So far the notes below are both redundant and very incomplete. I am searching for a good compromise between suggestive imagery and rigor. I summarize the conjecture here and consider specific cases. Better justification is found below. The horizontal rules are discontinuities in the development.

This analysis suggests linked branes in higher dimensions. What’s a brane? An m-brane and n-brane may be linked in a m+n+1 space. The ‘sphere’ in question is S^m+n whose content is kπ^k/2/(k/2)! where k=m+n+1. When m=1 and n=0 then we have the notions of winding number. S⁰ consists of two points and our integral is some multiple of 2π which says if the loop separates one point of S⁰ from the other.

Shifting a bit towards more careful math we come to the following description. An m sphere and n sphere are each embedded in a k = m+n+1 dimensional vector space. Let P be a smooth embedding of S^m, and Q be a smooth embedding of Sⁿ; both in ℝ^m+n+1. R(P) is the range of P and is a manifold in ℝ^m+n+1. Ditto R(Q). We refer to R(P) and R(Q) as the branes collectively. We will be concerned with the metric properties of and measures on the branes. We describe differential elements of each brane as exterior differential forms, dp for R(P) and dq for R(Q). These ‘differential forms’ are differential members of the exterior algebra where the ‘∧’ operator is defined. These geometric ideas were introduced by Grassmann in about 1850. The information captured by dp or dq is a magnitude and complete orientation of the tangent subspace at the point on the brane. ∫da = 0 when the integral is over a brane (oriented manifold) with no boundary. The integrand here denotes values from a graded Grassmann algebra.

Let x and y be points in the respective branes. (y−x)/|y−x| is a unit vector in ℝ^m+n+1. The space of unit vectors amounts to S^m+n. Indeed we here take members of S^m+n to be the unit vectors of ℝ^m+n+1. Depending on the embeddings we have thus a map from S^m×Sⁿ to S^m+n. We want to show that each part of S^m+n is in the map’s range net w times where w is the relative winding number of the two branes.

More relevant is the map from R(P) × R(Q) to S^m+n and where we must measure the elements of each brane. We must also show that this map’s Jacobian is ±1 each of these times. Given two elements, dp and dq, from the respective branes, and r which is their relative locations within ℝ^m+n+1, we consider content of the convex hull of dp and dq which is dp∧dq∧r. If dp and dq are both simplexes, then the hull is too.

We want the integral of dp∧dq∧r/|r|^k where r is the vector between the respective elements on the branes. The integral extends over all of R(P) × R(Q). The exponent in the denominator is necessary to make the integral invariant under similarity transformations of the branes. The hypothesis is that the integral is (the content of S^m+n) times the winding number of the embeddings. The winding number is 0 for unlinked spheres and ±1 for simple links.

The most precise definition of winding number that we offer here is highly analogous to the road metaphor introduced here. For each member w of S^m+n there are two complementary projections of ℝ^m+n+1:

• The projection T onto the m+n dimensional subspace orthogonal to w,
• The projection T' onto w.

T and T' are linear transformations and T+T' = I, the identity transformation. T(R(P)) ∪ T(R(Q)) is analogous to this 2D image. T(R(P)) ∩ T(R(Q)) is the set of ‘crossovers’ where m+n dimensions do not suffice for projections of linked branes to avoid each other. If (p, q) ∊ R(P)×R(Q) and T(p) = T(q) then T'(p) ≠ T'(q) and the larger of p⋅w and q⋅w determines which of p or q goes over the other. For an element p of a brane let U(p) be an oriented exterior value in the tangent plane of the brane at p. The winding number is the sum of the sign of U(p)∧U(q)∧w for that finite set of (p, q) ∊ R(P)×R(Q) where T(p) = T(q) and p⋅w > q⋅w. (This needs justification!!)

This sum does not vary as w ∊ S^m+n varies.

The above description is beginning to suffer from lack of short names for the various construct. We switch styles here and name everything as we recapitulate the above.

For m ≥ 0 we define, S^m = {<x₀, ... x_m> ∊ ℝ^m+1 | ∑ x_i² = 1}. We are concerned with the topology, metric and measure on S^m.

For some fixed m ≥ 0 and n ≥ 0 we have embeddings E and F
E: S^m ↪ ℝ^m+n+1 and F: Sⁿ ↪ ℝ^m+n+1
The ranges of E and F are disjoint. An ‘embedding’ is merely a (continuous) map—we use the word for its connotations. We do not need the injective hypothesis which normally accompanies the embedding notion.

If <x, y> ∊ S^m×Sⁿ then let G(x, y) = (E(y) − F(x))/|E(y) − F(x)| ∊ S^m+n and thus
G: S^m×Sⁿ → S^m+n.

We consider the projection P of the image (range) of E and F onto tangent planes of S^m+n taken as a subspace of ℝ^m+n+1. Whereas E and F have disjoint ranges, the projection of their ranges will collide at certain points in the tangent plane. If the tangent plane is at point z of S^m+n and <x, y>, <x', y'> ∊ S^m×Sⁿ and P(

We have assumed that out branes are images of Sⁿ but any manifold without border will do—it need not even be connected. S⁰ is indeed not connected as studied here. Our proofs, such as they are, seem to require smoothness.

Some pretty good numerical corroboration