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Tuesday, December 19, 2017

A naive constructible sheaf

In this post we describe a constructible sheaf over X=Ran valued in simplicial complexes, for a compact, smooth, connected manifold M. We note however that it does not capture all the information about the underlying space. Thanks to Joe Berner for helpful ideas.

Recall the category SC of simplicial complexes and simplicial maps, as well as the full subcategories SC_n of simplicial complexes with n vertices (the vertices are unordered). Let A = \bigcup_{k=1}^n SC_n with the ordering \leqslant_A as in a previous post ("Ordering simplicial complexes with unlabeled vertices," 2017-12-03), and f:X\to A the stratifying map. Let \{A_k\}_{k=1}^N be a cover of X by nested open sets of the type f^{-1}(U_S) = f^{-1}(\{T\in A\ :\ S\leqslant_A T\}), whose existence is guaranteed as A is finite. Note that f(A_1) is a singleton containg the complete simplex on n vertices.

Remark: For every simplicial complex S\in A, there is a locally constant sheaf over f^{-1}(S)\subseteq X. Given the cover \{A_k\} of X, denote this sheaf by \mathcal F_k \in \Shv(A_k\setminus A_{k-1}) and its value by S_k\in SC.

Let i^1:A_1\hookrightarrow A_2 and j^2:A_2\setminus A_1 \hookrightarrow A_2 be the natural inclusion maps . Note that A_1 is open and A_2\setminus A_1 is closed in A_2. The maps i^1,j^2 induce direct image functors on the sheaf categoriesi^1_*:\Shv(A_1) \to \Shv(A_2), \hspace{1cm} j^2_*:\Shv(A_2\setminus A_1) \to \Shv(A_2).The induced sheaves in \Shv(A_2) are extended by 0 on the complement of the domain from where they come. Note that since A_2\setminus A_1\subseteq A_2 is closed, j^2_* is the same as j^2_!, the direct image with compact support. We then have the direct sum sheaf i^1_*\mathcal F_1 \oplus j_*^2\mathcal F_2 \in \Shv(A_2), which we interpret as the disjoint union in SC. Then\left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)(U) = \begin{cases} S_1 & \text{ if }U\subseteq A_1, \\ S_2 & \text{ if }U\subseteq A_2\setminus A_1, \\ S_1\sqcup S_2 & \text{ else,} \end{cases} \hspace{1cm} \left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)_{(P,t)} = \begin{cases} S_1 & \text{ if } (P,t)\in A_1, \\ S_2 & \text{ if }(P,t)\in \text{int}(A_2\setminus A_1), \\ S_1\sqcup S_2 & \text{ else,} \end{cases}for U\subseteq A_2 open and (P,t)\in A_2. Generalizing this process, we get a sheaf on X. The diagram

may be helpful to keep in mind. We use the fact that direct sums commute with colimits (used in the definition of the direct image sheaf) to simplify notation. We then get sheaves\begin{array}{r c l} \mathcal F^1 & \in & \Shv(A_1), \\ i_*^1\mathcal F^1 \oplus j_*^2 \mathcal F^2 & \in & \Shv(A_2), \\ i_*^2i_*^1\mathcal F^1 \oplus i_*^2j_*^2 \mathcal F^2 \oplus j_*^3 \mathcal F^3 & \in & \Shv(A_3), \\ i_*^3i_*^2i_*^1\mathcal F^1 \oplus i_*^3i_*^2j_*^2 \mathcal F^2 \oplus i_*^3j_*^3 \mathcal F^3 \oplus j_*^4 \mathcal F^4 & \in & \Shv(A_4), \end{array}and finallyi_*^{N-1\cdots 1}\mathcal F^1 \oplus \left(\bigoplus_{k=2}^{N-1} i_*^{N-1\cdots k}j_*^k \mathcal F^k \right) \oplus j_*^N \mathcal F^N \in \Shv(A_N=X),where i_*^{N-1\cdots k} is the composition i_*^{N-1} \circ i_*^{N-2} \circ \cdots \circ i_*^k of direct image functors. Call this last sheaf simply \mathcal F \in \Shv(X). Each i_*^k extends the sheaf by 0 on an ever larger domain, so every summand in \mathcal F is non-zero on exactly one stratum as defined by f:X\to A. We now have a functor \mathcal F:Op(X) \to SC defined by\mathcal F(U) = \bigsqcup_{k=1}^N S_k \delta_{U,A_K\setminus A_{k-1}}, \hspace{1cm} \mathcal F_{(P,t)} = \bigsqcup_{k=1}^N S_k \delta_{(P,t),\text{cl}(,A_K\setminus A_{k-1})},where \delta_{U,V} is the Kronecker delta that evaluates to the identity if U\cap V \neq \emptyset and zero otherwise.

Remark: The sheaf \mathcal F is A-constructible, as \mathcal F|_{f^{-1}(S)} is a constant sheaf evaluating to the simplicial complex S\in A. However, if we want the cohomology groups to capture how the simplicial complexes change between strata, then we must use a different approach - all groups die when leaving a stratum because of the extension by zero construction.

References: nLab (article "Simplicial complexes")

Tuesday, December 5, 2017

Sheaves, derived and perverse

Let X,Y be topological spaces and f:X\to Y a continuous map. We let \Shv(X) be the category of sheaves on X, D(\Shv(X)) the derived category of sheaves on X, and D_b(\Shv(X)) the bounded variant. Recall that D(\mathcal A) for an abelian category \mathcal A is constructed first by taking C(\mathcal A), the category of cochains of elements of \mathcal A, quotienting by chain homotopy, then quotienting by all acylic chains.

Remark: Let \mathcal F\in \Shv(X). Recall:
  • a section of \mathcal F is an element of \mathcal F(U) for some U\subseteq X,
  • a germ of \mathcal F at x\in X is an equivalence class in \{s\in \mathcal F(U)\ :\ U\owns x\}/\sim_x,
  • s\sim_x t iff every neighborhood W of x in U\cap V has s|_W = t|_W, for s\in \mathcal F(U), t\in \mathcal F(V),
  • the support of the section s\in\mathcal F(U) is \supp(s) = \{x\in U\ :\ s \nsim_x 0\},
  • the support of the sheaf \mathcal F is \supp(\mathcal F) = \{x\in X\ :\ \mathcal F_x\neq 0\}.

Definition: The map f induces functors between categories of sheaves, called
\begin{array}{r r c l} \text{direct image} & f_*\ :\ \Shv(X) & \to & \Shv(Y), \\ & (U\mapsto \mathcal F(U)) & \mapsto & (V\mapsto \mathcal F(f^{-1}(V))),\\[15pt] \text{inverse image} & f^*\ :\ \Shv(Y) & \to & \Shv(X), \\ & (V\mapsto \mathcal G(V)) & \mapsto & \text{sh}\left(U\mapsto \text{colim}_{V\supseteq f(U)} \mathcal G(V)\right),\\[15pt] \text{direct image with compact support} & f_!\ :\ \Shv(X) & \to & \Shv(Y), \\ & (U\mapsto \mathcal F(U)) & \mapsto & \left(V\mapsto \left\{ s\in\mathcal F(f^{-1}(V))\ :\ f|_{\supp(s)} \text{ is proper}\right\}\right). \end{array}

Above we used that f:X\to Y is proper if f^{-1}(K)\subseteq X is compact, for every K\subseteq Y compact. Next, recall that a functor \varphi:\mathcal A\to \mathcal B induces a functor R\varphi:D(\mathcal A)\to D(\mathcal B), called the (first) derived functor of \varphi, given by R\varphi(A^\bullet) = H^1(\varphi(A)^\bullet).

Remark: Each of the maps f_*,f^*,f_! have their derived analogues Rf_*, Rf^*,Rf_!, respectively. For reasons unclear, Rf_! has a right adjoint, denoted Rf^!:D(\Shv(Y))\to D(\Shv(X)). This is called the exceptional inverse image.

We are now ready to define perverse sheaves.

Definition: Let A^\bullet \in D(\Shv(X)). Then:
  • the ith cohomology sheaf of A^\bullet is H^i(A^\bullet) = \ker(d^i)/\im(d^i),
  • A^\bullet is a constructible complex if H^i(A^\bullet) is a constructible sheaf for all i,
  • A^\bullet is a perverse sheaf if A^\bullet\in D_b(\Shv(X)) is constructible and \dim(\supp(H^{-i}(P))) \leqslant i for all i\in \Z and for P=A^\bullet and P=(A^\bullet)^\vee = (A^\vee)^\bullet the dual complex of sheaves.

We finish off with an example.

Example: Let X = \R be a stratified space, with X_0=0 the origin and X_1 = \R\setminus 0. Let \mathcal F\in \Shv(X) be an \R-valued sheaf given by \mathcal F(U) = \inf_{x\in U} |x|, and define a chain complex A^\bullet in the following way:
0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.
Note that for any U\subseteq \R, we have H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset if 0\not\in U, and 0 otherwise. Hence \supp(H^{-1}(A^\bullet)) = \R\setminus 0, whose dimension is 1. Next, H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0, and so \dim(\supp(H^0(A^\bullet))) = 0. Note that A^\bullet is self-dual and constructible, as the cohomology sheaves are locally constant. Hence A^\bullet is a perverse sheaf.

References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")

Sunday, December 3, 2017

Ordering simplicial complexes with unlabeled vertices

The goal of this post is to describe a partial order on the collection of simplical complexes with \leqslant n unlabeled vertices that is nice in the context of the space X=\Ran^{\leqslant n}(M)\times \R_{>0}.

First note that there is a natural order on (abstract) simplicial complexes, given by set inclusion. Interpreting elements of X as simplicial complexes induces a more restrictive order, as new vertices must "split off" from existing ones rather than just be introduced anywhere. Also note that the category usually denoted by SC of simplicial complexes and simplicial maps contains objects with unordered vertices. Here we assume an order on them and consider the action of the symmetric groups to remove the order.

Definition: Let SC_k, for some positive integer k, be the collection of simplicial complexes with k uniquely labeled vertices. This collection is a poset, with S\leqslant T iff \sigma\in T for every \sigma\in S.

The symmetric group on k elements acts on SC_k by permuting the vertices, and taking the image under this action we get SC_k/S_k, the collection of simplicial complexes with k unlabeled vertices. This set also has a partial order, with S\leqslant T in SC_k/S_k iff S'\leqslant T' in SC_k, for some S'\in q_k^{-1}(S) and T'\in q_k^{-1}(T), where q_k:SC_k \twoheadrightarrow SC_k/S_k is the quotient map.

Definition: For all i=1,\dots,k, let s_{k,i} be the ith splitting map, which splits the ith vertex in two. That is, if the vertices of S\in SC_k are labeled v_1,\dots,v_k, then s_{k,i} is defined by
\begin{array}{r c l} s_{k,i}\ :\ SC_k & \to & SC_{k+1}, \\ S & \mapsto & \left\langle S'\cup \{v_i,v_{i+1}\} \cup \displaystyle \bigcup_{\{v_i,w\}\in S'} \{v_{i+1},w\} \right\rangle , \end{array}where S' is S with v_j relabeled as v_{j+1} for all j>i, and \langle T\rangle is the simplicial complex generated by T.

By "generated by T" we mean generated in the Vietoris-Rips sense, that is, if \{v_a,v_b\}\in T for all a,b in some indexing set I, then \{v_c\ :\ c\in I\}\in \langle T\rangle. The ith splitting map is essentially the ith face map used for simplicial sets.

Let A = \bigcup_{k=1}^n SC_k/S_k. The splitting maps induce a partial order on A, with S\leqslant T, for S\in SC_k/S_k and T\in SC_{k+1}/S_{k+1}, iff s_{k,i}(S')\leqslant T' in SC_k, for some S'\in q_k^{-1}(S), T'\in q_{k+1}^{-1}(T), and i\in \{1,\dots,k\}. This generalizes via composition of the splitting maps to any pair S,T\in A, and is visually decribed by the diagram below.

Now, let M be a smooth, compact, connected manifold embedded in \R^N, and X=\Ran^{\leqslant n}(M)\times \R_{>0}. Let f:X\to A be given by (P,t)\mapsto VR(P,t), the Vietoris-Rips complex around the points of P with radius t.

Proposition: The map f:X\to A is continuous.

Proof: Let S\in A and U_S \subseteq A be the open set based at S. Take any (P,t)\in f^{-1}(U_S)\subseteq X, for which we will show that there is an open ball B\owns (P,t) completely within f^{-1}(U_S).

Case 1: t\neq d(P_i,P_j) for all pairs P_i,P_j\in P. Then set
\epsilon = \min\left\{t, \min_{i<j} |t-d(P_i,P_j)|, \min_{i<j} d(P_i,P_j) \right\}.Set B = B^{\Ran^{\leqslant n}(M)}_{\epsilon/4}(P) \times B^{\R_{>0}}_{\epsilon/4}(t), which is an open neighborhood of (P,t) in X. It is immediate that f(P',t'), for any other (P',t')\in B, has all the simplices of f(P,t), as \epsilon \leqslant |t-d(P_i,P_j)| for all i<j. If P_i has split in two in P', then for every simplex containing P_i in f(P,t) there are two simplices in f(P't'), with either of the points into which P_i split. That is, there may be new simplices in f(P',t'), but f(P',t') will be in the image of the splitting maps. Equivalently, f(P,t)\leqslant f(P',t') in A, so B\subseteq f^{-1}(U_S).

Case 2: t= d(P_i,P_j) for some pairs P_i,P_j\in P. Then set
\epsilon = \min\left\{t, \min_{i<j \atop t\neq d(P_i,P_j)} |t-d(P_i,P_j)|,\ \min_{i<j} d(P_i,P_j) \right\},and define B as above. We are using the definition of Vietoris-Rips complex for which we add an edge between P_i and P_j whenever t>d(P_i,P_j). Now take any (P',t')\in B such that its image and the image of (P,t) under f are both in SC_k/S_k. Then any points P_i,P_j \in P with d(P_i,P_j)=t that have moved around to get to P', an edge will possibly be added, but never removed, in the image of f (when comparing with the image of (P,t)). This means that we have f(P,t)\leqslant f(P',t') in SC_k/S_k, so certainly f(P,t)\leqslant f(P',t') in A. The same argument as in the first case holds if points of P split. Hence B\subseteq f^{-1}(U_S) in this case as well.  \square

This proposition shows in particular that X is poset-stratified by A