## Maps and Based maps

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

• The space of maps from $X$ to $Y$ is $Map(X,Y) ={f:X→Y | fis continuous}.$
• The wedge of $X$ and $Y$ is the subspace of $X×Y$ given by $X∨Y= (X×y0) ∪(x0×Y) and$
• Homotopy is the equivalence relation on $\mathrm{Map}\left(X,Y\right)$ given by $f1≃f2 if there exists F:X×[0,1] ⟶Y such that F(x,0)= f1(x) and F(x,1)= f2(x) ,$ for $x\in X$. Define $[X,Y]= Map(X,Y) ≃.$

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

• 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 $Map*(X,Y) = {f∈Map(X,Y) | f(x0) =y0},$ 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 $X×Y$ given by $X∧Y= X×Y ⟨(x,y0) = (x0,y0) = (x0,y)⟩$
• Based homotopy is the equivalence relation on ${\mathrm{Map}}_{*}\left(X,Y\right)$ given by $f1≃f2 if there exists F:X×[0,1] ⟶Y such that F(x,0)= f1(x) and F(x,1)= f2(x) ,$ for $x\in X$. (DO WE NEED TO ADD $F\left({x}_{0},t\right)={x}_{0}\right)$???) Define $[X,Y]= Map*(X,Y) ≃.$

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

## Fundamental group, loop space and suspension

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

• The $n$-sphere is $Sn =[0,1]n ⟨(s1,…, sn) = (0,…,0) if some si=0 or1⟩ so that S1= [0,1] ⟨0=1⟩ and Sn= S1∧⋯∧S1 &underbrace;nfactors .$
• The suspension of $X$ is $SX={S}^{1}\wedge X$.
• The loop space of $X$ is $ΩX= Map*(S1,X) with basepoint *: S1 → X θ↦x0 .$
• The fundamental group of $X$ is ${\pi }_{1}\left(X,{x}_{0}\right)$.
• The $n$th homotopy group of $X$ is $πn(X,x0) =[(Sn,s0) ,(X,x0)] with product (f1*f2) (s1,…, sn) = { f(2s1,… sn), 0≤s1≤ 12 , g(2s1-1, s2,…,sn ), 12 ≤s1≤1.$
• 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 $K\left(G,n\right)$ that has the homotopy type of a CW-complex, $πn(K(G,n), x0) =G and πi(K(G,n), x0)=0, fori≠n.$

HW: Show that $Map*(SX,Y) ≃ Map*(X,ΩY), ΩnX= Map*(Sn,X), SnX =Sn∧X, and [SnX,Y] =[X,ΩnY] .$

The Eilenberg-Maclane spaces are important because the group homology and cohomology of $G$ coincides with the homology and cohomology of $K\left(G,1\right)$, $Hn(G) ≃ Hn(K(G,1)) and Hn(G) ≃ Hn(K(G,1)) .$ Alternatively, $K\left(G,1\right)\simeq BG$.

## 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 $Y×\left[0,1\right]\stackrel{h}{\to }X$ extends to a lift of the entire homotopy, i.e. given $Y×0 ⟶f E ↓ ↓ p Y×[0,1] ⟶h X there exists H:Y×[0,1] →E$ 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 $the fibreX/Y of P⟶X where P={(y,ω) ∈Y× X[0,1] | f(y) =ω(1)}$ is the push out of $f$, $P ⟶pr2 X[0,1] ↓ pr1 ↓ Y ⟶f X (y,ω) ↦ ω ↓ ↓ y ↦ ω(1)=f(y)$ Explicitly, $X/Y=\left\{\left(y,\omega \right)\in Y×{X}^{\left[0,1\right]}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\omega \left(0\right)={x}_{0},\omega \left(1\right)=f\left(y\right)\right\}$ and $X/Y ⟶pr2 PX ↓ ↓ Y ⟶f X$

Let $\phi :X\to X$. A tricky way to view the fixed points of $\phi$, $Xφ = {x∈X | φ(x)=x} = {(x,x)∈X×X | φ(x)=x} = {(x1,x2) ∈X×X | (x1, φ(x1)) =(x2,x2)}$ is as the push out $Xφ ⟶ X ↓ ↓ Δ X ⟶(id,φ) X×X where Δ(x)=(x,x) .$ The map $\stackrel{\sim }{\Delta }:{X}^{\left[0,1\right]}\to X×X$ given by $Δ∼(ω) =(ω(0), ω(1)), for ω:[0,1] →X$ is homotopic to $\Delta$.

• The homotopy fixed points of $\phi :X\to X$, $Xφ ={(x,φ)∈ X×X[0,1] | ω(0)=x, ω(1)=φ(x)},$ is the pushout $Xhφ ⟶pr2 X[0,1] ↓ pr1 ↓ Δ∼ X ⟶(id,φ) X×X$

## Fibre bundles and classifying spaces

• A fibre bundle with fibre $F$ is a surjective map $E total space ↓p X base space such that ifx∈X then p-1(x) ≃F,$ and there is an open covering $\left\{{U}_{\alpha }\right\}$ of $X$ and homeomorphisms ${\phi }_{\alpha }$ with $Uα×F ⟶φα p-1(Uα) ↓ pr1 ↓p Uα = Uα$
• If $f:Y\to X$ and $E\stackrel{p}{\to }X$ is a fibre bundle the pullback ${f}^{*}\left(E\right)$ is $f*(E)⟶ f*(p)Y given by f*(E) ⟶pr2 E ↓ pr1 ↓ p Y ⟶f X$ so that ${f}^{*}\left(E\right)=\left\{\left(y,e\right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}f\left(y\right)=p\left(e\right)\right\}$.
• 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 ${\pi }_{1}\left(E,{e}_{0}\right)=0$ (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 $E×G\to G$.
• A universal $G$-bundle is a principal $G$-bundle $EG ↓ BG classifying space such that [X,BG] ⟶ {principalG -bundles onX} f ↦ f*(EG) is a bijection.$

HW: If $G$ is discrete $BG=K\left(G,1\right)$ 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 $X*Y = X×[0,1]×Y ⟨(x,0,y) =(x′,0,y) and (x,1,y) =(x,1,y′) ⟩$

The Milnor construction of the classifying space of $G$ is by letting $G$ act on $EG=G*G*⋯ ={ (t1g1, t2g2,…) | ti∈[0,1], ∑ti =1,most ti=0} {(t1g1, t2g2,… ) = ( t1g1′, t2g2′,… )if gi=gi′ for ti≠0}$ by $\left({t}_{1}{g}_{1},{t}_{2}{g}_{2},\dots \right)g=\left({t}_{1}{g}_{1}g,{t}_{2}{g}_{2}g,\dots \right)$ 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.