Classifying spaces

The goal of this section is to introduce the notion of classifying space $BG$$BG$ of a (Hausdorff) topological group $G$$G$. This space has the property that isomorphism classes of principal $G$$G$-bundles over a paracompact Hausdorff base space correspond bijectively to homotopy classes of maps from the base to $BG$$BG$, hence the name. More explicitly this correspondence is given by pullback of a universal principal $G$$G$-bundle $EG\to BG$$EG\to BG$.

The whole world in an example

We begin by considering the special case of $G=GL(r,\mathbb {C} )$$G=GL(r,\mathbb {C} )$. But first we need some preliminaries on Grassmannians. Denote by $Gr(k,\mathbb {C} ^{n})$$Gr(k,\mathbb {C} ^{n})$ the Grassmannian of $k$$k$-dimensional linear subspaces of $\mathbb {C} ^{n}$$\mathbb {C} ^{n}$. We then have a tautological short exact sequence of vector bundles on $Gr(k,\mathbb {C} ^{n})$$Gr(k,\mathbb {C} ^{n})$:

$$0\longrightarrow S(k,\mathbb {C} ^{n})\longrightarrow {\underline {\mathbb {C} ^{n}}}\longrightarrow U(k,\mathbb {C} ^{n})\longrightarrow 0.$$

$$0\longrightarrow S(k,\mathbb {C} ^{n})\longrightarrow {\underline {\mathbb {C} ^{n}}}\longrightarrow U(k,\mathbb {C} ^{n})\longrightarrow 0.$$

Here $S(k,\mathbb {C} ^{n})$$S(k,\mathbb {C} ^{n})$ denotes the tautological subspace bundle on $Gr(k,\mathbb {C} ^{n})$$Gr(k,\mathbb {C} ^{n})$, ${\underline {\mathbb {C} ^{n}}}$${\underline {\mathbb {C} ^{n}}}$ is the trivial bundle with fibre $\mathbb {C} ^{n}$$\mathbb {C} ^{n}$ over $Gr(k,\mathbb {C} ^{n})$$Gr(k,\mathbb {C} ^{n})$ and $U(k,\mathbb {C} ^{n})$$U(k,\mathbb {C} ^{n})$ is the obvious quotient bundle. If we define $Gr(\mathbb {C} ^{\infty },k):=q\varinjlim Gr(n-k,\mathbb {C} ^{n})$$Gr(\mathbb {C} ^{\infty },k):=q\varinjlim Gr(n-k,\mathbb {C} ^{n})$ and $U:=q\varinjlim U(n-k,\mathbb {C} ^{n})$$U:=q\varinjlim U(n-k,\mathbb {C} ^{n})$ we obtain a vector bundle $U\to Gr(\mathbb {C} ^{\infty },k)$$U\to Gr(\mathbb {C} ^{\infty },k)$ which we can think of as the tautological quotient bundle on the infinite quotient Grassmannian $Gr(\mathbb {C} ^{\infty },k)$$Gr(\mathbb {C} ^{\infty },k)$.

Now let $E\to M$$E\to M$ be a complex vector bundle of rank $r$$r$ over a manifold $M$$M$; for the present discussion it is irrelevant whether this is a smooth vector bundle or just a topological one. Denote by ${\underline {\Gamma (E)}}$${\underline {\Gamma (E)}}$ the trivial vector bundle with fibre $\Gamma (E)$$\Gamma (E)$ over $M$$M$, where $\Gamma (E)$$\Gamma (E)$ is the space of sections of $E$$E$. We then have a short exact sequence of vector bundles

$$0\longrightarrow \ker \phi \longrightarrow {\underline {\Gamma (E)}}{\overset {\phi }{\longrightarrow }}E\longrightarrow 0,$$

$$0\longrightarrow \ker \phi \longrightarrow {\underline {\Gamma (E)}}{\overset {\phi }{\longrightarrow }}E\longrightarrow 0,$$

where the map $\phi$$\phi$ is the obvious one given by evaluation of sections. Thinking of $\Gamma (E)$$\Gamma (E)$ as the vector space $\mathbb {C} ^{\infty }$$\mathbb {C} ^{\infty }$, this expresses $E$$E$ as a quotient of $M\times \mathbb {C} ^{\infty }$$M\times \mathbb {C} ^{\infty }$, i.e. every fibre of $E$$E$ is a $r$$r$-dimensional quotient of $\mathbb {C} ^{\infty }$$\mathbb {C} ^{\infty }$. This corresponds to having a map (continuous or smooth, depending on the chosen setting) $f\colon M\to Gr(\mathbb {C} ^{\infty },r)$$f\colon M\to Gr(\mathbb {C} ^{\infty },r)$ such that $E$$E$ is the pullback $f^{*}U$$f^{*}U$ of the tautological quotient bundle $U$$U$ under $f$$f$. When $E$$E$ is the tangent bundle of $M$$M$ we can think of this map $f$$f$ as a generalised Gauss map for the manifold. Denote by $\operatorname {Vect} _{\mathbb {C} }^{r}(M)$$\operatorname {Vect} _{\mathbb {C} }^{r}(M)$ the set of isomorphism classes of complex vector bundles of rank $r$$r$ over $M$$M$ and let $[M,Gr(\mathbb {C} ^{\infty },r)]$$[M,Gr(\mathbb {C} ^{\infty },r)]$ be the set of homotopy classes of maps $M\to Gr(\mathbb {C} ^{\infty },r)$$M\to Gr(\mathbb {C} ^{\infty },r)$. Since pullbacks under homotopic maps are isomorphic we have sketched the proof of a bijective correspondende

$$\operatorname {Vect} _{\mathbb {C} }^{r}(M)\longleftrightarrow [M,Gr(\mathbb {C} ^{\infty },r)].$$

$$\operatorname {Vect} _{\mathbb {C} }^{r}(M)\longleftrightarrow [M,Gr(\mathbb {C} ^{\infty },r)].$$

For the present case of $G=GL(r,\mathbb {C} )$$G=GL(r,\mathbb {C} )$ we write $BG:=qGr(\mathbb {C} ^{\infty },r)$$BG:=qGr(\mathbb {C} ^{\infty },r)$.

Example 5.1

Consider the case $r=1$$r=1$. By choosing inner products on the $\mathbb {C} ^{n}$$\mathbb {C} ^{n}$ (e.g. the obvious ones) we obtain an isomorphism $Gr(\mathbb {C} ^{\infty },1)\cong \mathbb {C} P^{\infty }$$Gr(\mathbb {C} ^{\infty },1)\cong \mathbb {C} P^{\infty }$. Combining this with what we have seen so far we obtain a bijective correspondence

$$\operatorname {Vect} _{\mathbb {C} }^{1}(M)\longleftrightarrow [M,\mathbb {C} P^{\infty }].$$

$$\operatorname {Vect} _{\mathbb {C} }^{1}(M)\longleftrightarrow [M,\mathbb {C} P^{\infty }].$$

 

Example 5.2

Now let $M=S^{n}$$M=S^{n}$ be the unit $n$$n$-sphere. One can show by means of clutching functions that $\operatorname {Vect} _{\mathbb {C} }^{r}(S^{n})$$\operatorname {Vect} _{\mathbb {C} }^{r}(S^{n})$ corresponds bijectively to $[S^{n-1},G]$$[S^{n-1},G]$. But by what we have seen now we also know that $\operatorname {Vect} _{\mathbb {C} }^{r}(S^{n})$$\operatorname {Vect} _{\mathbb {C} }^{r}(S^{n})$ corresponds to $[S^{n},BG]$$[S^{n},BG]$. Thus we conclude that there is an isomorphism

$$\pi _{n}(BG)\cong \pi _{n-1}(G)$$

$$\pi _{n}(BG)\cong \pi _{n-1}(G)$$

for every $n$$n$ (actually we only know that it is a bijection, we will however see shortly that it is indeed an isomorphism). Note that we are explicitly working with $G=GL(r,\mathbb {C} )$$G=GL(r,\mathbb {C} )$ and $BG=Gr(\mathbb {C} ^{\infty },r)$$BG=Gr(\mathbb {C} ^{\infty },r)$ here, but the same argument does work for general classifying spaces once we have defined them.

 

Note that The whole world in an example doesn't quite fit into the general statement given in the introduction: there are no principal bundles mentioned here. It turns out however that there is a somewhat canonical correspondence between complex vector bundles of rank $r$$r$ and principal $GL(r,\mathbb {C} )$$GL(r,\mathbb {C} )$-bundles (this works for other fields too). Namely we define the frame bundle of a given vector bundle $E\to M$$E\to M$ of rank $r$$r$ to be the principal $GL(r,\mathbb {C} )$$GL(r,\mathbb {C} )$-bundle $F(E)\to M$$F(E)\to M$ whose fibre over a point is the space of isomorphisms from $\mathbb {C} ^{r}$$\mathbb {C} ^{r}$ to the fibre of $E$$E$ over that point. Note that there is an obvious right action of $GL(r,\mathbb {C} )$$GL(r,\mathbb {C} )$ on $F(E)$$F(E)$ which preserves the fibres and acts freely and transitively on them. Noting that pulling back commutes with the frame bundle construction we can translate what we have seen so far into a correspondence

$$\{{\text{principal }}GL(r,\mathbb {C} ){\text{-bundles over }}M\}/{\text{isomorphism}}\longleftrightarrow [M,Gr(\mathbb {C} ^{\infty },r)].$$

$$\{{\text{principal }}GL(r,\mathbb {C} ){\text{-bundles over }}M\}/{\text{isomorphism}}\longleftrightarrow [M,Gr(\mathbb {C} ^{\infty },r)].$$

The correspondence here is given by pulling back the frame bundle of $U\to BG$$U\to BG$. We will denote it by $EG\to BG$$EG\to BG$ and say that it is the universal principal $G$$G$-bundle (precisely because of the above correspondence).

We now want to see that the space $EG$$EG$ is weakly contractible. This will be true in the general setting and is part of the allure of the universal principal bundle. Knowing that $BG=Gr(\mathbb {C} ^{\infty },r)$$BG=Gr(\mathbb {C} ^{\infty },r)$ is paracompact we obtain a long exact sequence of homotopy groups:

$$\cdots \longrightarrow \pi _{i}(G)\longrightarrow \pi _{i}(EG)\longrightarrow \pi _{i}(BG){\overset {\partial _{*}}{\longrightarrow }}\pi _{i-1}(G)\longrightarrow \cdots .$$

$$\cdots \longrightarrow \pi _{i}(G)\longrightarrow \pi _{i}(EG)\longrightarrow \pi _{i}(BG){\overset {\partial _{*}}{\longrightarrow }}\pi _{i-1}(G)\longrightarrow \cdots .$$

All the maps except $\partial _{*}$$\partial _{*}$ are the obvious ones. To construct $\partial _{*}$$\partial _{*}$ start by picking an element $[a]\in \pi _{i}(BG)$$[a]\in \pi _{i}(BG)$ and choosing a representative $a\colon S^{n}\to BG$$a\colon S^{n}\to BG$. We can think of this map as a map $a\colon D^{n}\to BG$$a\colon D^{n}\to BG$ on the closed unit $n$$n$-disk that is constant on its boundary $\partial D^{n}=S^{n-1}$$\partial D^{n}=S^{n-1}$. From the fact that $EG\to BG$$EG\to BG$ is a fibre bundle with paracompact base space we know that there is a lift ${\tilde {a}}\colon D^{n}\to EG$${\tilde {a}}\colon D^{n}\to EG$ of $a$$a$. Since the image of $S^{n-1}$$S^{n-1}$ under $a$$a$ is a single point we know that its image under ${\tilde {a}}$${\tilde {a}}$ must be contained in a fibre. Thus we have ${\tilde {a}}|_{S^{n-1}}\colon S^{n-1}\to G$${\tilde {a}}|_{S^{n-1}}\colon S^{n-1}\to G$ and we can define

$$\partial _{*}([a]):=q[{\tilde {a}}|_{S^{n-1}}]\in \pi _{n-1}(G).$$

$$\partial _{*}([a]):=q[{\tilde {a}}|_{S^{n-1}}]\in \pi _{n-1}(G).$$

The fact that this is well-defined follows from the homotopy lifting property for fibre bundles over a paracompact base. If we now knew that $\partial _{*}$$\partial _{*}$ is an isomorphism we'd be done. This is indeed true as the following exercise shows:

Exercise 5.1

Meditate on the fact that $\partial _{*}$$\partial _{*}$ is the same map as the one in (MISSING).

 

Exercise 5.2

Think about what the above construction of $\partial _{*}$$\partial _{*}$ looks like explicitly in the case of the Hopf fibration $S^{1}\hookrightarrow S^{3}\longrightarrow S^{2}$$S^{1}\hookrightarrow S^{3}\longrightarrow S^{2}$.

 

Exercise 5.3

Let $\mathbb {Z}$$\mathbb{Z}$ be the group of integers under addition. Show that $B\mathbb {Z} =S^{1}$$B\mathbb {Z} =S^{1}$ and that $E\mathbb {Z} \to B\mathbb {Z}$$E\mathbb {Z} \to B\mathbb {Z}$ is $\mathbb {R} \to S^{1}$$\mathbb {R} \to S^{1}$ with the map being the complex exponential. Do the above by noting that principal $\mathbb {Z}$$\mathbb{Z}$-bundles on $M$$M$ are in correspondence with $H^{1}(M;\mathbb {Z} )$$H^{1}(M;\mathbb {Z} )$ and that the latter is in correspondence with $[M,S^{1}]$$[M,S^{1}]$.

 

Exercise 5.4

Think about what the total space of the pullback of $E\mathbb {Z} \to B\mathbb {Z}$$E\mathbb {Z} \to B\mathbb {Z}$ under the double cover $S^{1}{\overset {z^{2}}{\longrightarrow }}S^{1}$$S^{1}{\overset {z^{2}}{\longrightarrow }}S^{1}$ looks like.

 

General groups

To define the classifying space and universal bundle for a general (Hausdorff) group we can start by constructing a principal $G$$G$-bundle $EG\to BG$$EG\to BG$ with $EG$$EG$ contractible. To do this we use the topological join construction:

Definition 5.1

Let $X,Y$$X,Y$ be topological spaces. The join $X\star Y$$X\star Y$ is then defined to be $(X\times Y\times [0,1])/\sim$$(X\times Y\times [0,1])/\sim$, where the equivalence relation $\sim$$\sim$ is generated by $(x,y_{1},0)\sim (x,y_{2},0)$$(x,y_{1},0)\sim (x,y_{2},0)$ and $(x_{1},y,1)\sim (x_{2},y,1)$$(x_{1},y,1)\sim (x_{2},y,1)$ for $x,x_{1},x_{2}\in X$$x,x_{1},x_{2}\in X$ and $y,y_{1},y_{2}\in Y$$y,y_{1},y_{2}\in Y$.

 

Note that the join of two spaces is just the space obtained by connecting every point of one space by the unit interval to every point of the other space. Moreover the construction is associative and commutative. We now define the universal bundle by setting $EG:=qG\star G\star G\star \cdots$$EG:=qG\star G\star G\star \cdots$ and $BG:=qEG/G$$BG:=qEG/G$. One has to verify that this is indeed a principal $G$$G$-bundle (in particular that it has local trivialisations), but that follows by thinking in an appropriately elegant way about $EG$$EG$.

Example 5.3

We have seen how to construct $EGL(r,\mathbb {C} )$$EGL(r,\mathbb {C} )$. If $G$$G$ has a faithful linear representation $G\hookrightarrow GL(r,\mathbb {C} )$$G\hookrightarrow GL(r,\mathbb {C} )$ then note that $G$$G$ acts freely on $EGL(r,\mathbb {C} )$$EGL(r,\mathbb {C} )$ and so we can just define $EG:=qEGL(r,\mathbb {C} )$$EG:=qEGL(r,\mathbb {C} )$ and $BG:=qEGL(r,\mathbb {C} )/G$$BG:=qEGL(r,\mathbb {C} )/G$.

 

We now have the following correspondence in general:

Theorem 5.1

Let $M$$M$ be a paracompact Hausdorff space and let $G$$G$ be a Hausdorff topological group. There is a bijective correspondence

$$\{{\text{principal }}G{\text{-bundles on M}}\}/{\text{isomorphism}}\longleftrightarrow [M,BG],$$

$$\{{\text{principal }}G{\text{-bundles on M}}\}/{\text{isomorphism}}\longleftrightarrow [M,BG],$$

where the map from right to left is given by pulling back $EG\to BG$$EG\to BG$ along a representative.

 

For Example 5.3 to fit into this general picture we have to mention another result:

Theorem 5.2

Let $G$$G$ be a Hausdorff topological group. Then every principal $G$$G$-bundle with contractible total space is a universal bundle in the sense of Theorem 5.1. In particular its total and base spaces are unique up to homotopy.

 

Although a rigorous proof of Theorem 5.1 in the general topological setting requires some care regarding technicalities, we can easily argue why this should be true for the case of $M$$M$ a smooth manifold and $G$$G$ a compact Lie group. Namely let $P\to M$$P\to M$ be a principal $G$$G$-bundle and consider the following diagram:

Here $EG\times _{G}P:=qEG\times P/\sim$$EG\times _{G}P:=qEG\times P/\sim$ where $(eg,x)\sim (e,gx)$$(eg,x)\sim (e,gx)$ for $e\in EG$$e\in EG$, $x\in P$$x\in P$ and $g\in G$$g\in G$. Now note that the bundle $EG\times P\to EG\times _{G}P$$EG\times P\to EG\times _{G}P$ is the pullback of $EG\to BG$$EG\to BG$ under the obvious map $\pi$$\pi$, and also that it is the pullback of $P\to M$$P\to M$ under the projection $p$$p$. But we also have that $EG\times _{G}P\to M$$EG\times _{G}P\to M$ is a fibre bundle with fibre $EG$$EG$ which is weakly contractible and so it has a global section $s\colon M\to EG\times _{G}P$$s\colon M\to EG\times _{G}P$. Finally it follows from what we have said so far that $P\to M$$P\to M$ is the pullback of $EG\to BG$$EG\to BG$ under $\pi \circ s$$\pi \circ s$.