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3.2.2 The Homotopy Groups of a Kan Complex

Let $X$ be a topological space and let $x \in X$ be a point. For every positive integer $n$, we let $\pi _{n}(X,x)$ denote the set of homotopy classes of pointed maps $( S^ n, x_0) \rightarrow (X,x)$, where $S^ n$ denotes a sphere of dimension $n$ and $x_0 \in S^ n$ is a chosen base point. The set $\pi _{n}(X,x)$ can be endowed with the structure of a group, which we refer to as the $n$th homotopy group of $X$ (with respect to the base point $x$). Note that the sphere $S^ n$ can be realized as the quotient space $| \Delta ^ n | / | \operatorname{\partial \Delta }^ n |$, obtained from the topological simplex $| \Delta ^ n |$ by collapsing its boundary to the point $q$. We can therefore identify pointed maps $(S^ n, x_0) \rightarrow (X,x)$ with maps of simplicial sets $f: \Delta ^ n \rightarrow \operatorname{Sing}_{\bullet }(X)$ which carry the boundary $\operatorname{\partial \Delta }^ n$ to the simplicial subset $\{ x\} \subseteq \operatorname{Sing}_{\bullet }(X)$. In [MR0111032], Kan elaborated on this observation to give a direct construction of the homotopy group $\pi _{n}(X,x)$ in terms of the simplicial set $\operatorname{Sing}_{\bullet }(X)$ (and the vertex $x$). Moreover, his construction can be applied directly to any Kan complex.

Notation 3.2.2.1. Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset. We let $B/A$ denote the pushout $B \coprod _{A} \{ q_0\} $, formed in the category of simplicial sets. We regard $B/A$ as a pointed simplicial set, with base point given by the vertex $q_0$.

Remark 3.2.2.2. Let $B$ be a simplicial set and let $A$ be a simplicial subset. Then the simplicial set $B/A$ can be described more informally as follows: it is obtained from $B$ by collapsing the simplicial subset $A \subseteq B$ to a single vertex $q_0$. Beware that this informal description is a bit misleading when $A = \emptyset $: in this case, the natural map $B \rightarrow B/A$ is not surjective (instead, $B/A$ can be described as the coproduct $B_{+} = B \coprod \{ q_0 \} $, obtained from $B$ by adding a new base point).

Example 3.2.2.3. For $n \geq 0$, the geometric realization $| \Delta ^ n / \operatorname{\partial \Delta }^ n |$ can be obtained from the topological $n$-simplex $| \Delta ^ n |$ by collapsing the boundary $| \operatorname{\partial \Delta }^ n |$ to a point (or by adding a new base point, in the degenerate case $n = 0$). It follows that $| \Delta ^ n / \operatorname{\partial \Delta }^ n |$ is homeomorphic to a sphere of dimension $n$.

Construction 3.2.2.4. Let $(X,x)$ be a pointed Kan complex and let $n$ be a nonnegative integer. We let $\pi _{n}(X,x)$ denote the set $[ \Delta ^ n / \operatorname{\partial \Delta }^ n, X]_{\ast }$ of pointed homotopy classes of maps from $\Delta ^ n / \operatorname{\partial \Delta }^ n$ to $X$ (Notation 3.2.1.8). For $n > 0$, we will refer to $\pi _{n}(X,x)$ as the $n$th homotopy group of $X$ with respect to the base point $x$ (see Theorem 3.2.2.10 below). In the special case $n = 1$, we refer to $\pi _{1}(X,x)$ as the fundamental group of $X$ with respect to the base point $x$.

Notation 3.2.2.5. Let $(X,x)$ be a pointed Kan complex and let $n$ be a nonnegative integer. Then the set of pointed morphisms $\Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow X$ can be identified with the set of $n$-simplices $\sigma : \Delta ^ n \rightarrow X$ having the property that $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$. In this case, we write $[\sigma ]$ for the image of $\sigma $ in the set $\pi _{n}(X,x)$. Note that, if $\tau $ is another $n$-simplex of $X$ for which $\tau |_{ \operatorname{\partial \Delta }^ n}$ is the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$, then the equality $[\sigma ] = [\tau ]$ holds in $\pi _{n}(X,x)$ if and only if there exists a homotopy $h: \Delta ^1 \times \Delta ^ n \rightarrow X$ satisfying $\sigma = h|_{ \{ 0\} \times \Delta ^ n }$, $\sigma ' = h|_{ \{ 1\} \times \Delta ^ n }$, and $h|_{ \Delta ^1 \times \operatorname{\partial \Delta }^ n }$ is the constant map taking the value $x$.

Example 3.2.2.6. Let $(X,x)$ be a pointed Kan complex. Then $\pi _{0}(X,x)$ can be identified with the set $\pi _0(X)$ of connected components of $X$ (Definition 1.1.6.8). Beware that, unlike the higher homotopy groups $\{ \pi _{n}(X,x) \} _{n \geq 1}$, there is no naturally defined group structure on $\pi _{0}(X,x)$.

Example 3.2.2.7. Let $X$ be a topological space and let $x \in X$ be a base point, which we identify with a vertex of the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$. For every positive integer $n$, we can identify $\pi _{n}( \operatorname{Sing}_{\bullet }(X), x)$ with the set $\pi _{n}(X,x)$ of (pointed) homotopy classes of maps from the sphere $S^ n \simeq | \Delta ^ n / \operatorname{\partial \Delta }^ n |$ into $X$.

Example 3.2.2.8. Let $X$ be a Kan complex, let $x$ be a vertex of $X$, and let $e,e': x \rightarrow x$ be edges of $X$ which begin and end at the vertex $x$. Then the equality $[e] = [e']$ holds in the fundamental group $\pi _{1}(X,x)$ if and only if $e$ is homotopic to $e'$ as a morphism in the $\infty $-category $X$ (in the sense of Definition 1.3.3.1); see Corollary 1.3.3.7.

Remark 3.2.2.9. Let $n$ be a nonnegative integer. By virtue of Corollary 3.1.7.2, there exists an anodyne morphism $f: \Delta ^ n / \operatorname{\partial \Delta }^ n \rightarrow Q$, where $Q$ is a Kan complex. Let $q \in Q$ denote the image of the base point $q_0$ of $\Delta ^ n / \operatorname{\partial \Delta }^ n$. If $(X,x)$ is a pointed Kan complex, then precomposition with $f$ induces a trivial Kan fibration $\operatorname{Fun}( Q, X ) \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n, X)$ (Proposition 3.1.6.10), hence also a trivial Kan fibration

\[ \operatorname{Fun}(Q,X) \times _{ \operatorname{Fun}( \{ q\} , X) } \{ x\} \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n ) \times _{ \operatorname{Fun}( \{ q_0 \} , X) } \{ x\} . \]

Passing to connected components, we see that $f$ induces a bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }}( Q, X ) \simeq \pi _{n}(X,x)$. In other words, the functor $(X,x) \mapsto \pi _{n}(X,x)$ is corepresentable (in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$) by the pointed Kan complex $(Q,q)$ (which can be regarded as a combinatorial incarnation of the $n$-sphere).

Theorem 3.2.2.10. Let $(X,x)$ be a pointed Kan complex and let $n$ be a positive integer. Then there is a unique group structure on the set $\pi _{n}(X,x)$ with the following properties:

$(a)$

Let $e: \Delta ^{n} \rightarrow \{ x\} \rightarrow X$ be the constant map. Then the homotopy class $[e]$ is the identity element of $\pi _{n}(X,x)$.

$(b)$

Let $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ be a morphism of simplicial sets, corresponding to a tuple $(\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ of $n$-simplices of $X$ (see Exercise 1.1.2.8). Assume that each restriction $\sigma _{i}|_{ \operatorname{\partial \Delta }^ n }$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$. Then $f$ extends to a map $\Delta ^{n+1} \rightarrow X$ if and only if the product

\[ [ \sigma _0 ]^{-1} [ \sigma _1 ] [ \sigma _2 ]^{-1} [ \sigma _3 ] \cdots [ \sigma _{n+1} ]^{ (-1)^{n} } \]

is equal to the identity element of $\pi _{n}(X,x)$.

Moreover, if $n \geq 2$, then the group $\pi _{n}(X,x)$ is abelian.

We will give the proof of Theorem 3.2.2.10 in ยง3.2.3.

Exercise 3.2.2.11. Show that when $n > 0$ is odd, condition $(a)$ of Theorem 3.2.2.10 follows from condition $(b)$ (beware that this is not true when $n$ is even).

Example 3.2.2.12. In the special case $n=1$, we can rewrite condition $(b)$ of Theorem 3.2.2.10 as follows:

  • Let $f$, $g$, and $h$ be edges of $X$ which begin and end at the vertex $x$. Then the equality $[h] = [g] [f]$ holds (in the fundamental group $\pi _{1}(X,x)$) if and only if there exists a $2$-simplex $\sigma $ of $X$ which witnesses $h$ as a composition of $f$ and $g$ (in the sense of Definition 1.3.4.1), as indicated in the diagram

    \[ \xymatrix@R =50pt@C=50pt{ & x \ar [dr]^{g} & \\ x \ar [ur]^{f} \ar [rr]^{h} & & x. } \]

It follows that the fundamental group $\pi _{1}(X,x)$ can be identified with the automorphism group of $x$ as an object of the fundamental groupoid $\pi _{\leq 1}(X) = \mathrm{h} \mathit{X}$.

Warning 3.2.2.13. Let $(X,x)$ be a pointed Kan complex, so that $\pi _{1}(X,x)$ can be identified with the set $\operatorname{Hom}_{\pi _{\leq 1}(X) }(x,x)$ of homotopy classes of paths from $x$ to itself. We have adopted the convention that the multiplication on $\pi _{1}(X,x)$ is given by composition in the homotopy category $\mathrm{h} \mathit{X}$. In other words, if $f,g: x \rightarrow x$ are edges which begin and end at $x$, then the product $[g] [f] \in \pi _{1}(X,x)$ is the homotopy class of a path which can be described informally as traversing the path $f$ first, followed by the path $g$. Beware that the opposite convention is also common in the literature (note that his issue is irrelevant for the higher homotopy groups $\{ \pi _{n}(X,x) \} _{n \geq 2}$, since they are abelian).

Remark 3.2.2.14. Let $(X,x)$ be a pointed Kan complex. For $n \geq 2$, the homotopy group $\pi _{n}(X,x)$ is abelian. We will generally emphasize this by using additive notation for the group structure on $\pi _{n}(X,x)$: that is, we denote the group law by

\[ +: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x) \quad \quad (\xi , \xi ') \mapsto \xi + \xi '. \]

With this convention, we can restate property $(b)$ of Theorem 3.2.2.10 as follows:

$(b)$

Let $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ be a morphism of simplicial sets, corresponding to a tuple $(\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ of $n$-simplices of $X$. Then $f$ extends to an $(n+1)$-simplex of $X$ if and only if the sum $\sum _{i = 0}^{n+1} (-1)^{i} [ \sigma _ i ]$ vanishes in $\pi _{n}(X,x)$.

Remark 3.2.2.15 (Functoriality). Let $f: X \rightarrow Y$ be a morphism of Kan complexes, let $x$ be a vertex of $X$, and set $y = f(x)$. For each $n \geq 1$, the morphism $f$ induces a homomorphism $\pi _{n}(f): \pi _{n}(X,x) \rightarrow \pi _{n}(Y,y)$, characterized by the formula $\pi _{n}(f)( [\sigma ] ) = [ f(\sigma ) ]$ for each $n$-simplex $\sigma $ of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$.

Remark 3.2.2.16. Let $X$ be a Kan complex and let $x$ be a vertex of $X$. Then $x$ can also be regarded as a vertex of the opposite simplicial set $X^{\operatorname{op}}$, which is also a Kan complex. For $n \geq 1$, we have an evident bijection $\varphi : \pi _{n}(X,x) \simeq \pi _{n}(X^{\operatorname{op}}, x)$. If $n \geq 2$, then this bijection is an isomorphism of abelian groups. Beware that, in the case $n=1$, it is generally not an isomorphism of groups: instead, it is an anti-isomorphism (that is, it satisfies the identity $\varphi ( \xi \xi ' ) = \varphi (\xi ') \varphi (\xi )$ for $\xi , \xi ' \in \pi _{1}(X,x)$; see Warning 3.2.2.13 above).

Remark 3.2.2.17. Let $(X,x)$ be a pointed Kan complex and let $n$ be a positive integer. Suppose that $\sigma , \sigma ': \Delta ^{n} \rightarrow X$ are $n$-simplices of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ and $\sigma '|_{ \operatorname{\partial \Delta }^{n} }$ are equal to the constant map $\operatorname{\partial \Delta }^{n} \rightarrow \{ x\} \subseteq X$. It follows from Theorem 3.2.2.10 that the equality $[ \sigma ] = [\sigma ']$ holds (in the homotopy group $\pi _{n}(X,x)$) if and only if there exists an $(n+1)$-simplex $\tau $ of $X$ such that $d_0(\tau ) = \sigma $, $d_1(\tau ) = \sigma '$, and $d_{i}(\tau )$ is the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $2 \leq i \leq n+1$.

Exercise 3.2.2.18 (Homotopy of Eilenberg-MacLane Spaces). Let $M_{\ast }$ be a chain complex of abelian groups and let $X = \mathrm{K}(M_{\ast })$ be the associated Eilenberg-MacLane space (Construction 2.5.6.3). Let $x \in X$ be the vertex corresponding to the zero element, and let $n$ be a positive integer. Note that a pointed map from $\Delta ^ n / \operatorname{\partial \Delta }^ n$ to $X$ can be identified with a map of chain complexes $\mathrm{N}_{\ast }( \Delta ^ n, \operatorname{\partial \Delta }^ n; \operatorname{\mathbf{Z}}) \simeq \operatorname{\mathbf{Z}}[n] \rightarrow M_{\ast }$: in other words, it can be identified with an $n$-cycle of the chain complex $M_{\ast }$, which we will denote by $\overline{\sigma }$.

$(1)$

Let $\sigma , \sigma ': \Delta ^ n \rightarrow X$ be $n$-simplices whose restriction to $\operatorname{\partial \Delta }^ n$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. Show that $[\sigma ] = [\sigma ']$ in $\pi _{n}(X,x)$ if and only if $\overline{\sigma }$ and $\overline{\sigma }'$ are homologous as $n$-cycles of $M_{\ast }$ (use Remark 3.2.2.17).

$(2)$

Show that the $[\sigma ] \mapsto [\overline{\sigma }]$ induces an isomorphism from $\pi _{n}(X,x)$ to the homology group $\mathrm{H}_{n}(M)$.

In particular, if $A$ is an abelian group and $m \geq 0$ is an integer, then the homotopy groups of the Eilenberg-MacLane space $X = \mathrm{K}(A,m)$ are given by

\[ \pi _{n}(X,x) = \begin{cases} A & \text{ if } n=m \\ 0 & \text{ otherwise. } \end{cases} \]