Deligne asphericity used for the relative complement

By [F6], the maps $(x,y)\mapsto x\otimes 1+y\otimes i$ and $v\otimes (a+ib)\mapsto (av,bv)$ are inverse real-linear identifications of $V\times V$ with $V_{\mathbb{C}}$. The formula for the inverse is balanced over $\mathbb{R}$, so it is well-defined on the tensor product. In a real basis they identify $V_{\mathbb{C}}$ with ${\mathbb{C}}^n$, where $n={\dim }_{\mathbb{R}}V$, and identify each $H{\otimes }_{\mathbb{R}}\mathbb{C}$ with a proper complex linear hyperplane. Such hyperplanes are closed, so $M$ is open. It is nonempty by [F16]: $V_{\mathbb{C}}$ is finite-dimensional over the infinite field $\mathbb{C}$, and the removed hyperplanes are a finite family of proper complex linear subspaces. For two distinct points of $M$, the complex affine line through them is contained in no removed hyperplane and meets each in at most one point. A complex line minus finitely many points is path-connected: a line-segment path can be modified by finitely many small circular detours. For coincident endpoints use the constant path. Thus $M$ is path-connected in the sense of [F15]. In dimension zero the collection of proper hyperplanes is empty and $M$ is the single point.

In dimension zero the point has its one-cell CW structure. For $n>0$, [F8] gives $M$ its standard smooth real $2n$-manifold structure. Its Euclidean subspace topology is Hausdorff. We check paracompactness by an explicit refinement. In real coordinates put $U=M\subseteq {\mathbb{R}}^{2n}$ and $K_j=\{u\in U:\parallel u\parallel \le j,\mathrm{dist}(u,U^c)\ge 1\mathrm{/}j\}$ for $j\ge 1$, using distance $+\mathrm{\infty }$ when $U^c$ is empty, and put $K_0=K_{-1}=\mathrm{\varnothing }$. These sets are compact, $K_j\subseteq {\mathrm{int}}_U{K}_{j+1}$, and their interiors cover $U$. For an arbitrary open cover of $U$, all rational balls whose closures lie in ${\mathrm{int}}_U{K}_{j+1}\setminus K_{j-2}$ and which lie in some cover member form a countable cover of the compact shell $K_j\setminus {\mathrm{int}}_U{K}_{j-1}$. Take the first finite list from this family covering that shell in a fixed enumeration of all finite lists of rational balls; take the empty list for an empty shell. These finite families cover $U$. They are locally finite, since ${\mathrm{int}}_U{K}_N$ meets none of the families with $j\ge N+2$. This is the refinement required by [F9], and the least-list rule makes the selections explicit.

For $n>0$, step 2.1 verifies the manifold hypotheses of [E1] recorded in [F5]. By step 1.1, $M$ is also a nonempty Euclidean open set, so the more precise [E2] in [F5] supplies a locally finite triangulation homeomorphic to $M$. Use the geometric conventions of [F13] and write its homeomorphism as $\mathrm{\Phi }:\mathrm{\mid }K\mathrm{\mid }\to M$. Each closed $d$-simplex is a disk with boundary its proper faces, as follows directly in barycentric coordinates. For $d>0$, put $c=(1\mathrm{/}(d+1),\dots ,1\mathrm{/}(d+1))$ in the standard simplex ${\mathrm{\Delta }}^d=\{\lambda \in {\mathbb{R}}^{d+1}:{\lambda }_i\ge 0,{\sum }_i{\lambda }_i=1\}$ and $L=\{u\in {\mathbb{R}}^{d+1}:{\sum }_i{u}_i=0\}$. For a unit vector $u\in L$, the ray $c+tu$ first meets the boundary at $t=\rho (u)=1\mathrm{/}((d+1){\max }_i(-u_i))$. The denominator is positive, and $\rho$ is continuous and bounded above and below by positive constants on the unit sphere. Thus $q(0)=c$ and $q(ru)=c+r\rho (u)u$ give a homeomorphism from the closed unit disk of $L$ onto ${\mathrm{\Delta }}^d$, taking its boundary to $\mathrm{\partial }{\mathrm{\Delta }}^d$; the inverse rescales each ray by $1\mathrm{/}\rho (u)$ and is continuous also at $c$ by those bounds. Identify $L$ isometrically with ${\mathbb{R}}^d$ and use the affine homeomorphism ${\mathrm{\Delta }}^d\to \sigma$ given by the vertices in [F13]. For $d=0$ use the map from a point to the vertex. Composing these disk maps with $\mathrm{\Phi }$ gives maps ${\chi }_{\sigma }:D^d\to M$ whose interiors map homeomorphically onto $\mathrm{\Phi }({\sigma }^{\circ })$ and whose boundaries lie in $X^{d-1}=\mathrm{\Phi }(\mathrm{\mid }K{\mathrm{\mid }}^{(d-1)})$. The common-face condition makes these open cells disjoint and their union all of $M$; each closed cell is the union of its finitely many open faces.

Each closed simplex is compact and closed in the Euclidean realization. If $A\subseteq \mathrm{\mid }K\mathrm{\mid }$ meets every closed simplex in a closed subset of that simplex, these intersections are closed in $\mathrm{\mid }K\mathrm{\mid }$ and form a locally finite family by [E2]. Their union $A$ is closed: near any point only finitely many intersections need be considered. The converse follows by restriction. The same reasoning applies to every skeleton with its subspace topology. In particular the vertex skeleton is discrete, since every union of singleton vertices is closed in it. For $d\ge 1$, the natural surjection from $X^{d-1}\bigsqcup {⨆}_{\dim \sigma =d}{D}^d$ onto $X^d=\mathrm{\Phi }(\mathrm{\mid }K{\mathrm{\mid }}^{(d)})$, using the maps ${\chi }_{\sigma }$ in step 3.1, identifies exactly the boundary attachments of [F14]. It is a quotient map: a subset of $X^d$ has closed preimage exactly when its intersection with $X^{d-1}$ and its inverse images in all the $d$-disks are closed, which is equivalent to having closed intersection with every simplex of that skeleton by the preceding closed-set criterion and the disk homeomorphisms. Hence the skeleton filtration is formed by attaching disks with those characteristic maps. The closed-cell criterion above gives the weak topology on all of $M$, and step 3.1 gives closure finiteness. Since $M$ is Hausdorff, [F10] now gives a CW structure on $M$ itself. For $n=0$ use the structure in step 2.1.

The given hyperplanes are linear, hence central, and all the finite simplicial hypotheses of [F1] are exactly those in the Statement. Consequently $M$ is aspherical; for the zero-dimensional point this is also immediate. By step 1.1 it is nonempty and path-connected, hence connected by [F15]. Choose a vertex $b$ of the CW structure from step 4.1 and set $\pi ={\pi }_1(M,b)$. By [F11], this loop-class multiplication is well-defined and $\pi$ is a group; its identity map preserves multiplication and is its own inverse, hence is a group isomorphism. With the groups interpreted by [F11] and [F12], these data satisfy [F4], proving the first assertion with the required based CW convention.

In the spherical case, the Coxeter data have the meaning in [F7]. By [F2], $({\mathrm{\Theta }}_J,{\mathcal{H}}^J)$ meets the finite central simplicial hypotheses just used, and [F3] identifies its complexified complement with ${\mathcal{X}}_J^{\mathbb{C}}$. Transport the CW structure and vertex basepoint through this identification. Step 4.1 then makes the relative complement a $K(\pi ,1)$ for its based fundamental group, with the identity identification.