The homotopy fiber of the map on classifying spaces

\(\require{amsCd}\newcommand{\colim}{\mathop{\mathrm{colim}}} \newcommand{\im}{\mathop{\mathrm{im}}}\)

Let \(f : G \to H\) be a continuous homomorphism of topological groups; applying the classifying space functor gives a map \(Bf: BG \to BH\).

The homotopy fiber of \(Bf\) is the homotopy orbit space \(H_{hG}\) where \(G\) acts on \(H\) via \(g \cdot h = f(g)h\).
The canonical map \(H_{hG} \to H/\im f\) has homotopy fiber given by \(BK\) where \(K = \ker f\).

Before sketching a proof, I’ll mention a couple of interesting special cases:

If \(f : G \to H\) is injective, the homotopy fiber of \(Bf : BG \to BH\) is the coset space \(H/G\).

The \(G\)-action on \(H\) is free, so \(H_{hG} \cong H/G\).

If \(1 \to K \to G \to H \to 1\) is a short exact sequence of topological groups, applying the classifying space functor gives a fiber sequence \(BK \to BG \to BH\).

Here \(H/\im f = \ast\), so \(H_{hG} = BK\).

Both parts of the proposition are special cases of the following lemma:

Let \(f : G \to H\) be a homomorphism and let \(\alpha : X \to Y\) be a map of \(H\)-spaces (spaces with a continuous \(H\)-action). We can also consider \(X\) and \(Y\) as \(G\)-spaces via \(g \cdot x = f(g) \cdot x\) and an analogous definition for \(Y\). There are canonical maps of the form \(X_{hG} \to X_{hH}\) giving a homotopy pullback square: \[ \begin{CD} X_{hG} @>>> Y_{hG} \\ @VVV @VVV \\ X_{hH} @>>> Y_{hH} \end{CD} \]

Apply the lemma to the map \(H \to \ast\) (where \(H\) acts on \(H\) by translation). Note that \(\ast_{hH} = BH\), \(\ast_{hG} = BG\), and \(H_{hH} \cong H/H = \ast\).
Apply the lemma again but this time, rather than using the homomorphism \(f\), use the group homomorphism \(G \to G/K\). Apply the lemma to the map of \(G/K\) spaces \(G/K \to H\) (which is isomorphic to the inclusion of the image of \(f\) into \(H\)). Note that the action of \(G/K\) on \(H\) is free, so \(H_{h(G/K)} \cong H/(G/K) = H/\im f\).

Also, \((G/K)_{hG} = BK\) as one can see by looking at the \(K \times G\) space \(G\) (letting \(K\) act on one side and \(G\) on the other) and computing \(G_{h(K \times G)}\) two ways: \(G_{h(K \times G)} = (G_{hK})_{hG} \cong (G/K)_{hG}\), and \(G_{h(K \times G)} = (G_{hG})_{hK} \cong \ast_{hK} = BK\).

The lemma in turn is the special case \(\mathcal{C} = BG\), \(\mathcal{D} = BH\), \(\mathcal{E} = \mathrm{Spaces}\) of the following lemma.

Let \(\mathcal{E}\) be an ∞-topos, \(F : \mathcal{C} \to \mathcal{D}\) be any functor of ∞-categories, and \(\alpha : X \to Y\) a Cartesian natural transformation¹ of functors \(X, Y : \mathcal{D} \to \mathcal{E}\). There are canonical morphisms of the form \(\colim X \circ F \to \colim X\) giving a pullback square: \[ \begin{CD} \colim X \circ F @>>> \colim Y \circ F \\ @VVV @VVV \\ \colim X @>>> \colim Y \end{CD} \]

The proof of this lemma uses the following defining properties of ∞-toposes:

Colimits in an ∞-topos are universal. If \(F : \mathcal{C} \to \mathcal{E}\) is any diagram with a cocone with vertex \(Y\) over it, and \(f : X \to Y\) is a morphism in \(E\) we can form the pullback diagram \(f^\ast F : \mathcal{C} \to \mathcal{E}\) given by \(f^\ast F(C) = \lim(F(C) \to Y \leftarrow X)\) and its colimit fits into a pullback square: \[ \begin{CD} \colim f^\ast F @>>> X \\ @VVV @VVV \\ \colim F @>>> Y \end{CD} \]
The generalized Mather cube theorem. Let \(F, G : \mathcal{C}^\triangleright \to \mathcal{E}\) be diagrams² in an ∞-topos \(\mathcal{E}\) and \(\alpha : F \to G\) a natural transformation between them. Assume \(F\) and \(G\) are both colimit diagrams and that \(\alpha\) restricted to \(F|_{\mathcal{C}} \to G|_{\mathcal{C}}\) is Cartesian. Then \(\alpha\) is also Cartesian, in particular, for every object of \(\mathcal{C}\) we have a pullback square \[ \begin{CD} F(C) @>>> \colim F \\ @V{\alpha_C}VV @V{\colim \alpha}VV \\ G(C) @>>> \colim Y \end{CD} \]

Warning: The first property is also true in ordinary (1-categorical) toposes, but not the second one.

We will show that pulling back the diagram \(Y \circ F\) along the morphism \(\colim \alpha : \colim X \to \colim Y\) gives the diagram \(X \circ F\), in which case universality of colimits gives us the result.

So we want to show that the following squares are pullbacks: \[ \begin{CD} X \circ F(C) @>>> \colim X \\ @V{\alpha_{F(C)}}VV @V{\colim \alpha}VV \\ Y \circ F(C) @>>> \colim Y \end{CD} \]

That follows directly from the generalized Mather cube property: since \(\alpha\) is Cartesian, for any object \(D \in \mathcal{D}\), whether or not \(D\) is in the image of \(F\) we have a pullback square: \[ \begin{CD} X(D) @>>> \colim X \\ @V{\alpha_D}VV @V{\colim \alpha}VV \\ Y(D) @>>> \colim Y \end{CD} \]

This means a natural transformation for which all of the naturality squares are pullbacks.

The notation \(\mathcal{C}^\triangleright\) means \(\mathcal{C}\) with a new terminal object added. If \(\mathcal{C}\) is the shape of some diagram \(F\), \(\mathcal{C}^\triangleright\) is the shape of a potential colimit diagram for \(F\).