In a previous blog post ("A constructible sheaf over the Ran space," 2017-06-24) it was claimed that there was a particular constructible sheaf over Ran⩽n(M)×R⩾0. However, the proof actually uses finite ordered subsets of M to make the stratification, rather than finite unordered subsets. This means that the sheaf is actually over M×n×R⩾0, and in this post we try to fix that problem.
Let Δn be the "fat diagonal" of M×n, that is, the collection of P∈M×n for which at least two coordinates are the same. For every k>0, there is an Sk action on M×k∖Δk, quotienting by which we get a map
M×k∖Δk qk →Rank(M)
to the Ran space of degree k. The stratification of M×k×R⩾0 given in the previous post will be pushed forward to a stratification of Rank(M)×R⩾0, for all 0<k⩽n. A large part of the work already has been done, it remains to put everything in the right order and check openness. The process is given as follows:
As stated in the proof of the Theorem, Ran⩾k(M)×R⩾0 is open inside Ran⩽n(M)×R⩾0, allowing us to make a stratification f:Ran⩽n(M)×R⩾0→A, where A is the poset
where the tail of an arrow is ordered lower than the head. The map f sends Rank(M)×R⩾0 to ak, which is a continuous map in the upset topology on A.
As stated in Definition 5, we have a stratification gk:(M×k∖Δk)×R⩾0→Bk, where Bk may be viewed as a directed graph Bk=(Vk,Ek). The vertex set is Vk={0,1}k(k+1)/2, whose elements are strings of 1 and 0, and the edge set Ek contains v→v′ iff dH(v,v′)=1 and dH(v,0)<dH(v′,0), for dH the Hamming distance. Let Uv⊂Bk denote the upset based at v, that is, all elements v′∈Bk with v⩽v′.
Order all distinct pairs (i,j)∈{1,…,k}2, of which there are k(k+1)/2. Under the stratifying map gk, each upset Uv based at the vertex v∈{0,1}k(k+1)/2 receives elements (P,t)∈(M×k∖Δk)×R⩾0 satisfying t>d(Pi,Pj) whenever the position representing (i,j) in v is 1. For example, when k=3,
To check that gk is continuous in the upset topology, we restate Lemma 2 in a clearer way.
Lemma 1: Let U⊂X be open and φ:X→A⊂R⩾0 continuous, with |A|<∞. Then
⋃x∈U{x}×(φ(x),∞) ⊆ X×(z′,∞)
is open, for any z′⩽z:=minx∈U{φ(x)}.
Proof: Consider the function
ψ : X×(z′,∞)→X×(−∞,z),(x,t)↦(x,φ(x)−t).
Since φ is continuous and subtraction is continuous, ψ is continuous (in the product topology). Since U×(−∞,0) is open in X×(−∞,z), the set ψ−1(U×(−∞,0)) is open in X×(z′,∞). For any x∈U and t=φ(x), we have φ(x)−t=0. For any x∈U and t→∞, we have φ(x)−t→−∞. It is immediate that all other t∈(φ(x),∞) give φ(x)−t∈(−∞,0). Hence ψ−1(U×(−∞,0)) is the collection of points (x,t) with t∈(φ(x),∞), which is then open in X×[0,z′). ◻
Applying Lemma 1 to U=X=M×k∖Δk and φ(P)=maxi≠j{d(Pi,Pj)}, which is continuous, gives that g−1k(U11⋯1)⊆M×k∖Δk is open. This also works to show that g−1k(Uv)⊆g−1k(Uv′) is open, for any v′⩽v, by limiting the pairs of indices iterated over by the function φ. Hence gk is continuous.
The symmetric group Sk acts on (M×k∖Δk)×R⩾0 by permuting the order of elements in the first factor. That is, for σ∈Sk, we have
σ(P={P1,…,Pk},t)=({Pσ(1),…,Pσ(k)},t).
Note that ((M×k∖Δk)×R⩾0)/Sk=Rank(M)×R⩾0.
Remark: Graph isomorphism for two graphs with k vertices may also be viewed as the equivalence relation induced by Sk acting on Γk={simple vertex-labeled graphs with k vertices}. First, let Gv be the (unique) graph first introduced at element v∈Bk by gk. That is, we have Gv=VR(P,t)1 (the ordered 1-skeleton of the Vietoris--Rips complex on the set P with radius t) whenever gk((P,t))∈Uv and gk((P,t))∉Uv′ for any v′⩽v, v′≠v. Then the elements of Bk are in bijection with the elements of Γk (given by v↔Gv), so we have Bk/Sk=B′k. Recall that v⩽v′ in Bk iff adding an edge to Gv gives Gv′. In Bk′, this becomes a partial order on equivalence classes [w]={v∈Bk : σGv=Gw for some σ∈Sk}. We write [w]⩽[w′] iff there is a collection of pairs {(v1,v′1),…,(vℓ,v′ℓ)} such that vi⩽v′i for all i, and {v1,…,vℓ}=[w] and {v′1,…,v′ℓ}=[w′] (there may be repetition among the vi or v′i).
By the universal property of the quotient, there is a unique map hk:Rank(M)×R⩾0→B′k that makes the following diagram commute.
This will be our stratifying map. To check that hk is continuous take U⊆B′k open. As π is the projection under a group action, it is an open map, so π−1(U)⊆Bk is open. Since gk is continuous in the upset topology, g−1k(π−1(U)) is open. Again, Sk↷ is the projection under a group action, so (Sk↷)(g−1k(π−1(U))) is open, giving continuity of hk.
Let Δn be the "fat diagonal" of M×n, that is, the collection of P∈M×n for which at least two coordinates are the same. For every k>0, there is an Sk action on M×k∖Δk, quotienting by which we get a map
M×k∖Δk qk →Rank(M)
to the Ran space of degree k. The stratification of M×k×R⩾0 given in the previous post will be pushed forward to a stratification of Rank(M)×R⩾0, for all 0<k⩽n. A large part of the work already has been done, it remains to put everything in the right order and check openness. The process is given as follows:
- Stratify Ran⩽n(M)×R⩾0 into n pieces, each being Rank(M)×R⩾0.
- Stratify (M×k∖Δk)×R⩾0 as in the previous post.
- Quotient by Sk-action to get stratification of Rank(M)×R⩾0.
Step 1
As stated in the proof of the Theorem, Ran⩾k(M)×R⩾0 is open inside Ran⩽n(M)×R⩾0, allowing us to make a stratification f:Ran⩽n(M)×R⩾0→A, where A is the poset
where the tail of an arrow is ordered lower than the head. The map f sends Rank(M)×R⩾0 to ak, which is a continuous map in the upset topology on A.
Step 2
As stated in Definition 5, we have a stratification gk:(M×k∖Δk)×R⩾0→Bk, where Bk may be viewed as a directed graph Bk=(Vk,Ek). The vertex set is Vk={0,1}k(k+1)/2, whose elements are strings of 1 and 0, and the edge set Ek contains v→v′ iff dH(v,v′)=1 and dH(v,0)<dH(v′,0), for dH the Hamming distance. Let Uv⊂Bk denote the upset based at v, that is, all elements v′∈Bk with v⩽v′.
Order all distinct pairs (i,j)∈{1,…,k}2, of which there are k(k+1)/2. Under the stratifying map gk, each upset Uv based at the vertex v∈{0,1}k(k+1)/2 receives elements (P,t)∈(M×k∖Δk)×R⩾0 satisfying t>d(Pi,Pj) whenever the position representing (i,j) in v is 1. For example, when k=3,
To check that gk is continuous in the upset topology, we restate Lemma 2 in a clearer way.
Lemma 1: Let U⊂X be open and φ:X→A⊂R⩾0 continuous, with |A|<∞. Then
⋃x∈U{x}×(φ(x),∞) ⊆ X×(z′,∞)
is open, for any z′⩽z:=minx∈U{φ(x)}.
Proof: Consider the function
ψ : X×(z′,∞)→X×(−∞,z),(x,t)↦(x,φ(x)−t).
Since φ is continuous and subtraction is continuous, ψ is continuous (in the product topology). Since U×(−∞,0) is open in X×(−∞,z), the set ψ−1(U×(−∞,0)) is open in X×(z′,∞). For any x∈U and t=φ(x), we have φ(x)−t=0. For any x∈U and t→∞, we have φ(x)−t→−∞. It is immediate that all other t∈(φ(x),∞) give φ(x)−t∈(−∞,0). Hence ψ−1(U×(−∞,0)) is the collection of points (x,t) with t∈(φ(x),∞), which is then open in X×[0,z′). ◻
Applying Lemma 1 to U=X=M×k∖Δk and φ(P)=maxi≠j{d(Pi,Pj)}, which is continuous, gives that g−1k(U11⋯1)⊆M×k∖Δk is open. This also works to show that g−1k(Uv)⊆g−1k(Uv′) is open, for any v′⩽v, by limiting the pairs of indices iterated over by the function φ. Hence gk is continuous.
Step 3
The symmetric group Sk acts on (M×k∖Δk)×R⩾0 by permuting the order of elements in the first factor. That is, for σ∈Sk, we have
σ(P={P1,…,Pk},t)=({Pσ(1),…,Pσ(k)},t).
Note that ((M×k∖Δk)×R⩾0)/Sk=Rank(M)×R⩾0.
Remark: Graph isomorphism for two graphs with k vertices may also be viewed as the equivalence relation induced by Sk acting on Γk={simple vertex-labeled graphs with k vertices}. First, let Gv be the (unique) graph first introduced at element v∈Bk by gk. That is, we have Gv=VR(P,t)1 (the ordered 1-skeleton of the Vietoris--Rips complex on the set P with radius t) whenever gk((P,t))∈Uv and gk((P,t))∉Uv′ for any v′⩽v, v′≠v. Then the elements of Bk are in bijection with the elements of Γk (given by v↔Gv), so we have Bk/Sk=B′k. Recall that v⩽v′ in Bk iff adding an edge to Gv gives Gv′. In Bk′, this becomes a partial order on equivalence classes [w]={v∈Bk : σGv=Gw for some σ∈Sk}. We write [w]⩽[w′] iff there is a collection of pairs {(v1,v′1),…,(vℓ,v′ℓ)} such that vi⩽v′i for all i, and {v1,…,vℓ}=[w] and {v′1,…,v′ℓ}=[w′] (there may be repetition among the vi or v′i).
By the universal property of the quotient, there is a unique map hk:Rank(M)×R⩾0→B′k that makes the following diagram commute.
This will be our stratifying map. To check that hk is continuous take U⊆B′k open. As π is the projection under a group action, it is an open map, so π−1(U)⊆Bk is open. Since gk is continuous in the upset topology, g−1k(π−1(U)) is open. Again, Sk↷ is the projection under a group action, so (Sk↷)(g−1k(π−1(U))) is open, giving continuity of hk.
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