### 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 $\mathbb{R}^N$:

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 $x$ and $y$ of a metric space are **adjacent** if
there’s no other point $p$ between them (that is, satisfying $d(x, y) =
d(x, p) + d(p, y)$). 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** $\rho X$ of a metric space $X$ is the set of all
points that are adjacent to *something* in $X$. 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 $conv$:

TheoremFor all closed $X \subseteq \mathbb{R}^N$,$conv(X) = X \cup conv(\rho X).$

(part of our Proposition 4.5). In other words, any point in the convex hull of $X$ that isn’t in $X$ 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 $X \subseteq \mathbb{R}^N$ is defined as

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

— the intersection of $X$ 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):

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

It’s important because it means that the inclusion $core(X) \hookrightarrow X$ has a distance-decreasing retraction $\pi$, defined by taking $\pi(x)$ to be the closest point of $core(X)$ to $X$. And having these maps $core(X) \leftrightarrows X$ 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 $X$ and $Y$ 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 $H_n(X)$ and $H_n(Y)$ are isomorphic for each $n$. 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 $X$ and $Y$ to be equivalent if there exists a map $X \to Y$ inducing an isomorphism in homology.

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

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

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 $n = 0$ because if not, that would make $X$ and $Y$ isometric and the whole thing would become trivial.

Main theoremLet $X$ and $Y$ be nonempty closed subsets of Euclidean space. Then $X$ and $Y$ 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 $\mathbb{R}^2$ 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.

### Postscript

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 CarathÃ©odory theorem on convex sets, due to the extraordinary Greek mathematician Constantin CarathÃ©odory. His famous theorem (well, one of them) is this:

CarathÃ©odory’s theoremLet $X$ be a subset of $\mathbb{R}^N$ and $a \in \conv(X)$. Then there exist $n \geq 0$ and affinely independent points $x_0, \ldots, x_n \in X$ such that$a \in conv\{x_0, \ldots, x_n\}.$

Since an affinely independent set in $\mathbb{R}^N$ can have at most $N + 1$ elements, it follows that any point in the convex hull of $X$ is in the convex hull of some $(N + 1)$-element subset of $X$. That’s actually the version of CarathÃ©odory’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 CarathÃ©odory theoremLet $X$ be a closed subset of $\mathbb{R}^N$ and $a \in \conv(X)$. Then there exist $n \geq 0$ and affinely independent points $x_0, \ldots, x_n \in X$ such that$a \in conv\{x_0, \ldots, x_n\}$

and$conv\{x_0, \ldots, x_n\} \cap X = \{x_0, \ldots, x_n\}.$

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

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