August 20, 2011

Fixed Point Indices for Groupoids

Posted by Mike Shulman

The fixed point index is a rare example of a concept in homotopy theory that is much easier to motivate, as far as I can tell, when you think of $\infty$-groupoids as presented by topological spaces. It is possible, however, to define it in purely categorical language (and quite simply, too, without referring to any complicated technology). I want to pose this as a puzzle: can you define the fixed-point index this way (just for 1-groupoids, to make it easy) — and better, can you motivate it?

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?

Posted at August 20, 2011 12:58 AM UTC

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Re: Fixed Point Indices for Groupoids

Let me try to generalize the question so as to help pin down the answer.

The base situtation is the identity map of a manifold, and the trivial section of the tangent bundle. Perturbing it, the Lefschetz number gives the Euler characteristic.

1) Instead of the Euler characteristic, we could compute all the Betti numbers. That would fix your positivity issues.
2) Instead of the identity map, we could use another map, and compute its Lefschetz number (as you describe).
3) Instead of the tangent bundle, we could use another bundle of the same dimension as M, and compute its Euler number. (Or even a smaller bundle, and compute its Euler class?)

Which pairs of these can be combined? Or can all three?

Posted by: Allen Knutson on August 20, 2011 6:31 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

Good question! There’s an obvious way to combine (1) and (2): the Lefschetz number of any map is the alternating sum of its traces in homology. Although that won’t “fix the positivity issues” in the same way as it does for Euler characteristics, since for a non-identity-map those individual traces could still be negative.

But I don’t know enough about Euler numbers/classes to say whether (3) could be combined with (1) or (2). Anyone?

Posted by: Mike Shulman on August 23, 2011 3:50 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

So a method which answers your description would be able to give you an index for endomaps of $B \mathbb{Z}$. These amount to multiplication of the integer labelled arrows by an integer. And multiplication by $n$ leads to an index $1 - n$.

A finitely generated free groupoid and mapping would contain all the information necessary to calculate the homology based Lefschetz number via abelianization and, in the one object case, calculating (1 - the trace).

But this is all very much not thinking just in terms of groupoids.

Posted by: David Corfield on August 20, 2011 11:05 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

If I understand correctly what you’re proposing, that’s close-ish to the technology I used to get my answer. Can you express the answer obtained in that way explicitly in terms of the finitely generated free groupoid you started with?

Posted by: Mike Shulman on August 23, 2011 3:52 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

So you take the image of each generator, $f$, and see what exponent of $f$ it contains. Then the sum of these exponents is the trace.

The index is the number of fixed objects $-$ the trace.

Posted by: David Corfield on August 23, 2011 9:32 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

So you take the image of each generator, $f$, and see what exponent of $f$ it contains. Then the sum of these exponents is the trace. The index is the number of fixed objects − the trace.

Exactly! Here’s (sort of) how I arrived at this formula. The geometric realization of the nerve of a groupoid, like that of any simplicial set, is a CW complex. For a finitely generated free groupoid, this CW complex is finite and 1-dimensional, with 0-cells corresponding to the objects of the groupoid and 1-cells corresponding to the generating morphisms. We can therefore calculate the fixed-point index of any endomap via the Lefschetz number in terms of its complex of cellular chains.

This chain complex is very simple; it has a generator in degree 0 for every object, and a generator in degree 1 for every generating morphism. The differential sends a morphism to the difference of its source and target. An endofunctor of our groupoid induces an endomap of this chain complex; it obviously sends each object $x$ to $f(x)$ in degree 0, and it sends a generating morphism $g$ to the sum of all the generators that appear in the morphism $f(g)$, with multiplicities. So if we have $f(g) = h g h k^{-1} g^{-1} h g$ then the chain map would send $g \mapsto h + g + h - k - g + h + g = g + 3 h - k$ Now the Lefschetz number of a chain map is the alternating sum of its traces; in this case that just means the trace in degree 0, minus the trace in degree 1. The trace in degree 0 is just the number of (literally) fixed objects. The trace in degree 1 is the sum over all generators $g$ of the net multiplicity with which $g$ occurs in $f(g)$; in the example above it would be 1. Then we subtract this sum from the number of fixed objects to get the fixed-point index.

For example, the identity map of $B\mathbb{Z}$ has one fixed object, and the one generating morphism is mapped to itself; thus we have $1-1=0$ as the Euler characteristic. The degree-two map also has one fixed object, but now the generating morphism is mapped to its square; thus we have $1-2 = -1$ as the fixed-point index.

But is there any sense in which this calculation counts something relating to “fixed points” from the perspective of the groupoid?

Posted by: Mike Shulman on August 26, 2011 6:01 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

My answer to your puzzle (“Can you…”) is “no”; at least, not right now. But let me say some stuff that’s closely related.

There’s a decent notion of Lefschetz number for endofunctors of more or less any finite category. The basic definitions and results are set out towards the end of Section 2 of The Euler characteristic of a category.

Given an endofunctor $F$ of a finite category $A$, let $Fix(F)$ denote the category whose objects are objects $a$ of $A$ such that $F(a) = a$, and whose maps are maps $f$ in $A$ such that $F(f) = f$. You might be shocked that I’m writing “$=$”, but that’s really what I mean. The Lefschetz number $L(F)$ of $F$ is the Euler characteristic of $Fix(F)$ (in so far as that Euler characteristic is defined).

The basic theorem is this. It’s Proposition 2.14 of the paper I just cited, but I’ll reproduce it here.

Theorem  Let $A$ be a finite category. Then:

(a) $L(1_A) = \chi(A)$, with one side defined if and only if the other is.

(b) If $B$ is another finite category and $F: A \to B$, $G: B \to A$ are functors then $L(G F) = L(F G)$, with one side defined if and only if the other is.

(c) Let $F: A \to A$ be a functor and write $|F|: |A| \to |A|$ for the induced map on classifying spaces. Suppose that $A$ is skeletal and contains no nontrivial idempotents. Then $L(F) = L(|F|)$, with both sides defined.

Obviously, the definition of the Lefschetz number of an endofunctor was topologically motivated. (Otherwise, I wouldn’t have called it “Lefschetz number”.) But it’s easy to imagine how to motivate it non-topologically. After all, the passage from the endofunctor $F$ to the category $Fix(F)$ is canonical: $Fix(F)$ is the limit of the diagram consisting of $F$ alone. (Or it’s the equalizer of $F$ and the identity.) And the definition of Euler characteristic can also be motivated non-topologically, though the name chosen was a topological one.

Posted by: Tom Leinster on August 22, 2011 5:45 PM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

Thanks for bringing this up again, Tom! I admit one reason I keep thinking about this stuff is the hope to bring together all the notions of Euler characteristic and trace. I haven’t really thought deeply about your notion of Lefschetz number before, but now that I am, I’m confused about your (c).

Consider the category $3 = (0 \to 1 \to 2)$. This is clearly finite, skeletal, and contains no nonidentity endomorphisms. Let $F\colon 3\to 3$ be defined by $F(0)=0$ and $F(1)=F(2)=2$. Then it seems as though your $Fix(F)$ is the discrete category on 0 and 2, which has Euler characteristic 2 — whereas ${|3|}$ is contractible and ${|F|}$ is homotopic to its identity map, hence has Lefschetz number 1. Am I missing something?

Posted by: Mike Shulman on August 23, 2011 3:59 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

The arrow $0 \to 2$ is fixed by $F$, right? So $Fix(F) = (0 \to 2)$, which has Euler characteristic $1$, as hoped.

One big difference between what you’re doing and what I’m doing is the isomorphism-invariance. Obviously, if $F, G: A \to B$ then we can have $F \cong G$ but $L(F) \neq L(G)$. I don’t know what to think about this.

Posted by: Tom Leinster on August 23, 2011 5:34 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

Oh, of course, silly me. I was thinking about the generating arrows, trying to connect this notion with the f.g. free groupoid case (where I think we have to think about generators, since the category as a whole isn’t finite).

I’m having some thoughts about a common generalization, now, but they’re only half-baked so far.

Posted by: Mike Shulman on August 23, 2011 6:06 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

I’m repeating myself (again), but the free category on a finite graph is naturally enriched in $Set^{\mathbb{N}} \simeq Set/\mathbb{N}$. (Every map in the free category has a length, a natural number.) So although the homsets in the free category are often infinite, they each have a ‘cardinality’ belonging to the polynomial rig $\mathbb{N}[x]$. This can be useful. However, you’re interested in free groupoids, where I can’t imagine anything similar being true — ultimately because the free groupoid monad is not cartesian.

Posted by: Tom Leinster on August 23, 2011 6:33 AM | Permalink | Reply to this

Re: Fixed Point Indices for Groupoids

Hmm, maybe that last comment about cartesian monads was a bit off-target. It sprang from the intuition that in a cartesian monad, the operations have a rigid shape to them. But of course when you take the free groupoid $G$ on a graph, each map in $G$ has a “length” belonging to $\mathbb{Z}$ (the free groupoid on the terminal graph).

Posted by: Tom Leinster on August 23, 2011 6:40 AM | Permalink | Reply to this

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