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\} $, formed in the category of simplicial sets. We regard $B/A$ as a pointed simplicial set, with base point given by the vertex $q$.

### 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.

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 $q$. 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\} $, 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 3.2.1.10). 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 $h: \Delta ^1 \times \Delta ^ n \rightarrow X$ such that $\sigma = h|_{ \{ 0\} \times \Delta ^ n }$, $\tau = h|_{ \{ 1\} \times \Delta ^ n }$, 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 1.2.1.8). 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 1.4.3.1); see Corollary 1.4.3.7.

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 of $\Delta ^ n / \operatorname{\partial \Delta }^ n$. If $(X,x)$ is a pointed Kan complex, then precomposition with $f$ induces a trivial Kan fibration $\operatorname{Fun}( Q, X ) \rightarrow \operatorname{Fun}( \Delta ^ n / \operatorname{\partial \Delta }^ n, X)$ (Theorem 3.1.3.5), hence also a trivial Kan fibration

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 Proposition 1.1.4.13). 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 1.4.4.1), 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 $\pi _{\leq 1}(X) = \mathrm{h} \mathit{X}$.

Example 3.2.2.13. Let $\mathcal{G}$ be a groupoid and let $x$ be an object of $\mathcal{G}$, which we identify with a vertex of the Kan complex $X = \operatorname{N}_{\bullet }( \mathcal{G} )$ (see Proposition 1.3.5.2). Then:

The set $\pi _{0}(X) = \pi _{0}(X,x)$ can be identified with the collection of isomorphism classes of objects of $\mathcal{G}$.

The fundamental group $\pi _{1}(X,x)$ can be identified with the automorphism group $\operatorname{Aut}_{ \mathcal{G}}(x)$ of $x$ as an object of $\mathcal{G}$.

The homotopy groups $\pi _{n}(X,x)$ are trivial for $n \geq 2$, since an $n$-simplex $\sigma : \Delta ^ n \rightarrow X$ is determined by the restriction $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ (see Exercise 1.3.1.5).

Warning 3.2.2.14. 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 3.2.2.15. 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

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 3.2.2.16 (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 $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. We can therefore regard the construction $(X,x) \mapsto \pi _{n}(X,x)$ as a functor from the category of pointed Kan complexes to the category of groups. Moreover, this functor preserves filtered colimits.

Remark 3.2.2.17 (Homotopy Invariance). In the situation of Remark 3.2.2.16, suppose that $f: X \rightarrow Y$ is a homotopy equivalence. It follows from Proposition 3.2.1.13 that the homotopy class $[f]$ determines an isomorphism from $(X,x)$ to $(Y,y)$ in the pointed homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}_{\ast }$. In particular, the induced map $\pi _{n}(X,x) \rightarrow \pi _{n}(Y,y)$ is an isomorphism of groups for all $n > 0$ (and a bijection of sets for $n = 0$).

Example 3.2.2.18 (Independence of Base Point). Let $X$ be a Kan complex and let $e: x \rightarrow y$ be an edge of $X$. Then evaluation at the vertices $0,1 \in \Delta ^1$ determines a diagram of pointed Kan complexes $(X,x) \xleftarrow { \operatorname{ev}_0 } ( \operatorname{Fun}( \Delta ^1, X), e) \xrightarrow { \operatorname{ev}_1 } (X,y)$, where the underlying maps are trivial Kan fibrations (Corollary 3.1.3.6). Applying For each $n > 0$, Remark 3.2.2.17 then supplies isomorphisms of homotopy groups

Warning 3.2.2.19. Let $X$ be a Kan complex and let $n > 0$ be an integer. It follows from Example 3.2.2.18 that if two vertices $x,y \in X$ belong to the same connected component of $X$, then the homotopy groups $\pi _{n}(X,x)$ and $\pi _{n}(X,y)$ are isomorphic. Beware that, in general, there is no *canonical* isomorphism between $\pi _{n}(X,x)$ and $\pi _{n}(X,y)$: the isomorphism constructed in Example 3.2.2.18 depends on (the homotopy class) of the chosen edge $e: x \rightarrow y$.

Remark 3.2.2.20. 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 3.2.2.14 above).

Remark 3.2.2.21. 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^{n+1}_0(\tau ) = \sigma $, $d^{n+1}_1(\tau ) = \sigma '$, and $d^{n+1}_{i}(\tau )$ is the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $2 \leq i \leq n+1$.

Exercise 3.2.2.22 (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 3.2.2.21).

- $(2)$
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