Processing math: 100%

Sunday, September 17, 2017

Stratifying correctly

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 Rann(M)×R0. 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×R0, and in this post we try to fix that problem.

Let Δn be the "fat diagonal" of M×n, that is, the collection of PM×n for which at least two coordinates are the same. For every k>0, there is an Sk action on M×kΔk, quotienting by which we get a map
M×kΔk qk Rank(M)
to the Ran space of degree k. The stratification of M×k×R0 given in the previous post will be pushed forward to a stratification of Rank(M)×R0, for all 0<kn. 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:
  1. Stratify Rann(M)×R0 into n pieces, each being Rank(M)×R0.
  2. Stratify (M×kΔk)×R0 as in the previous post.
  3. Quotient by Sk-action to get stratification of Rank(M)×R0.

Step 1


As stated in the proof of the Theorem, Rank(M)×R0 is open inside Rann(M)×R0, allowing us to make a stratification f:Rann(M)×R0A, where A is the poset
 
where the tail of an arrow is ordered lower than the head. The map f sends Rank(M)×R0 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×kΔk)×R0Bk, 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 vv iff dH(v,v)=1 and dH(v,0)<dH(v,0), for dH the Hamming distance. Let UvBk denote the upset based at v, that is, all elements vBk with vv.

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×kΔk)×R0 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 UX be open and φ:XAR0 continuous, with |A|<. Then
xU{x}×(φ(x),)  X×(z,)
is open, for any zz:=minxU{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 ψ1(U×(,0)) is open in X×(z,). For any xU and t=φ(x), we have φ(x)t=0. For any xU and t, we have φ(x)t. It is immediate that all other t(φ(x),) give φ(x)t(,0). Hence ψ1(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×kΔk and φ(P)=maxij{d(Pi,Pj)}, which is continuous, gives that g1k(U111)M×kΔk is open. This also works to show that g1k(Uv)g1k(Uv) is open, for any vv, 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×kΔk)×R0 by permuting the order of elements in the first factor. That is, for σSk, we have
σ(P={P1,,Pk},t)=({Pσ(1),,Pσ(k)},t).
Note that ((M×kΔk)×R0)/Sk=Rank(M)×R0.

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 vBk 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 vv, vv. Then the elements of Bk are in bijection with the elements of Γk (given by vGv), so we have Bk/Sk=Bk. Recall that vv in Bk iff adding an edge to Gv gives Gv. In Bk, this becomes a partial order on equivalence classes [w]={vBk : σGv=Gw for some σSk}. We write [w][w] iff there is a collection of pairs {(v1,v1),,(v,v)} such that vivi for all i, and {v1,,v}=[w] and {v1,,v}=[w] (there may be repetition among the vi or vi).

By the universal property of the quotient, there is a unique map hk:Rank(M)×R0Bk that makes the following diagram commute.
This will be our stratifying map. To check that hk is continuous take UBk open. As π is the projection under a group action, it is an open map, so π1(U)Bk is open. Since gk is continuous in the upset topology, g1k(π1(U)) is open. Again, Sk is the projection under a group action, so (Sk)(g1k(π1(U))) is open, giving continuity of hk.

No comments:

Post a Comment