The n-Category Café

First let me remind you about the fixed-point index in classical topology. Suppose $M$ is an $n$-dimensional manifold and $f\colon M \to M$ is a smooth map with isolated fixed points. For each fixed point $x_0\in M$, $f$ is close to the identity on a neighborhood of $x_0$, so that $v(x) = x - f(x)$ defines a nonvanishing vector field on that neighborhood. Restricting this vector field to an $(n-1)$-sphere around $x_0$ and normalizing it, we obtain an endomap of $S^{n-1}$; the fixed-point index of $x_0$ is the degree of this map. Finally, the fixed-point index of $f$ is the sum of those of all of its fixed points.

The geometry is perhaps most obvious in two dimensions. In the neighborhood of a fixed point, an endomap of a surface can do a number of things. (For each of these, you should draw a picture of the vector field and convince yourself that its degree of the induced endomap of $S^1$ is what I say it is.)

• It can stretch outwards in all directions. In this case the vector field $v$ points inward everywhere, and the index is 1.

• It can shrink inwards in all directions. In this case the vector field $v$ points outwards everywhere, and the index is again 1.

• It can rotate clockwise or counterclockwise around the fixed point. In this case the vector field is tangent to the sphere surrounding the fixed point, and the index is again 1.

• It can move steadily in one direction, only pausing briefly at the fixed point. In this case the vector field points constantly in that one direction, and the index is 0.

• It can shrink inwards in the left-right directions and stretch outwards in the up-down directions. In this case the vector field looks hyperbolic, and the index is -1.

You can also have fixed points of arbitrary integer index. If you’ve never done it, it’s a fun exercise to draw some fixed points of index 2, -2, 3, and -3.

For an example of a global endomap, consider the endomap of $S^2$ which “rotates the earth”, fixing the north and south poles. Both of these fixed points are rotational, having index 1, and so the total index of this endomap is 2.

It’s also worth thinking about one dimension. In the neighborhood of a fixed point, an endomap of a line can:

• stretch outwards in both directions; then $v$ points inward everywhere, and the index is -1.

• shrink inwards in both directions; then $v$ points outwards, and the index is 1.

• Move steadily in one direction with a pause; then $v$ points in that direction, and the index is 0.

Note that unlike in the 2-dimensional case, stretching and shrinking have different indices in one dimension. Moreover, it is impossible to have a single fixed point in one dimension with a degree other than 1, 0, or -1. This is because the 0-sphere $S^0$ is a bit degenerate (it consists of two points), and in particular it only has four endomaps: the identity (with degree 1), the switch map (with degree -1) and two constant maps (with degree 0).

Now amazingly (to me), the geometrically defined notion of fixed-point index turns out to be a homotopy invariant. In other words, if $f$ and $g$ are homotopic endomaps, they have the same fixed-point index. Notice that $f$ and $g$ could have different numbers of fixed points, but when you add up these integers associated to each of their fixed points, you always get the same thing.

(This leads to a nice theorem: if the fixed-point index of $f$ is nonzero, then $f$ is not homotopic to any fixed-point-free map; i.e. you cannot deform $f$ to get rid of all of its fixed points. This is closely related to the hairy ball theorem, and if you find a way to calculate the fixed-point index of $f$ homologically, you get the Lefschetz fixed point theorem.)

In particular, this means we can define the fixed-point index of a map $f$ whose fixed points are not isolated, by homotoping it to a map $g$ whose fixed points are isolated and calculating the fixed-point index there; homotopy invariance means it doesn’t matter what $g$ we choose. A particularly interesting map whose fixed points are not isolated is the identity map; and the fixed-point index we calculate for it in this way gives the Euler characteristic of the manifold $M$.

Okay, great; but now we have a homotopy invariant—that is, an invariant of $\infty$-groupoids—we ought to be able to define it in terms of $\infty$-groupoids, without referring to manifolds, vector fields, and degrees of endomaps of $S^{n-1}$! There is an abstract-nonsense way to do this, but let’s look for a concrete hands-on way to do it.

To make things simple, let’s restrict to 1-groupoids. We can’t expect to deal with all 1-groupoids, only those which are “like finite-dimensional manifolds” in some sense. It turns out that the classical fixed-point index can be defined for any finite CW complex, and the 1-groupoids corresponding to finite CW-complexes are finitely generated free groupoids: those freely generated (as groupoids) by some finite quiver. For instance, the one-object groupoid $B\mathbb{Z}$ with $\mathbb{Z}$ as its isotropy group is generated by the quiver with one object and one endo-arrow. This 1-groupoid corresponds to the manifold $S^1$.

Thus, suppose $G$ is a finitely generated free groupoid, and $f\colon G \to G$ an endomap; how do we define the fixed-point index of $f$? There are several naive things you might try, so let me shoot them all down immediately so you can see that it’s a tricky problem.

• We could just count the number of objects of $G$ that are literally fixed by $f$. But this is obviously not invariant under natural isomorphism (i.e. homotopy) of $f$.

• We could count the number of connected components of $G$ that are fixed by $f$. This is invariant under homotopy, but doesn’t agree with the classical notion. For instance, the Euler characteristic of $S^1$ is zero, but in this way we would obtain a “fixed-point index” of 1 for the identity (indeed, for any endomap) of $B\mathbb{Z}$.

• We could count the number of pairs $(x,\gamma)$ where $x$ is an object of $G$ and $\gamma\colon x\cong f(x)$ is an isomorphism—that is, all the “homotopy fixed points”. However, for the identity map of $B\mathbb{Z}$ this would give us infinity.

In fact, it’s obvious that no “just count” definition is going to work, because the classical fixed-point index can be negative, and this can happen in the world of 1-dimensional CW complexes (which are what model finitely generated free 1-groupoids). For instance, the degree-2 self-map of $S^1$ has one fixed point with index -1 (check this for yourself), and a wedge of two circles has Euler characteristic -1.

The failure of these three ideas also means that the categorical content of the fixed-point index is not clear! At least, it’s not clear to me. It doesn’t seem to correspond to any ordinary categorical way of talking about “fixed points”.

Puzzle: Can you define a notion of “fixed point index” for an endomap of a finitely generated free groupoid which is

1. Invariant under natural isomorphism and equivalence of groupoids, and
2. Reproduces the classical fixed-point index for corresponding endomaps of 1-dimensional CW complexes?

I know the answer (or at least an answer) coming from some complicated technology. But I’m posing it as a puzzle, firstly because I think it would be fun, and secondly because I’m curious whether this definition could have been invented by people studying (higher) groupoids but not knowing about manifolds (or complicated technology).

That doesn’t mean I’ll reject your answer if you use complicated technology to get to it, but to get full credit, you should express your answer in concrete terms, without reference to topology or technology. (In fact, without this caveat, there’s a trivial solution: “build the corresponding 1-dimensional CW complex and calculate the classical fixed-point index”.) What I really want to know is the answer to the extra-credit problem:

Question: How and why could someone have invented your definition, knowing only about groupoids?