Fix a manifold M along with an embedding of M into RN and set X=Ran(M)×R⩾. The goal of this post is to show that every (P,t)\in X has an open neighborhood that contains no points of the type (Q,d(Q_i,Q_j)), for some i\neq j. The collection of all such elements of X is called the singularity set of X, as the Vietoris-Rips complex at Q with such a radius changes at such elements.
Following Lurie, given a collection of open sets \{U_i\}_{i=1}^k in M, set
\Ran(\{U_i\}_{i=1}^k) = \left\{P\in \Ran(M)\ :\ P\subset \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i\right\}.
The topology on \Ran(M) is the smallest topology for which every \Ran(\{U_i\}_{i=1}^k) is open, for any \{U_i\}_{i=1}^k, for any k. The topology on the product X is the product topology.
Remark: Note that the Ran space \Ran(M) by itself can be split up into the pieces \Ran^k(M), with "singularities" viewed as when a point splits into two (or more) points, or two (or more) combine into one. Then every element of \Ran(M) is on the edge of the singularity set, as any neighborhood of a single point on the manifold contains two points on the manifold.
Fix (P,t)\in X not in the singularity set of X, with P=(P_1,\dots,P_k), for 1\leqslant k\leqslant n. Set
\mu = \min\left\{t,\min_{1\leqslant i<j\leqslant k}\left\{|t-d(P_i,P_j)|\right\}\right\},
with distance d being Euclidean distance in \R^N. The quantity \mu should be thought of as the upper bound on how "far" we may move from (P,t) without hitting the singularity set.
Proposition: Let (P,t) be as above and t,\alpha,\beta>0 such that \alpha+\beta=\mu. Then
U=\Ran\left(\{B(P_i,\alpha/2)\}_{i=1}^k\right) \times \left(t-\beta,t+\beta\right)
is an open neighborhood of (P,t) in X and does not contain any points of the singularity set of X.
If t=0, then having [0,\beta) as the second component of U, with \alpha+\beta=\min_{i,j}d(P_i,P_j) works as the open neighborhood of (P,t). The balls B(x,r) are N-dimensional in \R^N. The proof is mostly applications of the triangle inequality.
Proof: By construction we have that U is open in X and that it contains (P,t). For (Q,s)\in U any other element, we have three cases. We will show that the distance between any two Q_a,Q_b\in Q is never s. Fix distinct indices \ell,m\in \{1,\dots,k\}.
References: Lurie (Higher Algebra, Section 5.5.1)
Following Lurie, given a collection of open sets \{U_i\}_{i=1}^k in M, set
\Ran(\{U_i\}_{i=1}^k) = \left\{P\in \Ran(M)\ :\ P\subset \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i\right\}.
The topology on \Ran(M) is the smallest topology for which every \Ran(\{U_i\}_{i=1}^k) is open, for any \{U_i\}_{i=1}^k, for any k. The topology on the product X is the product topology.
Remark: Note that the Ran space \Ran(M) by itself can be split up into the pieces \Ran^k(M), with "singularities" viewed as when a point splits into two (or more) points, or two (or more) combine into one. Then every element of \Ran(M) is on the edge of the singularity set, as any neighborhood of a single point on the manifold contains two points on the manifold.
Fix (P,t)\in X not in the singularity set of X, with P=(P_1,\dots,P_k), for 1\leqslant k\leqslant n. Set
\mu = \min\left\{t,\min_{1\leqslant i<j\leqslant k}\left\{|t-d(P_i,P_j)|\right\}\right\},
with distance d being Euclidean distance in \R^N. The quantity \mu should be thought of as the upper bound on how "far" we may move from (P,t) without hitting the singularity set.
Proposition: Let (P,t) be as above and t,\alpha,\beta>0 such that \alpha+\beta=\mu. Then
U=\Ran\left(\{B(P_i,\alpha/2)\}_{i=1}^k\right) \times \left(t-\beta,t+\beta\right)
is an open neighborhood of (P,t) in X and does not contain any points of the singularity set of X.
If t=0, then having [0,\beta) as the second component of U, with \alpha+\beta=\min_{i,j}d(P_i,P_j) works as the open neighborhood of (P,t). The balls B(x,r) are N-dimensional in \R^N. The proof is mostly applications of the triangle inequality.
Proof: By construction we have that U is open in X and that it contains (P,t). For (Q,s)\in U any other element, we have three cases. We will show that the distance between any two Q_a,Q_b\in Q is never s. Fix distinct indices \ell,m\in \{1,\dots,k\}.
- Case 1: Q_a,Q_b\in B(P_\ell,\alpha/2). The situation looks as in the diagram below. Observe that d(Q_a,Q_b)\leqslant d((Q_a,P_\ell)+d(Q_b,P_\ell) <\alpha = \mu-\beta \leqslant t-\beta. Hence d(Q_a,Q_b)<s.
- Case 2: Q_a\in B(P_\ell,\alpha/2), Q_b\in B(P_m,\alpha/2), d(P_\ell,P_m)>t. The situation looks as in the diagram below. Observe that d(P_\ell,P_m)\leqslant d(P_\ell,Q_b)+d(P_m,Q_b)\leqslant d(P_\ell,Q_a)+d(Q_a,Q_b)+d(P_m,Q_b) < \alpha+d(Q_a,Q_b). Since d(P_\ell,P_m)>t, the definition of \mu gives us that \mu \leqslant d(P_\ell,P_m)-t, so combining this with the previous inequality, we get d(Q_a,Q_b) > d(P_\ell,P_m)-\alpha\geqslant \mu+t-(\mu-\beta)=t+\beta. Hence d(Q_a,Q_b)>s.
- Case 3: Q_a\in B(P_\ell,\alpha/2), Q_b\in B(P_m,\alpha/2), d(P_\ell,P_m)<t. The situation looks as in the diagram below. Observe that d(Q_a,Q_b)\leqslant d(P_m,Q_b) + d(P_m,Q_a) \leqslant d(P_\ell,Q_a)+d(P_\ell,P_m)+d(P_m,Q_a)<\alpha+d(P_\ell,P_m). Since d(P_\ell,P_m)<t, the definition of \mu gives us that \mu\leqslant t-d(P_\ell,P_m), so combining this with the previous inequality, we get d(Q_a,Q_b)<\mu-\beta+t-\mu = t-\beta. Hence d(Q_a,Q_b)<s. \square
References: Lurie (Higher Algebra, Section 5.5.1)
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