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June 18, 2024

Magnitude Homology Equivalence

Posted by Tom Leinster

My brilliant and wonderful PhD student Adrián Doña Mateo and I have just arXived a new paper:

Adrián Doña Mateo and Tom Leinster. Magnitude homology equivalence of Euclidean sets. arXiv:2406.11722, 2024.

I’ve given talks on this work before, but I’m delighted it’s now in print.

Our paper tackles the question:

When do two metric spaces have the same magnitude homology?

We give an explicit, concrete, geometric answer for closed subsets of N\mathbb{R}^N:

Exactly when their cores are isometric.

What’s a “core”? Let me explain…

To tell you what a core is, I need to take a run-up.

Two distinct points xx and yy of a metric space are adjacent if there’s no other point pp between them (that is, satisfying d(x,y)=d(x,p)+d(p,y)d(x, y) = d(x, p) + d(p, y)). For instance, if you view a graph as a metric space in which the distance between vertices is the number of edges in a shortest path between them, then two vertices are adjacent when they’re joined by an edge.

The inner boundary ρX\rho X of a metric space XX is the set of all points that are adjacent to something in XX. For instance, the inner boundary of a closed annulus is its inner bounding circle. Here’s the inner boundary of another closed subset of the plane, shown in thick blue:

This picture should help you believe the following theorem about convex hulls, which I’ll write as convconv:

Theorem   For all closed X NX \subseteq \mathbb{R}^N,

conv(X)=Xconv(ρX).conv(X) = X \cup conv(\rho X).

(part of our Proposition 4.5). In other words, any point in the convex hull of XX that isn’t in XX itself must be expressible as a convex combination of points in its inner boundary. I mention this result just as motivation for the concept of inner boundary: it tells us something that a convex geometer who’s never heard of magnitude homology might care about.

The core of X NX \subseteq \mathbb{R}^N is defined as

core(X)=conv(ρX)¯X core(X) = \overline{conv(\rho X)} \cap X

— the intersection of XX with the closed convex hull of its inner boundary. For instance, here’s the core of the set I just showed you, shaded:

It turns out that the core construction is idempotent: once you’ve shrunk a set down to its core, taking the core again doesn’t shrink it any further.

Here’s an important result on cores (a consequence of our Proposition 5.7):

Theorem   Let XX be a nonconvex closed subset of N\mathbb{R}^N. Then every point of XX has a unique closest point in core(X)core(X).

It’s important because it means that the inclusion core(X)Xcore(X) \hookrightarrow X has a distance-decreasing retraction π\pi, defined by taking π(x)\pi(x) to be the closest point of core(X)core(X) to XX. And having these maps core(X)Xcore(X) \leftrightarrows X is key to proving the main theorem on magnitude homology equivalence.

(The weird hypothesis “nonconvex” has to be there! A convex set has empty inner boundary, so its core is empty too, which means the theorem will fail for trivial reasons.)

So now we know what the core is.

But what do I mean by saying that two spaces XX and YY have the “same” magnitude homology? As for any homology theory of any kind of object, there are at least three possible interpretations:

  • We could just ask that the groups H n(X)H_n(X) and H n(Y)H_n(Y) are isomorphic for each nn. As with other homology theories, this seems to be too loose a relationship to be really interesting.

  • Or we could consider quasi-isomorphism, the equivalence relation on spaces generated by declaring XX and YY to be equivalent if there exists a map XYX \to Y inducing an isomorphism in homology.

  • More demandingly still, we could ask that there exist back-and-forth maps XYX \leftrightarrows Y that are mutually inverse in homology.

We take the third option, defining metric spaces XX and YY to be magnitude homology equivalent if there exist maps XYX \leftrightarrows Y whose induced maps H n(X)H n(Y)H_n(X) \leftrightarrows H_n(Y) are mutually inverse for all n1n \geq 1.

Here “map” means a map that is 1-Lipschitz, or short, or a contraction or distance-decreasing in the non-strict sense. When you view metric spaces as enriched categories, these are the enriched functors. And, incidentally, we exclude the case n=0n = 0 because if not, that would make XX and YY isometric and the whole thing would become trivial.

Main theorem   Let XX and YY be nonempty closed subsets of Euclidean space. Then XX and YY are magnitude homology equivalent if and only if their cores are isometric.

This is part — the most important part — of our Theorem 9.1, which also gives several other equivalent conditions.

I won’t say anything about the proof except that it makes crucial use of a theorem of Kaneta and Yoshinaga, which gives an explicit formula for the magnitude homology of subsets of Euclidean space.

So what does this theorem do for us?

For a start, it gives lots of examples of pairs of spaces that are magnitude homology equivalent. For instance, every closed set is magnitude homology equivalent to its core, so in the example

that we saw earlier, the whole grey set is magnitude homology equivalent to the shaded blue core. Or, all three of these closed subsets of the plane are magnitude homology equivalent:

(the last being the complement in 2\mathbb{R}^2 of the union of the disc and the square).

The main theorem also allows us to tell when two spaces are not magnitude homology equivalent. For instance, the set

is not magnitude homology equivalent to any of the three in the previous figure, simply because their cores are obviously not isometric. It’s an easy test to apply.


I just want to mention one more thing. In order to get to our main theorem, we needed to prove a strengthening for closed sets of the classical Carathéodory theorem on convex sets, due to the extraordinary Greek mathematician Constantin Carathéodory. His famous theorem (well, one of them) is this:

Carathéodory’s theorem   Let XX be a subset of N\mathbb{R}^N and aconv(X)a \in \conv(X). Then there exist n0n \geq 0 and affinely independent points x 0,,x nXx_0, \ldots, x_n \in X such that

aconv{x 0,,x n}.a \in conv\{x_0, \ldots, x_n\}.

Since an affinely independent set in N\mathbb{R}^N can have at most N+1N + 1 elements, it follows that any point in the convex hull of XX is in the convex hull of some (N+1)(N + 1)-element subset of XX. That’s actually the version of Carathéodory’s theorem that people most often use. But it’s the full statement that’s most relevant to us here.

Now here’s a stronger theorem for closed sets, which appears as Theorem 3.1 of our paper:

Closed Carathéodory theorem   Let XX be a closed subset of N\mathbb{R}^N and aconv(X)a \in \conv(X). Then there exist n0n \geq 0 and affinely independent points x 0,,x nXx_0, \ldots, x_n \in X such that

aconv{x 0,,x n}a \in conv\{x_0, \ldots, x_n\}


conv{x 0,,x n}X={x 0,,x n}. conv\{x_0, \ldots, x_n\} \cap X = \{x_0, \ldots, x_n\}.

In other words, when XX is closed, you can find affinely independent points of XX whose convex hull not only contains the point aa (as in the classical Carathéodory theorem), but also, only intersects XX where it absolutely has to.

This theorem is absolutely classical in flavour and could easily have been proved by Carathéodory himself. I imagine it’s in the literature somewhere. But despite reading survey papers on Carathéodory-like theorems and asking around (including on Mathoverflow), we haven’t been able to find it. So we included a proof in our paper.

Posted at June 18, 2024 8:46 AM UTC

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