A number that counts what it cannot see

Something always stays put began with Brouwer: every continuous map of a disc to itself has a fixed point. Where the fixed point escapes looked at spaces without that property, and it noticed that the answer depended on the holes in the space, in a way that a general theorem might capture. That theorem exists, and it is one of the most economical in topology. Solomon Lefschetz gave it in 1926, and Heinz Hopf put it in its modern form. It computes a single integer from the way a map acts on the holes of a space. If the integer is not nought, the map has a fixed point. More than that, the integer counts the fixed points, with signs.

Brouwer’s theorem, the hairy ball theorem, and the Euler characteristic of a surface as the sum of a vector field’s indices are all special cases, each obtained in one line. And the most far-reaching use of the same formula has nothing to do with continuity at all, since it is what counts solutions of polynomial equations over finite fields.

Crossings on a circle #

A map of the circle to itself can be drawn in a square, with the two ends of each side identified, and a fixed point is a place where the curve meets the diagonal. A map’s degree $d$ is the number of times it winds round. The degree-3 map $x \mapsto 3x$ goes up across the square three times, and it meets the diagonal at $x = 0$ and $x = 1/2$. The degree $-2$ map goes down twice and meets the diagonal at three points, at thirds of the way round.

Give each crossing a sign: minus one if the curve crosses the diagonal upwards, so that $f(x) - x$ is rising, and plus one if it is falling. Then the signed total depends only on the degree, and it equals $1 - d$. For degree 3, there are two crossings, each with sign minus one, giving $-2$. For degree $-2$, there are three, each with sign plus one, giving $+3$. The third picture is a degree-2 map with a strong bend added, where five crossings appear, with signs adding to $-1$, as they would for the straight map’s one crossing.

The reason is elementary. As $x$ goes once round, the quantity $f(x) - x$ changes by a total of $d - 1$, with its values read modulo one. It must pass through a whole number at least $|d - 1|$ times, and each extra crossing it makes has to be undone by one in the opposite direction. So $1 - d$ is forced as the net count, and every map of degree $d$ has at least $|1 - d|$ fixed points. A degree-1 map, such as a rotation, may have none. That is the case with a signed count of nought. It is the same logic as the winding number, and it is how this kind of argument always looks in one dimension.

Bending does nothing to the total #

Now bend one map harder and watch. For small $a$ there is one fixed point. Past a threshold near $a = 0.75$, the curve becomes tangent to the diagonal in two places at once, by symmetry, and two pairs are born, each one up and one down. More follow as $a$ grows. At $a = 3$ the map has 13 fixed points, and the signed total has held at exactly $-1$ at all 61 values of $a$ tested. This is the general behaviour of fixed points under deformation: they can be created and destroyed only in pairs of opposite sign. So the count with signs is a quantity that deformation cannot change, which is exactly what a homotopy invariant is. It is a topological quantity living inside what looks like a geometric one.

So which quantity is it? The answer involves how the map acts on holes. A circle has two homology groups: $H_0$, recording the fact that it is one piece, and $H_1$, recording its one loop. A map acts on each by multiplication by some number. It acts on $H_0$ by 1, since one piece goes to one piece, and on $H_1$ by the degree $d$. The Lefschetz number is the alternating sum of the traces:

$$L(f) = \sum_k (-1)^k \operatorname{tr}\!\left(f_* \text{ on } H_k\right) = 1 - d.$$

That is a formula which makes sense in every dimension and for every reasonable space. The theorem is that if $L(f) \ne 0$, then $f$ has a fixed point; and if the fixed points are isolated, then the sum of their indices is $L(f)$. Hopf’s version of the proof is a beautiful piece of bookkeeping. On a triangulated space, a map moved slightly can be arranged so that it sends each cell near itself, and then the trace on cells counts how many cells map to themselves. A short algebraic lemma, the Hopf trace formula, says that the alternating sum of traces is the same on the chains as on the homology. In the case of the identity map, that lemma is exactly the statement that faces minus edges plus vertices can be computed from homology.

The torus, where the count is exact #

The cleanest examples live on a torus. Let $A$ be a $2 \times 2$ integer matrix, and let it act on the unit square with opposite edges joined, by $x \mapsto Ax \pmod 1$. On $H_0$ it acts by 1, on $H_1 = \mathbb{Z}^2$ by the matrix $A$ itself, and on $H_2$ by $\det A$. So

$$L = 1 - \operatorname{tr} A + \det A = \det(I - A).$$

The derivative of this map is $A$ everywhere, so that all the fixed points have the same index, namely the sign of $\det(I - A)$. There is no cancellation. If $\det(I - A) \ne 0$, then the signed total is the count, and $A$ has exactly $|\det(I - A)|$ fixed points.

That is checkable. Fixed points satisfy $(I - A)x \in \mathbb{Z}^2$, so they are rational with denominator $|\det(I - A)|$, and they can be listed by trying all such points. It was done for all 2,401 matrices with entries from $-3$ to 3. Every one with a non-zero determinant has exactly that many. The cat map’s matrix $[2\ 1;\ 1\ 1]$ has one, the origin. The quarter turn has two, the origin and the centre, which are the only fixed points of a rigid quarter turn of the square torus. And $[5\ 2;\ 3\ 4]$ has six, in a row along a line. A question about solving a system of congruences is being answered by homology.

The list contains a small classification too. Of the 2,401, 232 have determinant $\pm 1$, so that they are invertible and give homeomorphisms of the torus. Among those of determinant 1, 40 have trace above 2 in size, like the cat map, and they stretch one direction and shrink another, with $L = 2 - \operatorname{tr} A$ never nought. Another 34 have trace $-1$, 0 or 1, and they are rotations of order 3, 4 or 6 in suitable coordinates, the only rotations a lattice can have. Their fixed points are 3, 2 and 1 in number. Those are the numbers familiar from wallpaper patterns: a threefold centre of rotation on a torus has three fixed points, a fourfold has two, and a sixfold has one. A determinant has given the arrangement of rotation centres in a repeating pattern, which is a fact any student of wallpaper groups would recognise, and which nobody would ordinarily expect to read off an alternating sum of traces on homology.

Even this much depends on the matrix being a true map of the torus, with whole-number entries and all, which is why a rotation by a generic angle has nothing to say here. The theorem is silent in 259 of those 2,401 cases, the warm bar, where $L = 0$. Those matrices have 1 as an eigenvalue, with a whole line or circle of fixed points or none at all once a translation is added. So the theorem is one-sided in general. If $L \ne 0$, fixed points exist, and if $L = 0$, nothing can be concluded. There is a sharper invariant, the Nielsen number, that groups fixed points into classes that cannot be merged, and it is a true lower bound on the minimum number of fixed points in the homotopy class. Boju Jiang and others have shown that it is sharp in many settings, and Jiang found the examples showing it is not always, on surfaces.

Periodic points by determinants #

A periodic point is a fixed point of an iterate. So the same formula applied to $A^n$ counts points of period dividing $n$.

For the cat map, $A = [2\ 1;\ 1\ 1]$, this gives 1, 5, 16, 45, 121, 320, 841, up to 103,680 at $n = 12$. Each of those is a Lucas number $L_{2n}$ minus 2, because $\det(I - A^n) = 2 - \operatorname{tr} A^n$ and the trace of $A^n$ is $\varphi^{2n} + \varphi^{-2n}$, a Lucas number. The first seven were confirmed by listing the actual rational points on the torus. So the growth rate of the periodic points is $\varphi^2 = 2.618$ a step, and its logarithm is the map’s entropy. It is a very pleasing result. Michael Artin and Barry Mazur showed in 1965 how to collect these counts into a zeta function, and Stephen Smale made it a standard tool in 1967. For an Anosov map like this, it is a rational function with one factor for each homology group. Counting periodic orbits is therefore the same as knowing some eigenvalues.

The same formula, over a finite field #

Here is where the idea goes furthest. Take a curve over a finite field, say $y^2 = x^3 + x + 1$ over the field with 5 elements. Its points over the larger field with $5^n$ elements are the fixed points of a map, the $n$-th power of the Frobenius map $x \mapsto x^5$. So counting solutions is a fixed-point problem. André Weil saw in the 1940s that if a cohomology theory existed for such curves, then the Lefschetz formula would give an exact count:

$$N_n = 1 - \operatorname{tr}(F^n \text{ on } H^1) + 5^n.$$

For an elliptic curve, $H^1$ has dimension two, and the trace of $F$ is the number $a = q + 1 - N_1$. Testing it: this curve has $N_1 = 9$ points over the field of 5 elements, so $a = -3$. Then the formula predicts $N_2 = 25 + 1 - (a^2 - 2 \cdot 5) = 27$, and direct enumeration over all 625 pairs in the field of 25 elements gives 27. A single number has predicted the count on a much larger field. That is exactly what the bound that follows from the size of those eigenvalues comes from. Alexander Grothendieck built the necessary cohomology theory, and Pierre Deligne finished the proof of the Weil conjectures in 1974. In that setting all the fixed points of Frobenius count with sign $+1$, as for the matrices on the torus, and for the same reason: no cancellation, only a count. It is among the strangest transfers in mathematics, since a formula about bending maps of spaces has become a formula about solving equations in a setting where nothing can be bent.

One formula, many old theorems #

The table reads as a list of corollaries. A disc has only $H_0$, so any map has $L = 1$, and that is Brouwer’s theorem, in one line. The sphere has $H_0$ and $H_2$, with degree $d$ on the latter. So $L = 1 + d$ on an even-dimensional sphere, and a map of even spheres with no fixed points must have degree $-1$, as the antipodal map does. The identity on a surface has $L$ equal to the Euler characteristic $2 - 2g$. Applied to a small deformation along a vector field, it gives the Poincaré–Hopf theorem, so that zeros of a field, counted with sign, add up to $2 - 2g$. The rational map $z^d$ on the Riemann sphere has $d - 1$ roots of unity, nought and infinity, making $d + 1$ fixed points, and it is the same count as for every holomorphic map of degree $d$, where all fixed points have positive sign when counted with multiplicity. It is this that gives the count of the fixed points of a rational map. So what are usually treated as separate theorems in different courses are one alternating sum.

What the figures cannot show #

The figures here are for maps that are simple enough to make their computation exact, and the crossings in the circle pictures are detected on a grid of 40,000 points, so that a tiny extra pair, born just past a tangency, could be missed. Their count is a numerical one, although the signed total is what is being verified, and any missed pair would not change it. All the cases shown are ones where the homology is already known. For a general space, computing the action on homology might be hard. Nothing here tests the theorem for a space with torsion, where the traces are taken over the rationals and the torsion goes unseen. And no picture can show why the total does not change for a map on a surface of higher genus, or how fixed points of mixed sign occur there.

What was computed and what was quoted #

All fixed points of integer matrices were enumerated exactly as rational points with an integer test, for all 2,401 matrices. Periodic counts to period seven were checked by listing, and to twelve by the determinant and a Lucas-number identity. The circle counts come from detecting sign changes on a fine grid, and the signed total was verified at every parameter. The elliptic curve example was counted directly over the fields of 5 and 25 elements.

Quoted: the Lefschetz and Hopf theorems, Nielsen theory and Jiang’s results, the zeta function of Artin, Mazur and Smale, the history of the Weil conjectures, and the statement about holomorphic maps. The sketch of Hopf’s proof is an outline.

Still open: fixed points that survive a symmetry #

How many fixed points must a symplectic map have? On a closed surface, an area-preserving map isotopic to the identity through suitable paths can have many more forced fixed points than the Lefschetz number suggests. Vladimir Arnold conjectured in the 1960s that the count is at least the sum of all Betti numbers, and not just their alternating sum. On a torus that is at least 4 rather than the 0 that Lefschetz gives, for such a map. Charles Conley and Eduard Zehnder proved it for the torus in 1983, and the general case led Andreas Floer to invent Floer homology. For non-degenerate maps it is been proved in general, by Fukaya and Ono and others, in rational form, while the degenerate case and the exact count over the integers remain open in many settings.

A second question lives at the other end of the spectrum. Given a map, how few fixed points can a map homotopic to it have? For surfaces, Jiang’s examples show that the Nielsen number is not always the answer, and the true minimum in general is not known. Theory gives a lower bound and a list of exceptions. The gap between the integer the homology sees and the true minimum in any given homotopy class is exactly what a finer invariant would have to capture.

A signed count of what stays #

The fixed points of a map, counted with signs, add up to an alternating sum of traces on homology, and that sum does not change when the map is bent. On a torus it is a determinant, and it gives the exact count, which was verified on 2,401 matrices and on the periodic points of the cat map.

So a global invariant has settled a local question, without any search through the space. The same formula, carried across to a finite field, turns counting the solutions of an equation into computing the trace of an operator, and that one count on a curve over five elements predicted 27 points on the larger field. A fixed point is a very local thing, and the number that detects it hardly knows where it is.