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## The homotopy groups are only groups

Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

• Given a Lie group $G$, its tangent space $T_e(G)$ at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{Z})$ is a priori a graded abelian group, but it ends up having the structure of a graded ring.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{F}_p)$ over $\mathbb{F}_p$ is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-$p$ Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups $\pi_n(X)$ of a space: that is, we will show that, in a suitable sense, each individual homotopy group $\pi_n(X)$ is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)

Extra structure on a functor

The setting in which we will work is the following. Suppose we have some functor $F : C \to D$ which a priori takes values in a category $D$. To what extent can we lift $F$ to a functor $F' : C \to D'$ taking values in a “more structured” category $D'$ equipped with a forgetful functor $D' \to D$ such that the obvious diagram commutes? As phrased, this question is incredibly general, so we will restrict ourselves to lifts which are described by taking into account structure coming from $n$-ary operations, as follows.

Suppose $D$ has finite products. Then we can consider natural transformations $F^n \to F$ to be $n$-ary operations (as in this previous post on Lawvere theories) on the outputs of the functor $F$ which equip the objects $F(c)$ with extra structure. More precisely, the full subcategory of the functor category $C \Rightarrow D$ on the objects $1, F, F^2, ...$ is a Lawvere theory, the endomorphism Lawvere theory $\text{End}(F)$ of $F$ (named in analogy with the endomorphism operad). Note that equipping an object in a category with finite products with the structure of a model of a Lawvere theory $T$ is equivalent to giving a morphism $T \to \text{End}(F)$ of Lawvere theories; in particular, $F$ itself is tautologically a model of $T$, and this model structure passes to $F(c), c \in C$. This lets us lift $F$ to a functor taking values in the category of $D$-valued models of $\text{End}(F)$, or more precisely the category of product-preserving functors $\text{End}(F) \to D$.

If $D = \text{Set}$, $F$ is representable by some object $f \in C$, and $C$ also has finite coproducts, then we can identify natural transformations $F^n \to F$ with morphisms $f \to f \sqcup ... \sqcup f$ by the Yoneda lemma. Consequently, we can identify $\text{End}(F)$ with $\text{End}(f^{op})$, where $f^{op}$ is $f$ regarded as an object in the opposite category $C^{op}$. There is a corresponding story where $F$ is a contravariant representable functor; here we just have $\text{End}(F) \cong \text{End}(f)$.

It may be hard to compute the entire endomorphism Lawvere theory of a functor, but any natural transformations $F^n \to F$ that we can find may already provide extra structure that wasn’t there before. More generally it is often possible to identify Lawvere theories $T$ and morphisms $T \to \text{End}(F)$ of Lawvere theories, which allow us to lift $F$ to the category of $D$-valued models of $T$. These kinds of observations are already enough to reproduce many familiar examples of extra structure, and generalize the observation that $\text{Hom}(c, d)$ is acted on from the left by the monoid of endomorphisms $d \to d$ and from the right by the monoid of endomorphisms $c \to c$.

Example. If $G$ is a group object in a category with finite products, then the group operation $G \times G \to G$ gives a morphism from the Lawvere theory of groups to $\text{End}(G)$. Hence $\text{Hom}(-, G)$ naturally acquires the structure of a group. (Conversely, by the Yoneda lemma, if $\text{Hom}(-, G)$ naturally has the structure of a group then $G$ is a group object.)

Example. Dually, if $c$ is a cogroup object in a category with finite coproducts, then the cogroup operation $c \to c \sqcup c$ gives a morphism from the Lawvere theory of groups to $\text{End}(c^{op})$. Hence $\text{Hom}(c, -)$ naturally acquires the structure of a group. (Again, conversely, by the Yoneda lemma, if $\text{Hom}(c, -)$ naturally has the structure of a group then $c$ is a cogroup object.)

Example. In the category of schemes over a base ring $R$, the endomorphism Lawvere theory of the affine line $\mathbb{A}^1_R \cong \text{Spec } R[x]$ is the Lawvere theory of polynomials over $R$, or equivalently the Lawvere theory of commutative $R$-algebras. Hence $\text{Hom}(-, \mathbb{A}^1_R)$ naturally acquires the structure of a commutative $R$-algebra. (We previously discussed the case $R = \mathbb{Z}$ for affine schemes in this blog post.)

Example. In the category of topological spaces, the space $\mathbb{R}$ admits addition and multiplication operations $+, \times : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ in addition to scalar multiplication operations $\mathbb{R} \to \mathbb{R}$, and these generate the Lawvere theory of polynomials over $\mathbb{R}$. Hence $\text{Hom}(-, \mathbb{R})$ naturally acquires the structure of a commutative $\mathbb{R}$-algebra.

Example. A distributive category is a category with finite products and coproducts such that the former naturally distribute over the latter; the standard example is $\text{Set}$, although $\text{FinSet}$ and more generally any cartesian closed category also qualify, and $\text{Top}$ and $\text{Sch}$ (the category of schemes) are important examples which are not cartesian closed.

In any distributive category, the endomorphism Lawvere theory of the object $2 = 1 + 1$ canonically admits a morphism from the Lawvere theory of Boolean algebras, or equivalently the Lawvere theory of Boolean rings, or equivalently the category of Boolean functions (the full subcategory of $\text{FinSet}$ on finite sets of size $2^n$). Hence $\text{Hom}(-, 2)$ naturally acquires the structure of a Boolean algebra, or equivalently a Boolean ring. In $\text{Top}$ this reproduces the lattice of clopen subsets of a topological space. In general I think it should be interpreted as something like the “lattice of decidable properties.”

Example. If $A$ is an abelian group, then the group operation $+ : A \times A \to A$ is itself a morphism in $\text{Ab}$, giving a morphism from the Lawvere theory of abelian groups to $\text{End}(A)$. Hence $\text{Hom}(-, A)$ naturally acquires the structure of an abelian group. (We discussed a more general setting in which such an abelian group structure exists in this previous post on semiadditive categories.)

The homotopy groups are groups

Recall that the pointed homotopy category $\text{hTop}_{\ast}$ is the category whose objects are pointed topological spaces $(X, x)$ and whose morphisms are homotopy classes of pointed continuous maps $(X, x) \to (Y, y)$ preserving the base point. Recall also that the homotopy groups are a sequence of functors $\pi_n(-) \cong \text{Hom}(S^n, -)$ naturally defined on this category and represented by the spheres $S^n$ with some choice of base point, which we will usually omit in our notation. That the homotopy groups are groups is equivalent to the statement that the spaces $S^n$, as objects of the pointed homotopy category, are all cogroup objects.

The basic idea is to observe that a pointed map from $S^n$ to a pointed space $(X, x)$ is the same thing as a map from the $n$-cube $I^n$ to $X$ such that the boundary $\partial I^n$ is sent to $x$. In general, morphisms from the $n$-cube can be glued together along any pair of $n-1$-dimensional faces provided that the images of those faces match. There are $n$ distinguished such gluings coming from gluing together each of the $n$ copies of $I$ in the product in the usual way that one glues two intervals together. These gluing operations are natural, associative, and have inverses up to homotopy. They give $n$ compatible group operations on $\pi_n(X)$ which, when $n \ge 2$, make it an abelian group by the Eckmann-Hilton argument.

The appearance of maps out of $I^n$ and multiple composition operations suggests a higher-category-theoretic perspective on the situation where we can think of $\pi_n(X)$ as a suitable automorphism group. More precisely, for any $n$ we can associate to an unpointed topological space $X$ its fundamental $n$-groupoid $\Pi_n(X)$, which is the $n$-category whose

• objects are the points of $X$,
• morphisms are the paths between points of $X$,
• $2$-morphisms are the homotopies between paths,
• $3$-morphisms are the homotopies between homotopies,

• $n$-morphisms are the homotopy classes of homotopies between homotopies between…

Note that a $k$-morphism can be thought of as a map $I^k \to X$, with its source and its target determined by its restriction to a suitable choice of two copies of $I^{k-1}$ in it. $k$-morphisms have $k$ notions of composition given by gluing along the $k$ coordinate directions, generalizing horizontal and vertical composition of $2$-morphisms in $2$-categories (in particular, of functors).

The homotopy group $\pi_n(X)$ of a pointed space $(X, x)$ can then be interpreted as the group of $n$-automorphisms of the identity $n-1$-endomorphism of the identity $n-2$-endomorphism of… of the identity endomorphism of $x$ in the fundamental $n$-groupoid.

The homotopy groups are only groups

We would like to show that the homotopy groups are only groups in the sense that the endomorphism Lawvere theories of the functors $\pi_n : \text{hTop}_{\ast} \to \text{Set}$ are generated by the Lawvere theory of groups. In fact we will be able to say slightly more than this.

Theorem: The endomorphism Lawvere theory of $\pi_1$ is precisely the Lawvere theory of groups.

Proof. By the Yoneda lemma, this means we want to show that the full subcategory of $\text{hTop}_{\ast}$ on the finite wedge sums $S^1 \vee ... \vee S^1$ of $S^1$ is equivalent, as a category with finite coproducts, to the full subcategory of $\text{Grp}$ on the finitely generated free groups. To show this it more or less suffices to show that the fundamental group of a wedge of $k$ circles is the free group $F_k$ generated by each circle (strictly speaking we should show that this identification can be made compatible with partial composition, but we already know this because we already know that the fundamental group is a group), but this follows from Seifert-van Kampen.

In the context of a more general result, $S^1 \vee ... \vee S^1$ not only has fundamental group $F_k$ but is an Eilenberg-MacLane space $K(F_k, 1)$, since its universal cover is a tree, and the subcategory of $\text{hTop}_{\ast}$ on Eilenberg-MacLane spaces (suitably pointed) is known to be equivalent to $\text{Grp}$, with the equivalence given by $\pi_1$. $\Box$

Theorem: The endomorphism Lawvere theory of $\pi_n, n \ge 2$ is precisely the Lawvere theory of abelian groups.

Proof. By the Yoneda lemma, this means we want to show that the full subcategory of $\text{hTop}_{\ast}$ on the finite wedge sums $S^n \vee ... \vee S^n$ is equivalent, as a category with finite coproducts, to the subcategory of $\text{Ab}$ on the finitely generated free abelian groups. To show this it more or less suffices to show that $\pi_n(S^n \vee ... \vee S^n)$ is the free abelian group $\mathbb{Z}^k$ generated by each inclusion of $S^n$ into the wedge (where there are $k$ spheres in the wedge) (and, again, strictly speaking we should show compatibility with partial composition, but we already know this).

Since $X = S^n \vee ... \vee S^n$ admits a CW-structure with a single $0$-cell and no $k$-cells, $1 \le k \le n-1$, it is $(n-1)$-connected by cellular approximation. By the Hurewicz theorem, it follows that the Hurewicz map $\pi_n(X) \to H_n(X)$ is an isomorphism, so to compute the former it suffices to compute the latter. But now $H_n(X) \cong \mathbb{Z}^k$ by Mayer-Vietoris. $\Box$

### 12 Responses

1. I agree with statement “homotopy groups are only groups”. Even the first two homotopy groups, with the second considered as a module over $\pi_1$, are but pale shadows of homotopy 2-types.

Compare that with “strict $n$-fold groupoids model weak homotopy $n$-types” (Loday). Grothendieck’s reaction was to exclaim: “That is absolutely beautiful!” Loday and I showed how you can do calculations of homotopy types using this model, and Ellis and Steiner developed a more concrete model called “crossed $n$-cubes of groups”. This relates quite clearly to classical work on $n$-ad homotopy groups. I.e. given an $n$-ad of spaces, one looks at all the sun $r$-ads for $r \laqslant n$ and all the generalised Whitehead products.

Now the homotopy groups exist somehow in the interior of these nonabelian models, so they may not be so easy to compute from a knowledge of the large model. But then few have worked with these models!

One moral is that these models don’r arise directly from a pointed space, but you have to construct a “resolution of the space by $n$-cubes of fibrations” (Richard Steiner has a nice account of this).

Perhaps the idea of cubical resolutions can be more generally applicable!

2. Homotopy groups in dimension > 1 are abelian groups, and in 1932 the first instinct of the top topologists was to tell Cech that his suggestion was therefore uninteresting, since they were wanting nonabelian higher dimensional versions of the fundamental group. Such structures can now be obtained if you move to higher groupoids, and accept that they are defined (as strict structures) for certain structured spaces, namely filtered spaces or n-cubes of spaces. So part of the difficulty is to accept that topological spaces are not adequate for many purposes. The case of space with base point is a bit confusing because it seems almost a space!

I’ve discussed this further in a talk “The intuitions for cubical sets in nonabelian algebraic topology”. I gave at the IHP in Paris in June, 2014, available on my preprint page. http://pages.bangor.ac.uk/~mas010/brownpr.html

3. […] Base from: Annoying Precision […]

4. I wouldn’t get too excited about this kind of result, and I think billing it as saying “you can’t naturally endow homotopy groups with extra structure” is a bit of false advertizing. You’re artificially restricting the kinds of extra structure to be not only algebraic, but also only adding new finitary operations and no new sorts. That’s pretty limiting: since Eilenberg-MacLane spaces exist, any new kind of structure that played by those rules would consist of hidden finitary operations defined on all groups but not derivable from just the group structure, which of course you wouldn’t expect to exist.

If you allow new sorts, but stay within finitary algebra there is the example you already mentioned of using the other homotopy groups and giving a π-algebra structure to the bunch of them, or at least, say adding the fundamental group and its action to a higher homotopy group.

But one kind of extra structure that’s very popular among mathematicians is to add a topology, and that is also not covered by the restrictions in your post. You can try to make the fundamental group into a topological group by, say, using the compact-open topology on maps from the circle into your space, and then taking the quotient by the relation of homotopy. This does not always give you a topological group if you start with arbitrary topological spaces, but (1) if you work instead, as algebraic topology usually does, in the category of compactly generated weakly Hausdorff spaces, or some other “convenient” category, it does work and you get a something like a topological fundamental group; (2) if you insist on the category of topological spaces, while multiplication is not always continous, it is continuous in each variable separately and taking inverses is continous as well (I think this is called being a quasi-topological group or something like that).

• Sure, but describing the result more accurately would’ve made the title less punchy.

• This is perhaps a bit off topic for the post, but I’d like to comment on Omar’s comment. One should not be adding topologies to homotopy groups without an intended application in mind. Different topologies could be useful for different things and there are many naturally arising choices. Some arise from shape theoretic constructions. The quotient topology is another but it leaves us with a quasitopological group in which multiplication may fail to be jointly continuous. Perhaps the most useful would be to take the finest group topology contained in the quotient topology – this is truly a “topological fundamental group” as most classical results translate directly into the topological group category (essential surjectivity of pi_1, van Kampen, Nielsen Schreier theorems, etc.) This last topology has provided (and continues to provide) solutions to long standing problems in topological group theory. From an algebraic topology standpoint it is often a good idea to work in a convenient category but in this particular case doing so is actually a bit naive.

5. What about the action by $\pi_1$ on the higher homotopy groups?

• That isn’t structure that one can talk about using only one homotopy group at a time; as I mentioned in the post, once you allow yourself to talk about more than one homotopy group at a time you get more structure (the corresponding multisorted Lawvere theory is very complicated since e.g. it contains the unstable homotopy groups of spheres as a subtheory!). The result in the post rules out the possibility that, say, $\pi_2$ is always canonically a module over some fixed ring $R$ (other than $\mathbb{Z}$), but it doesn’t rule out the possibility that it’s canonically acted on by something else that also depends on the space.

6. Nice! I think wedge sums are usually \vee rather than \wedge, though. (\wedge is usually the smash product.)

• Oops. That always confuses me.

• The mnemonic I use is that \vee is usually some kind of coproduct (join, logical or, etc.) and \wedge is usually some kind of product (meet, logical and, etc.).

• If we’re sharing mnemonics, mine is that \wedge is used for the exterior *product* of differential forms, so on spaces it should be something like the product, too.