Finite CW-models for spherical normaliser quotients

Since $W$ is finite, its normaliser subgroup and its quotient $G$ are finite. By [F1] the relative cone is the whole space $E$ and the arrangement is finite. By [F2], the action preserves this arrangement and its diagonal action on $X$ is free. Thus the finite-arrangement and action hypotheses of [F3] hold.

Take $K$ to be exactly the incident-pair order complex of [F5], with its weak realization. Applying [F3] to the arrangement and action in step 1.1 gives the $G$-equivariant homotopy equivalence $f:K\to X$ and the induced homotopy equivalence $\bar{f}:K\mathrm{/}G\to X\mathrm{/}G$, with both sources finite CW complexes under [F6]. This specifies the model in [F1] by the very construction used in [F3], rather than identifying an unspecified homotopy model with it.

Let $h:Z\to Y$ be either homotopy equivalence in step 2.1, where $Y$ is nonempty and path-connected by [F4]. Choose a homotopy inverse $r:Y\to Z$ and a homotopy $rh\simeq {\mathrm{id}}_Z$ using [F6]. Then $Z$ is nonempty since it contains $r(y)$ for any $y\in Y$. For $z,z^{\mathrm{\prime }}\in Z$, the homotopy tracks join $rh(z)$ to $z$ and $rh(z^{\mathrm{\prime }})$ to $z^{\mathrm{\prime }}$. A path in $Y$ from $h(z)$ to $h(z^{\mathrm{\prime }})$, mapped by $r$, and those tracks (the first reversed) concatenate to a path from $z$ to $z^{\mathrm{\prime }}$. Thus $Z$ is path-connected.

For every $n\ge 1$, functoriality and the moving-basepoint formula in [F7], applied to $rh\simeq {\mathrm{id}}_Z$, show that $r_{\ast }{h}_{\ast }:{\pi }_n(Z,z)\to {\pi }_n(Z,rh(z))$ is an isomorphism: it is the identity followed by transport along the homotopy track. Likewise $h_{\ast }{r}_{\ast }:{\pi }_n(Y,y)\to {\pi }_n(Y,hr(y))$ is an isomorphism using $hr\simeq {\mathrm{id}}_Y$. Injectivity of these composites makes $h_{\ast }$ and $r_{\ast }$ injective at every basepoint. At $y=h(z)$, surjectivity of $r_{\ast }{h}_{\ast }$ and injectivity of $r_{\ast }$ then give surjectivity of $h_{\ast }$: for any $b\in {\pi }_n(Y,h(z))$, choose $a$ with $r_{\ast }{h}_{\ast }(a)=r_{\ast }(b)$ and cancel $r_{\ast }$. Hence $h_{\ast }:{\pi }_n(Z,z)\to {\pi }_n(Y,h(z))$ is an isomorphism in every positive degree.

Apply steps 3.1 and 4.1 first to $f:K\to X$ and then to $\bar{f}:K\mathrm{/}G\to X\mathrm{/}G$. By [F4], both targets are nonempty and path-connected, their fundamental groups are respectively $N(P,J)$ and $N(A,J)$ up to isomorphism, and their higher homotopy groups vanish. The same properties therefore hold for the respective sources. Together with their finite CW structures from step 2.1, this proves the asserted finite classifying-space models. Choice enters through [F4]'s group identifications.