Does $\dim(\varnothing) <0 $ make any sort of sense?

Question

I am currently exploring the world of fractals in Fractal Geometry - Mathematical Foundations and Applications (Falconer).

In Chapter $8$ (Intersection of fractals), it says:

In $\mathbb{R}^n$, if smooth manifolds $E$ and $F$ intersect at all, then in general they intersect in a sub-manifold of dimension $\dim E +\dim F-n\;$ unless this number is negative, in which case they typically do not intersect.

For example, in $\mathbb{R}^2$, let $E$ be a point and $F$ a line that do not intersect (i.e., $E\cap F =\varnothing$). Applying the above equation yields $$ \dim \varnothing = 0+1 -2 = -1<0 $$

This is consistent as we assumed $E$ and $F$ do not intersect. But does $\dim \varnothing <0$ make any sort of sense? Is there a way to interpret this, or is it complete non sense coming from the fact $E$ and $F$ do not intersect and therefore the above equation shouldn't even be considered?

Answer 1

I think the part on the statement that is important to understand is the in general. I am pretty sure that in this case this means that if two manifold intersect, they could do so non-transversally (in which case the dimension of the intersction would be greater than expected), but if you look at the "space of perturbations" of the manifolds (i.e. you move them around just a little bit), almost all of them (this is the "in general") will give you a transversal intersection with the stated dimension for the intersection.

Now for the question about the empty set, I have already seen various places where $\dim\emptyset$ is set formally to $-1$, mainly for reasons of coherence with various results. For example (if I remember correctly), in algebraic topology, $\emptyset$ is sometimes considered as the $(-1)$-dimensional sphere.

Answer 2

The quoted passage is being prosaic, rather than precise, and I think you're inferring the wrong formalization of it. It could be rephrased something like this:

Let $e$, $f$, and $n$ be nonnegative integers.

1. If $e+f≥n$, then if $E$ and $F$ are smooth submanifolds of $\mathbb{R}^n$ of dimension $e$ and $f$ respectively, and if $E$ and $F$ intersect transversely, then $\dim (E\cap F) = e+f-n$.

2. If $e+f<n$, then if $E$ and $F$ are smooth submanifolds of $\mathbb{R}^n$ of dimension $e$ and $f$ respectively, which are in general position, then $E \cap F = \varnothing$.

The devil is in the details (“transversely”...“general position”...), but the important thing is that $\dim(E\cap F)$ is referred to only when $e+f\geq n$, and $E$ and $F$ are assumed to intersect in the first place. So your second statement that the equation shouldn't be considered in the case of an empty intersection is the right way to think about it.