $BG=EG/G$.