What homology forgets about a loop

There are two ways of asking what holes a space has, and both have been followed here before. One draws loops and asks which can be pulled tight: the group a space has at a point, where a loop round a hole and a loop round it twice are different elements and the order in which two holes are visited matters. The other is linear algebra: chains of edges and faces, a boundary matrix, and a hole is a cycle that bounds nothing — a kernel modulo an image, computable by row reduction, and commutative by construction because it is built from sums.

The two are not unrelated, and the relation is exact. Take the loop group and decree that any two loops commute; what remains is the first homology group. That is the first case of a theorem of Witold Hurewicz from 1935, though the observation in this case goes back to Poincaré, who introduced the loop group and the counts of holes that became homology in the same paper of 1895. This essay is about what the decree throws away — which loops homology cannot tell from nothing — and about how much that is. On the simplest space where the two differ, the answer is: almost all of them.

The loop that bounds and does not shrink #

The figure eight is two circles joined at a point. A loop starting at the join is a word in four letters: $a$ for once round the left circle, $a^{-1}$ for once round it backwards, $b$ and $b^{-1}$ for the right. Cancelling a letter against its inverse when they stand next to each other is pulling a backtrack tight, and once no such pair is left the word is reduced. Two loops can be deformed into each other exactly when they reduce to the same word, so the loop group is the free group on $a$ and $b$, and no two different reduced words are equal in it — the fact the Cayley graph of this group shows by being a tree.

Now take the loop $aba^{-1}b^{-1}$: round the left circle, round the right, back round the left, back round the right.

The word is reduced, so the loop is not trivial. It cannot be shrunk to its base point however it is deformed, because every deformation is a sequence of insertions and deletions of $aa^{-1}$-type pairs and none of those changes the reduced word.

And yet it uses $a$ once forwards and once backwards, and $b$ the same, so it goes round each hole a net zero times. In the chain picture it is the edge sum $a + b - a - b = 0$, a cycle that is literally zero before any boundary is quotiented out. The square on the right says the same thing geometrically: label its four sides $a$, $b$, $a^{-1}$, $b^{-1}$ in order and its boundary is the loop, so the loop bounds a surface. Glue the two $a$ sides together and the two $b$ sides together and the square becomes a torus — except that the four corners are where the loop runs, so it is a torus with a hole, and the loop is the rim of the hole.

That is the general shape of the gap. A loop that bounds a surface is nought in homology, because homology counts a boundary as nothing. A loop is trivial in the loop group only if it bounds a disc. Between those, the surface can have handles, and the commutator $aba^{-1}b^{-1}$ is the loop round the rim of exactly one handle.

The commutative version is a lift to the grid #

Making the letters commute has a picture as concrete as the tree. Draw the square grid in the plane, and read a word as a path: a step right for $a$, left for $a^{-1}$, up for $b$, down for $b^{-1}$. In the grid, going right then up arrives at the same point as going up then right — the two letters commute — and the endpoint of a word’s path records exactly the net number of times it used each letter.

The grid is not a decoration. It is a cover of the figure eight: wrap the plane down onto the figure eight by sending every horizontal edge to the circle $a$ and every vertical edge to $b$, and every grid point to the join. A loop on the figure eight lifts to a path in the grid starting at the origin, and the lift is closed precisely when the loop lies in the subgroup this cover corresponds to. Under the dictionary of every cover is a subgroup, the grid is the cover belonging to the subgroup generated by all commutators — the commutator subgroup — and the symmetries of the cover, the translations by whole-number vectors, form the group $\mathbb{Z}^2$. That group is the first homology of the figure eight, and the endpoint of a lift is a loop’s homology class.

So the first figure’s three closed paths are three loops in the kernel of the map from loops to homology: $aba^{-1}b^{-1}$, $a^2ba^{-2}b^{-1}$, $ab^2a^{-1}b^{-2}$. Each is reduced and non-trivial, and each lift returns to where it began.

An area the lift still remembers #

The closed lifts in the figure are not the same, though: the first encloses one square, the second and third two. The signed area of a closed lift — the area measured by walking round it, positive when the walk goes anticlockwise — is a second invariant, and it is a real one. A backtrack $aa^{-1}$ adds a step out and the same step back and encloses nothing, so inserting or deleting cancelling pairs does not change the area; it depends only on the reduced word. And since the trivial loop has area zero, any closed lift with non-zero area proves its loop non-trivial — a proof, for $aba^{-1}b^{-1}$, that does not need the theory of reduced words at all.

The area is information homology has discarded. It is also not the whole of what was discarded.

The third word is the product of two commutators, $aba^{-1}b^{-1}$ followed by $ab^{-1}a^{-1}b$. Its lift traces one square anticlockwise and the square below it clockwise, and the areas cancel. The word is reduced — nothing cancels at the join, where $b^{-1}$ meets $a$ — so the loop is not trivial, and neither the endpoint nor the area can see it.

This is the start of a tower rather than the end of a list. Homology is the loop group modulo commutators. The area is a coordinate on the next layer up, the loop group modulo commutators of commutators, which is a group of $3 \times 3$ triangular matrices called the Heisenberg group. The third word lies in that next layer — it is a product of commutators of letters with commutators — so it is invisible to the area and visible one layer further. Each layer is a finer quotient of the free group, every non-trivial loop is detected by some layer — a theorem of Magnus — and no finite number of layers detects them all.

The kernel is a free group with a generator for every square #

The loops homology calls nought form a subgroup, and the grid says what it is. A subgroup of the loop group is the loop group of its cover, and the cover here is the grid itself — an infinite graph of edges and corners. A graph’s loop group is free, with one generator for every independent cycle, and in a planar graph the independent cycles are the faces: one loop round each unit square, joined to the origin by a path. So the commutator subgroup of the free group on two letters is itself free, and it needs infinitely many generators, one for each square of the plane.

That is a strange thing for a subgroup of a group with two generators to be, and it is the Nielsen–Schreier theorem behaving as it must: a subgroup of infinite index in a free group need not be finitely generated, and this one is not. A tidier basis is known — the commutators $[a^m, b^n] = a^m b^n a^{-m} b^{-n}$ for every pair of non-zero whole numbers, each of which lifts to the boundary of an $m \times n$ rectangle with a corner at the origin. Two of them were drawn in the first lift figure: $a^2ba^{-2}b^{-1}$ is the rectangle two wide and one high, and $ab^2a^{-1}b^{-2}$ the one standing on its end.

The signed area is then read off a basis element at a glance — $mn$, the rectangle’s own area — and it adds when loops are multiplied, since walking one closed path and then another encloses what each enclosed. It is a homomorphism from the kernel to the whole numbers, and its own kernel, the loops whose closed lifts enclose no area, is the next floor of the tower below.

How much the decree throws away #

The tower makes the loss plausible; counting makes it precise. Ask how many different loops on the figure eight can be written with at most $n$ letters, first in the loop group, then after letting the letters commute.

In the loop group, the different elements are the reduced words. There is one of length nought, four of length one, and each later letter can be any of the four except the inverse of the letter before it, so there are $4 \cdot 3^{k-1}$ of length $k$, and $2 \cdot 3^n - 1$ of length at most $n$. In the commutative version, an element is a grid point, and the points reachable in $n$ steps are those with $|x| + |y| \le n$ — a diamond holding $2n^2 + 2n + 1$ points.

On the logarithmic scale the first count is a straight line and the second bends over: one is exponential and one polynomial. At eight letters there are 13,121 loops and 145 homology classes, so on average ninety loops share each class, and the average doubles and redoubles with every few letters. Homology keeps a polynomial’s worth of an exponential.

That count is about distinct elements. The more pointed question is about the one class that matters most — nought — and asks of a loop that homology says is nothing how likely it is that it really is nothing.

Of the loops homology calls nought, how many shrink #

Choose a word of $2n$ letters at random, each letter one of the four with equal chance. It is nought in homology when its lift to the grid closes, which is a question about a walk on the plane coming home, and it is the trivial loop when it reduces to the empty word, which is a question about a walk on the four-branched tree coming home. Both chances can be computed exactly — the grid’s from a binomial coefficient squared, the tree’s from a recursion on distance from the root that the essay on how rarely a walk on a group comes home works through — and their quotient is the share of homologically trivial loops that are trivial loops.

At two letters the share is one: the only closed two-letter words are $aa^{-1}$, $a^{-1}a$, $bb^{-1}$ and $b^{-1}b$, and every one cancels. At four letters there are 36 closed words and 28 of them reduce to nothing; the other eight are $aba^{-1}b^{-1}$, the three words obtained from it by starting at a different letter, and the four inverses of those. The commutator appears at the first length where it can, and a check of all 256 four-letter words confirms there are exactly eight.

After that the share falls steadily, by close to three quarters for every two letters, and by sixty letters it is about $1.5 \times 10^{-4}$. The reason is the difference between the two walks. A walk on the plane is recurrent, and comes home with chance about $1/(\pi n)$ after $2n$ steps; a walk on the tree is pushed outward, three branches leading away for every one leading back, and comes home with chance falling like $(\sqrt{3}/2)^{2n}$. Almost every closed lift is the shadow of a loop that has wandered far out along the tree and returned in the plane only because the plane cannot tell order apart.

So on the figure eight, “nought in homology” is a statement about almost no loops’ actual shape. It says the loop bounds a surface, and the surface almost always has handles.

Which surface a loop bounds #

That last sentence has a precise version. A loop that is nought in homology is a product of commutators — that is what lying in the commutator subgroup means — and a product of $g$ commutators $[x_1, y_1] \cdots [x_g, y_g]$ is the boundary of a surface with $g$ handles and one rim, built from a polygon of $4g$ sides exactly as the square was built for $g = 1$. Conversely, a loop that bounds a surface of genus $g$ in the space can be written as a product of $g$ commutators. So the least number of commutators a loop needs, its commutator length, is the least genus of a surface it bounds — Marc Culler’s 1981 reading, and the reason the same number turns up in the surface a knot bounds as the genus of a knot.

The loop group sees the genus; homology sees only that some surface exists. A loop of genus zero in this sense bounds a disc and shrinks. A loop of genus one bounds a handle and does not.

The example worth having in mind is not on the figure eight, where everything is free and so nearly every loop is non-trivial, but on a closed surface, where the loop group has relations and the gap is narrower.

The surface of genus two is a sphere with two handles, and cutting it into pieces shows its loop group has four generators and a single relation: the product of the two commutators is trivial. Its first homology is $\mathbb{Z}^4$, because once letters commute that relation says nothing. The dashed curve is the commutator $[a_1, b_1]$. It separates the surface into two halves, each a handle with a rim, so it bounds a surface and is nought in homology. But it does not bound a disc on either side, and the single relation of the group is not enough to make $[a_1, b_1]$ trivial — it is the curve on which homology and homotopy part company, and the separating curves are exactly where they do. On the torus, with one handle, the commutator of the two generating loops is the rim of the very square the torus is glued from, so it bounds a disc and shrinks; the loop group there is already commutative, and the loops on a torus lose nothing to homology.

What the forgetting buys #

All of this reads like a list of losses, and the reasons to make them anyway are the reasons homology is used far more than the loop group.

The first is that the base point disappears. A loop group depends on its base point, and moving the point along a path changes every element by conjugation, $x \mapsto p\,x\,p^{-1}$. Once letters commute, $p\,x\,p^{-1} = x$, and the dependence is gone. Homology is a property of the space rather than of the space with a marked point, which is why its classes can be added up across a space with no preferred centre.

The second is computability. Deciding whether two words name the same element of a group given by generators and relations is, in general, impossible — there are finitely presented groups for which no algorithm can do it. Deciding whether two vectors are equal in a quotient of $\mathbb{Z}^n$ is integer row reduction. Homology trades a question nobody can always answer for one a matrix answers.

The third is that the forgetting is principled rather than arbitrary. The map from loops to homology is the largest commutative quotient: any homomorphism from the loop group to a commutative group factors through it. So any commutative invariant of loops — a winding number, a count of crossings with sign, the degree of a map to the circle — is already a function of the homology class, and to see anything finer, like the area, one has to leave commutative groups behind.

The same theorem higher up #

The theorem has a second half, and it is where the name usually attaches. Replace loops by spheres. The higher homotopy groups were shown always to be commutative, so there is nothing to make commutative, and Hurewicz’s theorem says that for a space with no holes of lower dimension — every loop shrinks, every sphere of lower dimension shrinks — the first non-vanishing homotopy group is the corresponding homology group, with nothing forgotten. The first case, the one drawn here, is the only one where something is.

That is why the loop group is the stubborn one. It is the only homotopy group that can fail to commute, so it is the only one where homology, which is commutative by construction, cannot simply be equal to it — and on a space as small as two circles joined at a point, the difference is almost everything.