Branched Covering and Profinite Completion

Weil conjectured the existence of some cohomology theory for nonsigular projective varieties over finite fields. According to the Lefschetz fixed-point formula and the Poincaré duality for ordinary cohomology theory, he further conjectured several statements which were later called the famous Weil Conjectures.

Grothendieck defined the étale topology for schemes and the associated étale cohomology theory was later defined by Artin-Grothendieck-Verdier, exposited in the famous SGA notes [1], or independently by Lubkin ([6]). The étale cohomology provides the Lefschetz fixed-point formula and the Poincaré duality. Hence the proof of most of the Weil conjectures is straightforward, except the Riemann hypothesis, which was completed by Deligne ([4]).

Out of the étale topology of schemes, Artin-Mazur ([2]) also defined the étale homotopy type of a scheme. They generalized the Cěch nerve of an open cover to define a hypercovering of a (pointed) Grothendieck-Verdier site by a (pointed) simplicial object satisfying some refinement condition at each level. Then all the hypercoverings of a site form a pro system up to homotopy. When the site is locally connected, levelwise connected components of a hypercovering form a simplicial set. Then one gets a pro-space out of hypercoverings.

They also defined the profinite completion of a pro-space by the left adjoint of the inclusion functor from the pro homotopy category of finite spaces to the pro homotopy category of pointed CW complexes. Later on, Sullivan ([8]) proved that the pro-space of Artin-Mazur’s profinite completion has a homotopy inverse limit.

If one considers the étale site of some ‘nice’ scheme, each hypercovering is a finite space and the pro-space induced from all hypercoverings is called the étale homotopy type of the scheme. Assuming that the scheme is finite type over $\mathbb{C}$, Artin-Mazur proved that its étale homotopy type is homotopy equivalent to the profinite completion of its analytification.

This paper aims to extend Arin-Mazur’s construction to a piecewise linear pseudomanifold and prove the generalized Riemann existence theorem for pseudomanifolds. In the postscript of [8], Sullivan already claimed it and suggested a method for proof via transversality and the symmetric product approximation of an Eilenberg-Maclane space. However, the method of this paper is quite different.

Let us first explain why we use pseudomanifolds and branched coverings. Roughly speaking, a pseudomanifold $X$ is a manifold with singularity of real dimension at least $2$. It is like a piecewise-linear generalization of a complex variety. Moreover, an étale morphism into a complex variety is almost a finite covering map onto a Zariski open subset. An appropriate piecewise-linear generalization is a finite covering map onto the complement of a real codimension at least $2$ subcomplex of the pseudomanifold. We can always complete such a map to a piecewise linear branched covering map.

Apply Artin-Mazur’s construction to the Grothendieck site of all branched cover over a pseudomanifold, i.e., the pro-space $X_{\operatorname{\acute{e}t}}$ formulated by the levelwise connected components of all hypercoverings. By Artin-Mazur’s arguments, we prove that each space in the pro-system is a finite space and then deduce that

The technical part is the comparison theorem of cohomologies under ordinary topology and étale topology, i.e., any degree at least $1$ cocycle of a finite local system vanishes after passing to each branched covering in some ‘open’ covering of the étale site.

We use the complement of the codimension $2$ skeleton or the codimension $2$ coskeleton to form the étale cover. Each of them deformation retracts onto some $1$-dimensional complex and any cocycle vanishes after passing to some finite cover of the $1$-dimensional complex.

I want to thank my thesis advisor Dennis Sullivan, who led me to this fantastic area. I also want to thank Jason Starr for disucssions in the Riemann existence theorem and Siqing Zhang for explaining the algebro-geometric proof of the comparison theorem in the étale theory.

This chapter is a review of Artin-Mazur’s definition [2] of profinite completion.

There is a more concrete way to represent a pro-morphism ([2]).

In practice, we can avoid the set theoretic issue by isomorphism classes of objects and thus we can assume $\mathcal{C}$ is small.

Let $\textbf{W}_{0}$ be the category of based connected CW complexes and let $\textbf{H}_{0}$ be its homotopy category. A (pro) space is a (pro) object of $\textbf{H}_{0}$.

Let $A$ be an abelian group. The homology of a pro-space $X=\{(X_{i},x_{i})\}$ with coefficient $A$ is defined by the pro-group $H_{n}(X;A)=\{H_{n}(X_{i};A)\}$ and the cohomology is the group $H^{n}(X;A)=\underset{i}{\varinjlim}H^{n}(X_{i};A)$. Similarly, define the homotopy group $\pi_{n}X$ by the pro-group $\{\pi_{n}(X_{i},x_{i})\}$.

Let $\mathcal{C}\textbf{W}_{0}$ be the full subcategory of $\textbf{W}_{0}$ consisting of the CW complexes with all homotopy groups in $\mathcal{C}$. Let $\mathcal{C}\textbf{H}_{0}$ be the corresponding homotopy category.

The idea is analogous to the case of groups, namely, the pro-space $\widehat{X}$ is constructed out of all homotopy classes of pro-maps $f:X\rightarrow F$ with $F\in\mathcal{C}\textbf{H}_{0}$.

It is not hard that

In this chapter, we review the definition of sites and Artin-Mazur’s definition of hypercoverings for a site. We also revisit the necessary lemmas used for our main result.

For simplicity, we always assume that the underlying category of a site admits finite limits and finite coproducts.

Artin-Mazur defined hycoverings of a site so that they can further define a pro homotopy type for a ‘nice’ site.

A simplicial morphism of hypercoverings $K_{*}\rightarrow K^{\prime}_{*}$ is called a refinement if for each level $n$ the map $K_{n}\rightarrow K^{\prime}_{n}$ is a covering.

Let $X$ be an object of C and $S$ be a finite set. Define the object $X\times S$ of C by $\bigsqcup_{S}X$.

Let $I_{*}$ be the simplicial set of the unit interval. Then for any simplicial object $K_{*}$ of C we can form the simplicial object $K_{*}\times I_{*}$ by $(K_{*}\times I_{*})_{n}=K_{n}\times I_{n}$.

Two simplicial maps $f,g:K_{*}\rightarrow K^{\prime}_{*}$ of simplicial objects are strictly homotopic if there is a map $F:K_{*}\times I_{*}\rightarrow K^{\prime}_{*}$ connecting them, i.e., $F\circ j_{0}=f$ and $F\circ j_{1}=g$, where $j_{0},j_{1}:K_{*}\rightarrow K_{*}\times I_{*}$ are induced by the maps $[0],[1]:\operatorname{pt}\rightarrow I_{*}$. Call $f$ and $g$ homotopic if they are connected by a finite chain of strict homotopies.

Let $HR(\textbf{C})$ be the category of hypercoverings of the (pointed) site C, whose morphisms are homotopy classes of simplicial morphisms.

There is a natural connected component functor $\pi:\textbf{C}\rightarrow\mathbf{Set}$ for a locally connected site C defined by mapping an object to the index set of its connected components. Then for any hypercovering $K_{*}$ of C, $\pi(K_{*})$ is a simplicial set. If C is pointed, then $\pi(K_{*})$ is a pointed/based simplicial set; if C is connected, then the simplicial set $\pi(K_{*})$ is connected.

Since $HR(\textbf{C})$ is cofiltering, we get a pro-system of (homotopy) simplicial sets $\pi C=\{\pi(K_{*})\}_{K_{*}\in HR(\textbf{C})}$.

Let us skip the definition of fibered categories and only define the descent data in the form that we will use.

For each object $X$ of a site C, let $\mathbf{F}(X)$ be the small category of objects

where morphisms are of the form $X\times S\rightarrow X\times S^{\prime}$ so that the following diagram commutes.

Any morphism $f:X\rightarrow Y$ in C induces a natural functor $f^{*}:\mathbf{F}(Y)\rightarrow\mathbf{F}(X)$.

If C is locally connected, then a locally constant covering consists of a hypercovering $K_{*}$, a finite set $S$ and a $1$-cocycle $\sigma$ of $\pi(K_{*})$ with values in the symmetric group $\operatorname{Sym}(S)$. Two locally constant coverings $(K_{*},S,\sigma)$ and $(K^{\prime}_{*},S^{\prime},\sigma^{\prime})$ are isomorphic if $S=S^{\prime}$ and there exists a common refinement $K^{\prime\prime}$ with $\phi_{0}:K^{\prime\prime}\rightarrow K$ and $\phi_{1}:K^{\prime\prime}\rightarrow K^{\prime}$ such that $\phi_{0}^{*}\sigma$ and $\phi_{1}^{*}\sigma^{\prime}$ are cohomologous.

Let $G$ be a finite group. Define a principal bundle over C with fiber $G$ by a locally constant sheaf with stalk $|G|$ left acted by $G$ over a site C, namely, a locally constant covering $(K_{*},|G|,\sigma)$ with $\sigma\in G\subset\operatorname{Sym}(|G|)$. Hence,

Finally, let us review the results of étale homotopy theory for schemes. Let $\mathcal{C}$ be the class of finite groups.

The proof is based on the following key lemma.

For a scheme $X$ over $\mathbb{C}$ of finite type, let $X_{\operatorname{cl}}$ be the complex algebraic set with analytic topology.

Artin-Mazur first proved that $X_{\operatorname{cl}}\rightarrow X_{\operatorname{\acute{e}t}}$ is a C-equivalence. To show that the map is indeed a profinite homotopy equivalence, the followings are needed.

Let $f$ be a pointed morphism of pointed connected sites $\textbf{C}\rightarrow\textbf{C}^{\prime}$ induced by the functor $F:\textbf{C}^{\prime}\rightarrow\textbf{C}$. Then for any hypercovering $K_{*}$ of $\textbf{C}^{\prime}$, consider all maps $L_{*}\rightarrow F(K_{*})$ of hypercoverings in C. They induce a morphism of pro-spaces

Call a pseudomanifold $X$ normal if the link of each $i$-simplex is connected for $i<n-1$.

Let $X$ be a pseudomanifold of dimension $n$ with a fixed triangulation $T$. Let $T^{\prime}$ be the barycentric subdivision of $T$. Each simplex of $T^{\prime}$ is uniquely represented by a sequence $(\sigma_{0},\sigma_{1},\cdots,\sigma_{k})$, where each $\sigma_{i}$ is a simplex of $T$ and each $\sigma_{i+1}$ is a face of $\sigma_{i}$.

An $i$-dimensional dual cone $C^{i}(\sigma^{n-i})$ of $X$ associated to a simplex $\sigma^{n-i}\in T$ is defined by the union of all the closed simplices in $T^{\prime}$ like $(\sigma_{0},\sigma_{1},\cdots,\sigma_{k}=\sigma^{n-i})$.

The link $L(\sigma^{n-i})$ of $\sigma^{n-i}$ is the subcomplex consisting of the simplices $(\sigma_{0},\sigma_{1},\cdots,\sigma_{k}\neq\sigma^{n-i}))$ in $C^{i}(\sigma^{n-i})$.

The $i$-skeleton of $X$ is the union of all simplices of dimension at most $i$ and the $i$-coskeleton of $X$ is the union of all dual cones of dimension at most $i$.

Notice that the $i$-skeleton intersects the $(n-i)$-coskeleton transversally.

It is obvious that the link of each codimension $1$ simplex is $S^{0}$ and the link of each codimension $2$ simplex is a finite number of $S^{1}$.

Let $B$ be a subpolyhedron of a pseudomanifold $X$. Let $C(B,X)$ be the union of simplices in $T^{\prime}$ disjoint from $B$. Notice that $C(B,X)$ is also the union of dual cones that is disjoint from $B$.

In particular,

On the other hand, let $D$ be a full subcomplex of $T^{\prime}$ so that $D$ consists of dual cones. Let $S(D,X)$ be the union of simplicies of $T$ that does not intersect $D$. Notice that each simplex is a cone over its boundary about its barycenter and the barycentric subdivision respects the cone structure. Apply the same argument for the skeletons and we get

The lemmas are also true for more general cases, such as finite simplicial complexes.

Similarly, we can also define dual cones and links for a pseudomanifold with boundary and prove the following.

Likewise, 4.5 and 4.3 are also true.

Recall the definition of regular neighborhoods. Let $B$ be a full subcomplex of some triangulation $T$ of $X^{n}$. Define a regular neighborhood $N(B,X)$ of $B$ by $\overline{X-C(X,B)}$. Indeed, $N(B,X)$ consists of all closed dual cones that have nonempty intersetion with $B$.

The argument in the proof also shows that

In this chapter, we prove a similar theorem for pseudomanifolds as the generalized Riemann existence theorem for complex varieties.

Let $\textbf{S}_{\operatorname{cl}}(X)$ be the site of open subsets of $X$. To establish a connection between $\textbf{S}_{\operatorname{cl}}(X)$ and $\textbf{S}_{\operatorname{\acute{e}t}}(X)$, we need a common refinement. Let $\textbf{S}^{\prime}(X)$ be the site consisting of finite covering maps onto some open subset of $X$. Then we have morphisms of pointed sites

Notice that each covering of $S^{\prime}(X)$ is dominated by a covering of $S_{\operatorname{cl}}(X)$, i.e., we can cover a finite covering map by homeorphisms. Hence, $\pi(\textbf{S}^{\prime}(X))\rightarrow\pi(\textbf{S}_{\operatorname{cl}}(X))\rightarrow\operatorname{Sing}(X)$ is a weak equivalence of pro-spaces.

Let $G$ be a finite group. Any principal bundle $\mathfrak{G}$ with fiber $G$ over $\pi(\textbf{S}_{\operatorname{\acute{e}t}}(X))$ is also a principal bundle over $\textbf{S}^{\prime}(X)$. So it induces a homomorphism $\pi_{1}(X)\rightarrow G$. On the other hand, the kernel of any homomorphism $\pi_{1}(X)\rightarrow G$ corresponds to a finite covering space $X^{\prime}$ of $X$ and hence the trivial bundle over $X^{\prime}$ with fiber $G$ gives a principal bundle $\mathfrak{G}$ with fiber $G$ over $\pi(\textbf{S}_{\operatorname{\acute{e}t}}(X))$. Due to 3.15, we get that

Let $A$ be a finite abelian group. Let $\mathfrak{A}$ be a locally constant sheaf with stalk $A$ on $\textbf{S}_{\operatorname{\acute{e}t}}(X)$, which induces a local system $\widetilde{A}$ on $X$.