Homotopy theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 31 May 2012

Maps and Based maps

Let $X$ and $Y$ be topological spaces.

• The space of maps from $X$ to $Y$ is $$\mathrm{Map}(X,Y)=\{f:X\to Y|f\text{is continuous}\}.$$
• The wedge of $X$ and $Y$ is the subspace of $X\times Y$ given by $$X\vee Y=(X\times y_0)\cup (x_0\times Y)\text{and}$$
• Homotopy is the equivalence relation on $\mathrm{Map}(X,Y)$ given by $$f_1\simeq f_2\text{if there exists}F:X\times [0,1]⟶Y\text{such that}F(x,0)=f_1\left(x\right)\text{and}F(x,1)=f_2\left(x\right),$$ for $x\in X$. Define $$[X,Y]=\frac{\mathrm{Map}(X,Y)}{\simeq }.$$

HW: Show that $\mathrm{Map}(X\vee Y,Z)=\mathrm{Map}(X,\mathrm{Map}(Y,Z\left)\right)$

• A based space is a topological space $X$ with a distinguished point $x_0$, the basepoint of $X$.

Let $X$ and $Y$ be based spaces.

• The space of based maps from $X$ to $Y$ is $${\mathrm{Map}}_{*}(X,Y)=\{f\in \mathrm{Map}(X,Y\left)\right|f\left(x_0\right)=y_0\}\text{,}$$ where $x_0$ is the basepoint of $X$ and $y_0$ is the basepoint of $Y$.
• The smash of $X$ and $Y$ is the quotient space of $X\times Y$ given by $$X\wedge Y=\frac{X\times Y}{⟨(x,y_0)=(x_0,y_0)=(x_0,y)⟩}$$
• Based homotopy is the equivalence relation on ${\mathrm{Map}}_{*}(X,Y)$ given by $$f_1\simeq f_2\text{if there exists}F:X\times [0,1]⟶Y\text{such that}F(x,0)=f_1\left(x\right)\text{and}F(x,1)=f_2\left(x\right),$$ for $x\in X$. (DO WE NEED TO ADD $F(x_0,t)=x_0)$???) Define $$[X,Y]=\frac{{\mathrm{Map}}_{*}(X,Y)}{\simeq }.$$

HW: Show that ${\mathrm{Map}}_{*}(X\wedge Y,Z)={\mathrm{Map}}_{*}(X,{\mathrm{Map}}_{*}(Y,Z\left)\right).$

Fundamental group, loop space and suspension

Let $X$ be a based???? space.

• The $n$-sphere is $${\displaystyle S^n=\frac{{[0,1]}^n}{⟨(s_1,\dots ,s_n)=(0,\dots ,0)\text{if some}{s}_i=0\text{or}1⟩}\text{so that}{S}^1=\frac{\left[\mathrm{0,1}\right]}{⟨0=1⟩}\text{and}{S}^n=\underset{\underset{n\text{factors}}{\&underbrace;}}{S^1\wedge \cdots \wedge S^1}}.$$
• The suspension of $X$ is $SX=S^1\wedge X$.
• The loop space of $X$ is $$\Omega X={\mathrm{Map}}_{*}(S^1,X)\text{with basepoint}\begin{array}{cccc}*:& S^1& \to & X\\ & \theta & \mapsto & x_0.\end{array}$$
• The fundamental group of $X$ is ${\pi }_1(X,x_0)$.
• The $n$th homotopy group of $X$ is $${\pi }_n(X,x_0)=\left[\right(S^n,s_0),(X,x_0\left)\right]\text{with product}(f_1*f_2)(s_1,\dots ,s_n)=\{\begin{array}{ll}f(2s_1,\dots s_n),& 0\le s_1\le \frac{1}{2},\\ g(2s_1-1,s_2,\dots ,s_n),& \frac{1}{2}\le s_1\le 1.\end{array}$$
• The space $X$ is $n$-connected if ${\pi }_i\left(X\right)$, for $0\le i\le n$.
• The based space $X$ is simply connected if ${\pi }_1\left(X\right)=0$ and ${\pi }_0\left(X\right)=0$.
• The based space $X$ is path connected if ${\pi }_0\left(X\right)=0$.
• Let $G$ be a group. The Eilenberg-Maclane space is a space $K(G,n)$ that has the homotopy type of a CW-complex, $${\pi }_n\left(K\right(G,n),x_0)=G\text{and}{\pi }_i\left(K\right(G,n),x_0)=0,\text{for}i\ne n.$$

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

The Eilenberg-Maclane spaces are important because the group homology and cohomology of $G$ coincides with the homology and cohomology of $K(G,1)$, $$H_n\left(G\right)\simeq H_n\left(K\right(G,1\left)\right)\text{and}{H}^n\left(G\right)\simeq H^n\left(K\right(G,1\left)\right).$$ Alternatively, $K(G,1)\simeq BG$.

Fibrations

• A continuous function $E\stackrel{p}{\to }X$ 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 $$\begin{array}{ccc}Y\times 0& \stackrel{f}{⟶}& E\\ \downarrow & & \downarrow p\\ Y\times [0,1]& \stackrel{h}{⟶}& X\end{array}\text{there exists}H:Y\times [0,1]\to E$$ making the diagram commute.
• A Hurewicz fibration is $E\stackrel{p}{\to }X$ such that the homotopy lifting property holds for all topological 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 $f:Y\to X$ is $$\text{the fibre}X/Y\text{of}P⟶X\text{where}P=\left\{\right(y,\omega )\in Y\times X^{[0,1]}|f\left(y\right)=\omega \left(1\right)\}$$ is the push out of $f$, $$\begin{array}{ccc}P& \stackrel{{\mathrm{pr}}_2}{⟶}& X^{[0,1]}\\ \downarrow {\mathrm{pr}}_1& & \downarrow \\ Y& \stackrel{f}{⟶}& X\end{array}\begin{array}{ccc}(y,\omega )& \mapsto & \omega \\ \downarrow & & \downarrow \\ y& \mapsto & \omega \left(1\right)=f\left(y\right)\end{array}$$ Explicitly, $X/Y=\left\{\right(y,\omega )\in Y\times X^{[0,1]}|\omega \left(0\right)=x_0,\omega \left(1\right)=f\left(y\right)\}$ and $$\begin{array}{ccc}X/Y& \stackrel{{\mathrm{pr}}_2}{⟶}& \mathrm{PX}\\ \downarrow & & \downarrow \\ Y& \stackrel{f}{⟶}& X\end{array}$$

Let $\phi :X\to X$. A tricky way to view the fixed points of $\phi$, $$\begin{array}{rcl}{X}^{\phi }& =& \{x\in X|\phi \left(x\right)=x\}\\ & =& \left\{\right(x,x)\in X\times X|\phi \left(x\right)=x\}\\ & =& \left\{\right(x_1,x_2)\in X\times X|(x_1,\phi (x_1\left)\right)=(x_2,x_2)\}\end{array}$$ is as the push out $$\begin{array}{ccc}{X}^{\phi }& ⟶& X\\ \downarrow & & \downarrow \Delta \\ X& \stackrel{(\mathrm{id},\phi )}{⟶}& X\times X\end{array}\text{where}\Delta \left(x\right)=(x,x).$$ The map $\stackrel{\sim }{\Delta }:X^{[0,1]}\to X\times X$ given by $$\stackrel{\sim }{\Delta }\left(\omega \right)=\left(\omega \right(0),\omega (1\left)\right),\text{for}\omega :[0,1]\to X$$ is homotopic to $\Delta$.

• The homotopy fixed points of $\phi :X\to X$, $$X^{\phi }=\left\{\right(x,\phi )\in X\times X[0,1]|\omega \left(0\right)=x,\omega \left(1\right)=\phi \left(x\right)\},$$ is the pushout $$\begin{array}{ccc}{X}^{h\phi }& \stackrel{{\mathrm{pr}}_2}{⟶}& X^{[0,1]}\\ \downarrow {\mathrm{pr}}_1& & \downarrow \stackrel{\sim }{\Delta }\\ X& \stackrel{(\mathrm{id},\phi )}{⟶}& X\times X\end{array}$$

Fibre bundles and classifying spaces

• A fibre bundle with fibre $F$ is a surjective map $$\begin{array}{cc}E& \text{total space}\\ \downarrow p& \\ X& \text{base space}\end{array}\text{such that}\text{if}x\in X\text{then}{p}^{-1}\left(x\right)\simeq F,$$ and there is an open covering $\left\{U_{\alpha }\right\}$ of $X$ and homeomorphisms ${\phi }_{\alpha }$ with $$\begin{array}{ccc}{U}_{\alpha }\times F& \stackrel{{\phi }_{\alpha }}{⟶}& p^{-1}\left(U_{\alpha }\right)\\ \downarrow {\mathrm{pr}}_1& & \downarrow p\\ U_{\alpha }& =& U_{\alpha }\end{array}$$
• If $f:Y\to X$ and $E\stackrel{p}{\to }X$ is a fibre bundle the pullback $f^{*}\left(E\right)$ is $$f^{*}\left(E\right)\stackrel{f^{*}\left(p\right)}{⟶}Y\text{given by}\begin{array}{ccc}{f}^{*}\left(E\right)& \stackrel{{\mathrm{pr}}_2}{⟶}& E\\ \downarrow {\mathrm{pr}}_1& & \downarrow p\\ Y& \stackrel{f}{⟶}& X\end{array}$$ so that $f^{*}\left(E\right)=\left\{\right(y,e\left)\right|f\left(y\right)=p\left(e\right)\}$.
• 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 $$\begin{array}{cc}EG& \\ \downarrow & \\ BG& \text{classifying space}\end{array}\text{such that}\begin{array}{ccc}[X,BG]& ⟶& \left\{\text{principal}G\text{-bundles on}X\right\}\\ f& \mapsto & f^{*}\left(EG\right)\end{array}\text{is a bijection.}$$

HW: If $G$ is discrete $BG=K(G,1)$ and $EG$ is the universal cover of $BG$.

HW: $EG$ is the unique, up to homotopy, contractible space on which $G$ acts freely.

HW: $\Omega BG\simeq G$.

Let $X$ and $Y$ be topological spaces.

• The join of $X$ and $Y$ is $$X*Y=\frac{X\times [0,1]\times Y}{⟨(x,0,y)=(x\prime ,0,y)\text{and}(x,1,y)=(x,1,y\prime )⟩}$$

The Milnor construction of the classifying space of $G$ is by letting $G$ act on $$EG=G*G*\cdots ={\displaystyle \frac{\left\{\right(t_1{g}_1,t_2{g}_2,\dots \left)\right|t_i\in [0,1],\sum t_i=1,\text{most}{t}_i=0\}}{\{(t_1{g}_1,t_2{g}_2,\dots )=(t_1{g}_1^{\prime },t_2{g}_2^{\prime },\dots )\text{if}{g}_i=g_i^{\prime }\text{for}{t}_i\ne 0\}}}$$ by $(t_1{g}_1,t_2{g}_2,\dots )g=(t_1{g}_{1}g,t_2{g}_{2}g,\dots )$ and $BG=EG/G$.

Notes and References

These notes were written?????????

References

[Bou] N. Bourbaki, Algèbre, Chapitre ?: ??????????? MR?????.

[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.

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