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## Euler characteristic as homotopy cardinality

Let $X$ be a finite CW complex with $c_0$ vertices, $c_1$ edges, and in general $c_i$ different $i$-cells. The Euler characteristic

$\displaystyle \chi(X) = \sum_{i \ge 0} (-1)^i c_i$

is a fundamental invariant of $X$, and the observation that it is homotopy invariant is the appropriate generalization of Euler’s formula $V - E + F = 2 = \chi(S^2)$ for a convex polyhedron. But where exactly does this expression come from? The modern story involves the homology groups $H_i(X, \mathbb{Q})$, but actually one can work on a more intuitive level characterized by the following slogan:

The Euler characteristic is a homotopy-invariant generalization of cardinality.

More precisely, the above expression for Euler characteristic can be deduced from three simple axioms:

1. Cardinality: $\chi(\text{pt}) = 1$.
2. Homotopy invariance: If $X \sim Y$, then $\chi(X) = \chi(Y)$.
3. Inclusion-exclusion: Suppose $X$ is the union of two subcomplexes $A, B$ whose intersection $A \cap B$ is a subcomplex of both $A$ and $B$. Then $\chi(X) = \chi(A) + \chi(B) - \chi(A \cap B)$.

Of course, this isn’t enough to conclude that there actually exists an invariant with these properties. Nevertheless, it’s enough to motivate the search for a proof that such an invariant exists.

Some examples

First, a word about the type of spaces we’re dealing with. It is too restrictive to work with simplicial complexes, since the axiom that two simplices intersect in a face is quite strong; for example, two points joined by two edges is not a simplicial complex, although the above result applies just fine to it. The ideas we’re describing apply to a generalization of simplicial complexes in which two simplices are allowed to intersect in a union of faces. For practical purposes, we’re dealing with spaces that can be constructed from a finite number of simplices by gluing along faces (or which are homotopy equivalent to such things), and we’ll refer to the interior of an $i$-simplex as an $i$-cell. For example, a solid tetrahedron is a $3$-simplex and consists of one $3$-cell, four $2$-cells, six $1$-cells, and four $0$-cells.

As a sanity check, the inclusion-exclusion axiom implies that $\chi(X \sqcup Y) = \chi(X) + \chi(Y)$, so in particular the Euler characteristic of a finite set of points is just its cardinality.

The interval $I = [0, 1]$ is contractible, hence $\chi(I) = 1 = 2 - 1$. Another way to see this is that the interval can be written as the union of two intervals whose intersection is a point, so $\chi(I) = 2 \chi(I) - 1$, hence $\chi(I) = 1$. Note that $I$ can be written as a collection of $n+1$ points joined by $n$ edges, and $(n+1) - n = 1$.

The circle $S^1$ can be written as the union of two intervals whose intersection consists of two points, hence $\chi(S^1) = 2 - 2 = 0$. If you believe that the Euler characteristic exists, it follows that $S^1$ is not contractible. Note that $S^1$ can be written as a collection of $n$ points joined by $n$ edges, and $n - n = 0$.

Now let $G$ be a connected finite $1$-complex (graph) with $c_0$ vertices and $c_1$ edges. Any such graph has a spanning tree, and contracting along that spanning tree gives a wedge sum of $c_1 - c_0 + 1$ circles. This wedge sum in turn can be written as $A \cup B$ where $A$ consists of an interval taken from each circle, $B$ is the closure of the complement of $A$, which is contractible, and $A \cap B$ consists of $2(c_1 - c_0 + 1)$ points. It follows that

$\chi(G) = (c_1 - c_0 + 1) + 1 - 2(c_1 - c_0 + 1) = c_0 - c_1$.

Intuitively, any edge can potentially cancel out a point via a homotopy, so it makes sense that edges count for $-1$. However, note that we cannot extend the Euler characteristic to spaces like the open interval $(0, 1)$ without losing homotopy invariance.

Two-dimensional examples

The sphere $S^2$ can be written as the union of two disks $A, B$ (the northern and southern hemispheres), which are contractible, whose intersection is a circle, hence $\chi(S^2) = 1 + 1 - 0 = 2$. Note that the sphere can be obtained by gluing a single point to a $2$-cell, so the $2$-cell appears to be contributing $+1$.

Why might $2$-cells do this? One intuition is that $2$-cells act as “higher edges” between edges. Take the space $X$ obtained from two points with two edges between them by gluing a $2$-cell in between the edges. This space is the disk, but by contracting the $2$-cell we get a deformation to an interval: in other words, the $2$-cell has been used to cancel out a $1$-cell, so it counts for minus what a $1$-cell counts for, or $-(-1) = 1$. In the lingo of higher category theory, if $1$-cells are morphisms between $0$-cells, then $2$-cells are $2$-morphisms between $1$-cells.

The torus $T^2$ can be written as the quotient of a square $I \times I$ obtained by identifying opposite edges in the same orientation. (Intuitively this corresponds to curling up the square once into a cylinder, then joining the two ends of the cylinder together.) We can therefore write it as $A \cup B$ where $A$ is a small disc cut out of the middle of the square, $A \cap B$ is $S^1$, and $B$ deformation retracts onto its $1$-skeleton (the complex obtained by only looking at the cells of dimension $1$ or lower). This turns out to be the wedge $S^1 \vee S^1$ of two circles, whence $\chi(B) = -1$ and $\chi(T^2) = 1 - 1 - 0 = 0$. Note that the torus has a CW-structure with $c_0 = 1, c_1 = 2, c_0 = 1$, and $1 - 2 + 1 = 0$.

Let $S_g$ denote the $g$-holed torus. By pinching off a part containing one hole, we can write it as $A \cup B$ where $A$ is a torus minus a disc, $A \cap B \cong S^1$, and $B$ is $S_{g-1}$ minus a disc. Inclusion-exclusion gives $\chi(B) = \chi(S_{g-1}) - 1$, and we know that $A \cong S^1 \vee S^1$ satisfies $\chi(A) = -1$, so it follows by inclusion-exclusion that $\chi(S_g) = \chi(S_{g-1}) - 2$. Together with the base case $\chi(S_1) = 0$, we conclude by induction that

$\chi(S_g) = 2 - 2g.$

It turns out that we can obtain $S_g$ by identifying certain pairs of sides in a $4g$-gon. It follows that it has a CW-structure with $c_0 = 1, c_1 = 2g, c_2 = 1$, and $\chi(S_g) = 1 - 2g + 1$ as expected.

The proof

Having built some intuition, we are now ready to dive into the proof. Our current intuition is that $i$-cells can at least potentially cancel out $(i-1)$-cells up to homotopy, hence if Euler characteristic behaves additively on cells, an $i$-cell should count for $(-1)^i$ by induction. To carry out this intuition we will work by induction and use inclusion-exclusion to remove all of the top-dimensional cells. Thus let $X$ be an $n$-dimensional finite complex with $c_i$ different $i$-cells. We want to remove a small ball $B^n$ from each $n$-cell; balls are contractible, so $\chi(B^n) = 1$. The intersection of this ball and the closure of its complement is the sphere $S^{n-1}$, and after removing a small ball we can retract each $n$-cell onto the lower-dimensional cells it’s attached to, so it follows that, if $X_{(n-1)}$ denotes the $n-1$-skeleton of $X$, we have

$\displaystyle \chi(X) = \chi(X_{(n-1)}) + c_n (1 - \chi(S^{n-1}))$

so it remains to compute $\chi(S^n)$. Again, by looking at two hemispheres, we can write $S^n$ as $A \cup B$ where $A, B$ are both homeomorphic to $B^n$ and $A \cap B$ is the equator, which is homeomorphic to $S^{n-1}$. Explicitly, if $S^n = \{ (x_1, ... x_{n+1}) \in \mathbb{R}^{n+1} : \sum x_i^2 = 0 \}$, then we can take $A = \{ x : x_1 \ge 0 \}, B = \{ x : x_1 \le 0 \}, C = \{ x : x_1 = 0 \}$. It follows that

$\displaystyle \chi(S^n) = 2 - \chi(S^{n-1})$

and since $\chi(S^0) = 2$ it follows by induction that $\chi(S^n) = 1 + (-1)^n$, which gives

$\displaystyle \chi(X) = \chi(X_{(n-1)}) + (-1)^n c_n$

and the result follows by induction.

Existence

So how do we prove that the Euler characteristic exists? Given the existing discussion, a standard proof proceeds by showing that the Euler characteristic is equal to the alternating sum

$\displaystyle \sum_{i \ge 0} (-1)^i \dim H_i(X, \mathbb{Q})$

where the $H_i$ are, depending on your preferences, the singular or simplicial homology groups, which are homotopy-invariant. This is usually done with a mildly unenlightening computation, but given what we know we can cheat: the above alternating sum satisfies inclusion-exclusion because of the Mayer-Vietoris sequence (and the fact that the Euler characteristic of an exact sequence vanishes), and it obviously satisfies the cardinality axiom, so by homotopy invariance it must be the Euler characteristic.

Of course there is much more to say on this subject. The Euler characteristic figures in many fundamental theorems of topology and geometry, including the Poincare-Hopf theorem and the Gauss-Bonnet theorem. I hope to say more about the Euler characteristic in future posts.

### 10 Responses

1. […] Associated to any essentially finite groupoid is a rational number, its groupoid cardinality , which is uniquely determined by the following four properties, analogous to the properties uniquely specifying Euler characteristic: […]

2. My understanding of the history is that Steve Schanuel was the first to realize and really push the idea that “the Euler characteristic is a homotopy-invariant generalization of cardinality”. Dan Klain and Gian-Carlo Rota credit him with this in the introduction to their book Introduction to Geometric Probability, and I haven’t seen evidence to the contrary.

Schanuel hasn’t written about this idea extensively, as far as I’m aware, but you’ll find it developed in the following nice papers:

Stephen H. Schanuel, Negative sets have Euler characteristic and dimension. Lecture Notes in Mathematics 1488, Springer, 1991, pp. 379-385.

Stephen H. Schanuel, What is the length of a potato? An introduction to geometric measure theory. Lecture Notes in Mathematics 1174, Springer, 1986, pp. 118-126.

The first paper is the more relevant. It also covers non-compact spaces, perhaps addressing some of Theo’s concerns.

• Thanks, Tom. Awhile back I read a few papers on this subject (including some of yours!) starting from the relevant TWFs; it’s quite interesting. I was going to write a follow-up post about Euler characteristic of chain complexes (where there are nice easy ways to show that the Euler characteristic is well-defined and homotopy invariant) but got distracted…

3. […] Qiaochu Yuan: Euler characteristic as homotopy cardinality […]

4. You got this up pretty sharpish after finishing exams! How did they go?

• Could’ve been better. I’m just glad they’re over and I can think about other things now.

5. Did you write this post because of a question on math.stackexchange?

• No. I was just trying to convince myself that the signs in the definition of the Euler characteristic are there for a good reason.

6. I think this discussion “gets” Euler characteristic on the nose for compact spaces. But I’ve never liked this definition of Euler characteristic for non-compact spaces. In particular, there are many situations where you want a topological invariant that assigns $-1$ to the open interval. One way to build such an invariant is to ask for the dimension of the compactly supported cohomology. Note that the request that the open interval have weight $-1$ means in particular that any such invariant is not a homotopy invariant.

Here’s one place in my work where this type of request comes up. Suppose you have a finite-dimensional vector space $X$ with Lebesgue measure $dx$. Then you can ask for Gaussian integrals of the form $\int \exp( -A x^2/2) dx$, where $A$ is a positive-definite symmetric square matrix (and so eats the exterior tensor square of the vector $x$). It has some formula with determinants. Now consider rescaling $A \to \lambda A$ for positive $\lambda$. The value of the integral changes by some multiplicative factor, and this factor is precisely $\lambda^{-\dim X / 2}$ — in particular, knowing it is the same as knowing the dimension of $X$.

In quantum field theory, you often want to write down similar expressions, where $X$ is some space of functions on a manifold $M$, usually with prescribed (asymptotic) boundary values at the ends of $M$. Of course, in infinite dimensions it’s very rare to be able to define Gaussian integrals generally. Rather, what you do is guess an answer and check that it satisfies whatever functional equations it would satisfy if it came from an integral (i.e. check that you have a “differentiate inside the integral” rule). What you find out is that quite often, the role of $\dim X$ in scaling laws (etc.) is played by the compactly-supported Euler characteristic of $M$. (Or, really, one works with manifolds with boundary, functions with prescribed boundary values, and the correct Euler characteristic is the characteristic of the closed thing minus the characteristic of the boundary, which is to say the compactly-supported characteristic of the interior.)

This is particularly appropriate if you are going to have a cutting-and-gluing formula in quantum field theory. One of the reasons that the path-integral approach is attractive is that, if it existed, it would satisfy a “Fubini”-style theorem for gluings of manifolds along boundaries. For such a Fubini theorem to hold, a necessary condition is that the scaling rules satisfy this “Euler-characteristic” formula.

• “Characteristic of the closed thing minus the characteristic of the boundary” definitely seems natural, and I was annoyed that I couldn’t directly refer to it in my post because I wasn’t sure what properties it had. The fact that such a thing isn’t homotopy invariant suggests to me that homotopy invariance and inclusion-exclusion together are just stand-ins for some functoriality condition, but I haven’t at all sorted out my thoughts about this stuff.