I am not sure what a brane is yet for my purposes. It is at least a continuous map of a compact boundaryless oriented topological manifold into a finite dimensional real vector space. This is enough for a Grassmann algebra. We may need a vector space with a positive definite quadratic form which defines a norm. It is permissible for the map f to collide with itself: f(x)=f(y) and x ≠ y. The manifold need not be connected but it and its mapped image should be bounded, which follows, I think, from previous assumptions. Neither need the manifold be differentiable. S1 is not simply connected and S0 is not even connected. Genus seems irrelevant. The case at hand is an m-brane and n-brane each mapped into an m+n+1 dimensional vector space. Branes are allowed to collide with them-self but not with each other for my conjecture.

The proof imagery seems to require a differentiable map from m-space to k-space, but I conjecture that this limitation is unneeded. The Jordan curve theorem requires only a continuous map without collisions. In so far as such Jordan curves are the limit of smooth curves, and integrals are defined over the smooth curves, and the integrals approach the same limit for each such limiting sequence of smooth curves, then this limit of integrals can be taken as the integral of the limiting unsmooth curve. I think that these hypotheses are true for our integrals. Integrals defining the length of the smooth curves do not converge in general, but I believe that all of the integrals here do converge. I should examine the proof of the general Jordan curve theorem but our conjectures are not generalizations of the Jordan curve theorem and do not require non-colliding maps.

The ranges of the above maps do not suffice for the conjecture, for the same reason that winding numbers relate maps of an oriented circle into the complex plane, with points in that plane. The range alone is a point set which lacks an orientation. When the map collides with itself it is possible to assign orientations to portions of the range in many different ways and arrive at inconsistent winding numbers.

Here is a notion of fine mesh for a topological manifold:

A sequence of meshes in an n dimensional topological manifold. They ‘become fine’ if for every covering of the manifold, there is a member of the sequence such that every simplex of the mesh is contained in some particular member of the covering.

This notion fails for defining area of an embedded manifold, but does not fail for our integrals, I think.