Contents

Idea

The homotopy groups $\pi_n(X,x)$ of a pointed topological space $(X,x)$ are a sequence of groups that generalise the fundamental group $\pi_1(X,x)$ to higher homotopies.

The $n$th homotopy group $\pi_n(X,x)$ has as elements equivalence classes of spheres $\gamma : S_*^n \to X_*$ in $X$ where two such are regarded as equivalent if there is a left homotopy $\gamma \Rightarrow \gamma'$ between them, fixing the base point. The group operation is given by gluing of two spheres at their basepoint.

In degree 0, $\pi_0(X,x)$ is not a group but merely a pointed set. In degree $n \geq 2$ all homotopy groups are abelian groups. Only $\pi_1(X,x)$ may be an arbitrary group. In general, $\pi_n(X,x)$ is an $n$-tuply groupal set.

Lower homotopy groups act on higher homotopy groups; the nonabelian group cohomology of this gives the Postnikov invariants of the space. All of this data put together allows one to reconstruct the original space, at least up to weak homotopy type, through its Postnikov system.

These definitions only depend on the homotopy type of $X$, by definition: a weak homotopy equivalence between topological spaces is a continuous map that induces an isomorphism on all homotopy groups.

Accordingly, homotopy groups are defined for all other models of homotopy types, notably for simplicial sets. See at simplicial homotopy group for more.

There are generalizations of the concept to stable homotopy groups of spectra and to equivariant homotopy groups for topological G-spaces and equivariant spectra.

Definition

For topological spaces

Let $X$ be a topological space, let $x : * \to X$ be a point, to be called the base point.

For $n \in \mathbb{N}$, let $S^n$ be the pointed $n$-sphere.

Definition

The underlying set of $\pi_n(X,x)$ is the set of equivalence classes of basepoint-preserving continuous functions

$\gamma : S^n \to X$

where two such are regarded as equivalent if there is a basepoint-preserving left homotopy between them.

Now we will put some structure on that set.

Definition

There are $n$ independent equators through the basepoint of $S^n$. Given two maps $f, g: S^n \to (X,x)$, form their copairing in the category of pointed spaces to get a map

$S^n \vee S^n \to (X,x)$

(where $\vee$ indicates the wedge sum); then combine this with a map $S^n \to S^n \vee S^n$ that maps the $i$th equator to the basepoint and each hemisphere to one copy of the sphere. The result is a map $S^n \to (X,x)$, called the $i$th concatenation of $f$ and $g$:

$S^n \to_i S^n \vee S^n \stackrel{[f,g]}{\to} (X,x) \,.$

One can check that each of these operations respects homotopy equivalence and hence equips $\pi_n(X,x)$ with the structure of a magma.

Proposition

These magmas are in fact groups; in particular:

• the constant function that maps all of $S^n$ to $x$ represents the null element of $\pi_n(X,x)$, which is an identity for every concatenation.
Remark

This is closely related to the statement that the positive dimension spheres are H-cogroup objects (see there) in the homotopy category of pointed topological spaces.

Definition

This may seem like quite a complicated kind of structure, but it is actually quite simple up to homotopy. First of all, all $n$ concatenations of given maps $f$ and $g$ are homotopic, so we speak of simply a single concatenation for $n \geq 1$ (and none for $n = 0$). By the Eckmann-Hilton argument, this concatenation will be commutative up to homotopy for $n \geq 2$. In any case, it is associative and invertible up to homotopy, and the null element is an identity up to homotopy.

Proposition

The result is that the set $\pi_n(X,x)$ of equivalence classes is an abelian group for $n \geq 2$, a group for $n = 1$, and a pointed set for $n = 0$ (when the null element is the only structure).

Proposition

If $X$ is path-connected, then all of the $\pi_n(X,a)$ are isomorphic. Accordingly, it's traditional to just write $\pi_n(X)$ in that case. (This is why we must use $\Pi_n(X)$ for the homotopy $n$-groupoid.) However, there may be many different isomorphisms between $\pi_n(X,a)$ and $\pi_n(X,b)$ (given by $\pi_{n+1}(X)$), so a more careful treatment requires keeping track of the basepoint even in the connected case.

For objects in a general $\infty$-stack $(\infty,1)$-topos

Top is the archetypical (∞,1)-topos.

The definition of homotopy groups for objects in Top is just a special case of a general definition of homotopy groups of objects of ∞-stack (∞,1)-toposes.

This is described in detail at

Truncated and connected spaces

Often it is useful to talk about spaces whose homotopy groups are all trivial above or below a certain degree, for instance in the context of Postnikov towers and Whitehead towers.

For $n = -1, 0, 1, 2, \ldots, \infty$:

• a space is $n$-truncated? if all homotopy groups above degree $n$ are trivial.

• a space is $n$-connected (or $n$-simply connected) if it is inhabited and all homotopy groups at or below degree $n$ are trivial.

Vacuously, every space is $\infty$-truncated, and precisely the inhabited spaces are $(-1)$-connected. On the other end, precisely the weakly contractible spaces are $\infty$-connected, and a space is $(-1)$-truncated iff it is weakly contractible if inhabited. (So classically, using excluded middle, a space is $(-1)$-truncated iff it is either empty or weakly contractible.)

To extend one step further in negative thinking, every space (even the empty space) is $(-2)$-connected, and precisely a weakly contractible space (but not the empty space) is $(-2)$-truncated.

• A pointed space is a degree-$n$ Eilenberg?MacLane space if it is both $n$-truncated as well as $(n-1)$-connected, i.e. if its only nontrivial homotopy group is in degree $n$.

A weakly contractible space is an Eilenberg–MacLane space in every degree, and these are the only Eilenberg–MacLane spaces in degree $\infty$ or $-1$. In degree $0$, they are the pointed discrete spaces (and those weakly homotopy equivalent to such). In degree $1$, they are (up to weak homotopy equivalence) precisely the classifying spaces of groups. And so on.

Examples

In low dimensions

The $0$th homotopy ‘group’ $\pi_0(X,a)$ can be identified with the set of all path components of $X$, with the component containing $a$ as the basepoint. Similarly, the fundamental 0-groupoid $\Pi_0(X)$ is the set of all path components without a chosen basepoint. Note that $\Pi_0(X)$ is traditionally written $\pi_0(X)$, even without a basepoint.

The $1$st homotopy group $\pi_1(X,a)$ is precisely the fundamental group of $X$ at $a$. This is the original example from which all others derived. It was once written simply $\pi(X,a)$ with the $\pi$ standing for Poincaré, who invented it.

At least, that's where I think that it comes from … —Toby

Of the circle

The first homotopy group of the circle $S^1$ is the group of integers.

$\pi_1(S^1) \simeq \mathbb{Z} \,.$

A formalization of a proof of this in homotopy type theory is in (Shulman).

History

In the early years of the 20th century it was known that the nonabelian fundamental group $\pi_1(X,a)$ of a space $X$ with base point $a$ was useful in geometry and complex analysis. It was also known that the abelian homology groups $H_n(X)$ existed for all $n \geq 0$ and that if $X$ is connected then $H_1(X)$ is isomorphic to the abelianisation of any $\pi_1(X,a)$.

Consequently it was hoped to generalise the fundamental group to higher dimensions, producing nonabelian groups whose abelianisations would be the homology groups.

In 1932, E. Čech proposed a definition of higher homotopy groups using maps of spheres, but the paper was rejected for the Zurich ICM since it was found that these groups $\pi_n(X,a)$ were abelian for $n \geq 2$, and so do not generalise the fundamental group in the way that was originally desired. Nonetheless, they have proved to be extremely important in homotopy theory, although more difficult to compute in general than homology groups. See weak homotopy equivalence.

Properties

It was early realised that the fundamental groupoid $\Pi_1(X)$ operates on the family of groups $\{\pi_n(X,a) | a \in X\}$ which should thus together be regarded as a module over $\pi_1(X,a)$.

A key property of homotopy groups is the Whitehead theorem: if $f:X \to Y$ is a map of connected m-cofibrant spaces (spaces each of the homotopy type of a CW complex), and $f$ induces isomorphisms $\pi_n(X,a) \to \pi_n(Y,f(a))$ for some $a$ and all $n \geq 1$, then $f$ is a homotopy equivalence.

However, the homotopy groups by themselves, even considering the operations of $\pi_1$, do not characterise homotopy types. See also algebraic homotopy theory.

Some general nonsense

Using the Eckmann-Hilton duality between cohomology and homotopy (as an operation) one may discuss homotopy groups along the same lines as the discussion of cohomology groups (see there).

From that perspective we might say that:

for $B, X$ any two objects in an (∞,1)-topos $\mathbf{H}$, the “homotopy of $X$ with co-coefficients $B$” is the hom-set

$\pi_H(X,B) := H(B,X) := \pi_0 \mathbf{H}(B,X) \,,$

where $\pi_H$ denotes the homotopy category of $\mathbf{H}$.

For the special case that the object $B$ here is a co-group object, this homotopy set $\pi(X,B)$ naturally inherits the structure of a group.

The standard example is that where $B = S^n$ is the $n$-sphere. This naturally comes with an co-group structure up to homotopy, which is precisely the structure underlying the co-category structure of the interval object and more generally that underlying the mechanism of the Trimble n-category.

As opposed to cohomology where people are used to talking about generalized cohomology, “homotopy” usually just means this ordinary homotopy for $B = S^n$.

General

With an eye towards application in mathematical physics:

Formalization in HoTT

Homotopy groups and their properties can naturally be formalized in homotopy type theory. In this context a proof that $\pi_1(S^1)\simeq \mathbb{Z}$ is in

and a proof that

$\pi_k(S^n) \simeq \left\{ \array{ 0 & for \;k \lt n \\ \mathbb{Z} & for\; k = n} \right.$

is in