Proposition

Proposition 2 ([Topological comparison]). In the contour coupling, the projections π_m may be chosen so that the following conclusions hold on the sphere and on every buffered compact part of the plane. Each conclusion is eventual for each …

Proposition 2 ([Topological comparison]). In the contour coupling, the projections πm may be chosen so that the following conclusions hold on the sphere and on every buffered compact part of the plane. Each conclusion is eventual for each fixed choice of margins. If the projections of two primal vertices approach each other, they can be joined by primal paths with arbitrarily small projected diameter. This is uniform over all lifts in a fixed compact region. The same statement holds on the dual graph and for surface paths. In particular fixed tubes with clearance contain paths following them on either graph. All lifts of a sufficiently small closed disk or other compact well can be included in a connected set over any fixed open enlargement. A loop projecting into a small disk can be filled over a slightly larger disk, provided that both have the prescribed clearance. The projection has degree +1 or -1 in a compact spherical completion. Robustly transverse primal and dual tube crossings intersect at an edge and its dual. Projected circuits with nonzero winding difference about two separated wells separate all the corresponding lifted fibers. Integer winding differences about two wells agree upstairs and downstairs up to the common degree sign, when loops and wells have fixed separation. These statements retain repeated edge and face incidences.

Prove 1Disprove 0Cited as proposition 4Chain graphAdd Your Proof

Add Your Proof1

Prove
Proposition 1 ([Sewing and continuum background]). The following couplings and…Theorem 1.1 ([Linear clock and conditional laws]). For every finite T>0 ,…Proposition
The construction and the first three assertions are the construction preceding Lemma ``Uniform topological joining'', that lemma, Lemma ``Exterior connection and small-loop filling'', Proposition ``Degree and robust planar topology'', and Lemma ``Separation of entire fibers'' in . We explain their applicability to both graphs and to the plane. The hypotheses are uniform contour convergence, vanishing projected cell mesh, and approximation of the limiting fiber generators by discrete links with small projected connecting paths. For a Brownian excursion above a level, use last and first crossings of nearby levels on the two sides of an interior time. Their discrete analogues approximate the chord. At a triple, apply this on the two excursions on either side of its middle minimum. The fiber description above therefore supplies every generator required for topological joining. Interchanging the coordinates gives the same argument for the dual. Applying planar separation to the two embedded graph paths gives the mixed crossing assertion: a primal path and a dual path can meet only through a primal edge and its dual, with the usual corner subdivision making the intersection literal on the surface. For a bounded plane region, choose a window containing all its visits and its fixed additional star layers. Both contour coordinates make strictly lower visits on both sides of a sufficiently large window. Lemma ``Protected capping of a bounded window'' in then supplies a spherical completion preserving all these incidences. Its continuum completion is absolutely continuous with respect to a Brownian excursion, which supplies the sphere quotient needed for the preceding topological results. For the integer assertion, connect all lifts of each well away from the tested loop, using the first two assertions. A two-chain bounded by that loop has constant multiplicity along each such connected fiber. Projection multiplies the difference of its multiplicities at the two fibers by π_m=±1 . This is exactly the equality of winding differences. Equivalently, one can use relative homology after a simplicial perturbation inside the fixed margins.
Show More →