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3.2.3 The Group Structure on $\pi _{n}(X,x)$

Let $(X,x)$ be a pointed Kan complex and let $n \geq 2$ be an integer, which we regard as fixed throughout this section. Our goal is to give a proof of Theorem 3.2.2.10, which supplies a group structure on the set $\pi _{n}(X,x) = [ \Delta ^ n / \operatorname{\partial \Delta }^ n, X]_{\ast }$ (note that the case $n=1$ of Theorem 3.2.2.10 is subsumed in our construction of the homotopy category $\pi _{\leq 1}(X) = \mathrm{h} \mathit{X}$, by virtue of Example 3.2.2.12).

Notation 3.2.3.1. Let $\Sigma $ denote the collection of all $n$-simplices $\sigma : \Delta ^ n \rightarrow X$ having the property that the restriction $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is equal to the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \subseteq X$. We let $e \in \Sigma $ denote the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$. Note that an $(n+2)$-tuple $\vec{\sigma } = ( \sigma _0, \sigma _1, \ldots , \sigma _{n+1} )$ of elements of $\Sigma $ can be identified with a map of simplicial sets $f: \operatorname{\partial \Delta }^{n+1} \rightarrow X$, having the property that the restriction of $f$ to the $(n-1)$-skeleton of $\operatorname{\partial \Delta }^{n+1}$ is equal to the constant map $\operatorname{sk}_{n-1}( \operatorname{\partial \Delta }^{n+1} ) \rightarrow \{ x\} \subseteq X$ (see Exercise 1.1.2.8). We will say that a tuple $\vec{\sigma }$ bounds if $f$ can be extended to an $(n+1)$-simplex of $X$: that is, if there exists an $(n+1)$-simplex $\tau $ of $X$ satisfying $\sigma _{i} = d_ i(\tau )$ for $0 \leq i \leq n+1$.

The construction $\sigma \mapsto [ \sigma ]$ then determines a surjective map $\Sigma \twoheadrightarrow \pi _{n}(X,x)$. We will say that a pair of elements $\sigma , \sigma ' \in \Sigma $ are homotopic if $[\sigma ] = [ \sigma ' ]$ (that is, if there is a homotopy from $\sigma $ to $\sigma '$ which is constant along the boundary $\operatorname{\partial \Delta }^ n$).

Lemma 3.2.3.2. Let $\vec{\sigma } = (\sigma _0, \sigma _1, \ldots , \sigma _{n+1})$ be an $(n+2)$-tuple of elements of $\Sigma $. The condition that $\vec{\sigma }$ bounds depends only on the sequence of homotopy classes $\{ [ \sigma _ i ] \in \pi _{n}(X,x) \} _{0 \leq i \leq n+1}$. In other words, if $\vec{\sigma }' = (\sigma '_{0}, \sigma '_{1}, \ldots , \sigma '_{n+1})$ is another $(n+2)$-tuple of elements of $\Sigma $ satisfying $[\sigma '_ i] = [\sigma _ i]$ for $0 \leq i \leq n+1$ and $\vec{\sigma }$ bounds, then $\vec{\sigma }'$ also bounds.

Proof. Let us identify $\vec{\sigma }$ and $\vec{\sigma }'$ with morphisms of simplicial sets $f,f': \operatorname{\partial \Delta }^{n+1} \rightarrow X$ (carrying the $(n-1)$-skeleton of $\operatorname{\partial \Delta }^{n+1}$ to the vertex $x$). For $0 \leq i \leq n+1$, the equality $[\sigma _ i] = [ \sigma '_{i} ]$ allows us choose a homotopy $h_{i}: \Delta ^{1} \times \Delta ^{n} \rightarrow X$ from $\sigma _{i}$ to $\sigma '_{i}$ which carries $\Delta ^{1} \times \operatorname{\partial \Delta }^{n}$ to the vertex $\{ x\} \subseteq X$. These maps can be amalgamated to a homotopy $h$ from $f$ to $f'$: that is, an edge joining $f$ to $f'$ in the simplicial set $\operatorname{Fun}( \operatorname{\partial \Delta }^{n+1}, X)$. If $\vec{\sigma }$ bounds, then $f$ can be extended to an $(n+1)$-simplex $\tau : \Delta ^{n+1} \rightarrow X$. Since $X$ is a Kan complex, the restriction map $\operatorname{Fun}( \Delta ^{n+1}, X) \rightarrow \operatorname{Fun}( \operatorname{\partial \Delta }^{n+1}, X)$ is a Kan fibration (Corollary 3.1.3.3), so $h$ can be extended to a homotopy $\widetilde{h}$ from $\tau $ to another map $\tau ': \Delta ^{n+1} \rightarrow X$ satisfying $\tau '|_{ \operatorname{\partial \Delta }^{n+1} } = f'$. It follows that the tuple $\vec{\sigma }'$ also bounds. $\square$

Remark 3.2.3.3. Let $\vec{\eta } = (\eta _0, \eta _1, \ldots , \eta _{n+1} )$ be an $(n+2)$-tuple of elements of $\pi _{n}(X,x)$, so that we can write $\eta _ i = [ \sigma _ i ]$ for some $n$-simplex $\sigma _{i} \in \Sigma $. We will say that the tuple of homotopy classes $\vec{\eta }$ bounds if the tuple of simplices $\vec{\sigma } = ( \sigma _0, \sigma _1, \ldots , \sigma _{n+1} )$ bounds, in the sense of Notation 3.2.3.1. By virtue of Lemma 3.2.3.2, this condition is independent of the choice of $\vec{\sigma }$.

With this terminology, Theorem 3.2.2.10 asserts (in the case $n \geq 2$) that there is a unique abelian group structure on the set $\pi _{n}(X,x)$ with the following pair of properties:

$(a)$

The identity element of $\pi _{n}(X,x)$ is the homotopy class $[e]$.

$(b)$

An $(n+2)$-tuple $\vec{\eta } = (\eta _0, \eta _1, \ldots , \eta _{ n+1} )$ bounds if and only if the sum $\sum _{i=0}^{n+1} (-1)^{i} \eta _ i$ vanishes in $\pi _{n}(X,x)$.

Lemma 3.2.3.4. Let $0 \leq i \leq n+1$, and suppose we are given a collection of homotopy classes $\{ \eta _ j \in \pi _{n}(X,x) \} _{0 \leq j \leq n+1, j \neq i}$. Then there is a unique element $\eta _{i} \in \pi _{n}(X,x)$ for which the tuple $\vec{\eta } = ( \eta _0, \eta _1, \ldots , \eta _{n+1} )$ bounds.

Proof. For $j \neq i$, choose an element $\sigma _{j} \in \Sigma $ satisfying $[ \sigma _ j ] = \eta _ j$. Then the tuple of $n$-simplices $( \sigma _0, \ldots , \sigma _{i-1}, \bullet , \sigma _{i+1}, \ldots , \sigma _{n+1} )$ determines a map of simplicial sets $f_0: \Lambda ^{n+1}_{i} \rightarrow X$ (see Exercise 1.1.2.14). Since $X$ is a Kan complex, we can extend $f_{0}$ to an $(n+1)$-simplex $\tau $ of $X$. Then $\eta _{i} = [ d_ i( \tau ) ]$ has the property that the tuple $\vec{\eta } = ( \eta _0, \eta _1, \ldots , \eta _{n+1} )$ bounds. This proves existence. To prove uniqueness, suppose we are given another element $\eta '_{i} \in \pi _{n}(X,x)$ for which the tuple $( \eta _0, \ldots , \eta _{i-1}, \eta '_{i}, \eta _{i+1}, \ldots , \eta _{n+1} )$ bounds. Write $\eta '_{i} = [ \sigma '_{i} ]$ for some $\sigma '_{i} \in \Sigma $, so that we can choose a simplex $\tau ': \Delta ^{n+1} \rightarrow X$ satisfying

\[ d_{j}( \tau ' ) = \begin{cases} \sigma '_{i} & \text{ if $j = i$ } \\ \sigma _{j} & \text{ otherwise. } \end{cases} \]

Since the inclusion $\Lambda ^{n+1}_{i} \hookrightarrow \Delta ^{n+1}$ is anodyne, so the restriction map $\operatorname{Fun}( \Delta ^{n+1}, X) \rightarrow \operatorname{Fun}( \Lambda ^{n+1}_{i}, X)$ is a trivial Kan fibration (Corollary 3.1.6.12). It follows that there exists a homotopy from $\tau $ to $\tau '$ which is constant along the subset $\Lambda ^{n+1}_{i} \subseteq \Delta ^{n+1}$, so that $\eta _ i = [ d_ i( \tau ) ] = [ d_ i( \tau ' ) ] = \eta '_ i$. $\square$

As a special case of Lemma 3.2.3.4, we obtain several potential candidates for the composition law on $\pi _{n}(X,x)$:

Lemma 3.2.3.5. Fix $1 \leq i \leq n$. Then there is a unique function $m_ i: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x)$ with the following property:

$(\ast )$

Let $\eta _{i-1}$, $\eta _{i}$, and $\eta _{i+1}$ be elements of $\pi _{n}(X,x)$. Then the $(n+2)$-tuple

\[ ( [e], \ldots , [e], \eta _{i-1}, \eta _{i}, \eta _{i+1}, [e], \ldots , [e] ) \]

bounds if and only if $\eta _{i} = m_ i( \eta _{i-1}, \eta _{i+1} )$.

Example 3.2.3.6. Let $\sigma $ be an element of $\Sigma $, and let $1 \leq i \leq n$. Then the degenerate $(n+1)$-simplex $\tau = s_{i}(\sigma )$ satisfies $d_{j}(\tau ) = \begin{cases} \sigma & \text{ if } j \in \{ i, i+1\} \\ e & \text{ otherwise.} \end{cases}$ It follows that the multiplication map $m_ i: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x)$ of Lemma 3.2.3.5 satisfies the identity $m_ i( [e], [\sigma ] ) = [\sigma ]$. A similar argument shows that $m_ i( [\sigma ], [e] ) = [\sigma ]$.

Lemma 3.2.3.7. Let $\vec{\eta } = ( \eta _0, \eta _1, \ldots , \eta _{n+1} )$ be an $(n+2)$-tuple of elements of $\pi _{n}(X,x)$, let $1 \leq i \leq n$ be an integer, and let $\alpha $ be another element of $\pi _{n}(X,x)$. If $\vec{\eta }$ bounds, then the tuple $( \eta _0, \ldots , \eta _{i-2}, m_ i( \alpha , \eta _{i-1}), m_ i( \alpha , \eta _{i} ), \eta _{i+1}, \ldots , \eta _{n+1} )$ also bounds.

Proof. For $0 \leq i \leq n+1$, choose an element $\sigma _{i} \in \Sigma $ satisfying $[ \sigma _ i ] = \eta _ i$. Since $\vec{\eta }$ bounds, we can choose an $(n+1)$-simplex $\overline{\sigma }$ of $X$ satisfying $\sigma _ i = d_ i( \tau )$ for $0 \leq i \leq n+1$. Choose $\tau \in \Sigma $ satisfying $[\tau ] = \alpha $. Since $X$ is a Kan complex, we can choose $(n+1)$-simplices $\rho , \rho ': \Delta ^{n+1} \rightarrow X$ satisfying the identities

\[ d_{j}(\rho ) = \begin{cases} e & \text{ if } 0 \leq j < i-1 \\ \tau & \text{ if } j = i-1 \\ \sigma _{i-1} & \text{ if } j = i+1 \\ e & \text{ of } i+1 < j \leq n+1. \end{cases} \quad \quad d_{j}(\rho ') = \begin{cases} e & \text{ if } 0 \leq j < i-1 \\ \tau & \text{ if } j = i-1 \\ \sigma _{i} & \text{ if } j = i+1 \\ e & \text{ of } i+1 < j \leq n+1. \end{cases} \]

The definition of the multiplication $m_ i$ then gives $m_{i}( \alpha , \eta _{i-1} ) = [ d_ i(\rho ) ]$ and $m_ i( \alpha , \eta _ i ) = [ d_ i(\rho ')]$. The tuple $( s_{i}(\sigma _0), \ldots , s_{i}( \sigma _{i-2}), \rho , \rho ', \bullet , \overline{\sigma }, s_{i+1}(\sigma _{i+2}), \ldots , s_{i+1}( \sigma _{n+1} ) )$ then determines a map of simplicial sets $\Lambda ^{n+2}_{i+1} \rightarrow X$ (Exercise 1.1.2.14). Since $X$ is a Kan complex, this map can be extended to an $(n+2)$-simplex of $X$. Let $\overline{\sigma }'$ denote the $(i+1)$st face of this simplex. By construction, we have

\[ d_{j}( \overline{\sigma }' ) = \begin{cases} d_ i(\rho ) & \text{ if } j = i-1 \\ d_ i( \rho ' ) & \text{ if } j = i \\ \sigma _{j} & \text{otherwise,} \end{cases} \]

so that $\overline{\sigma }'$ witnesses that the tuple $( \eta _0, \ldots , \eta _{i-2}, m_ i( \alpha , \eta _{i-1}), m_ i( \alpha , \eta _{i} ), \eta _{i+1}, \ldots , \eta _{n+1} )$ bounds. $\square$

Lemma 3.2.3.8. Let $\alpha $, $\beta $, and $\gamma $ be elements of $\pi _{n}(X,x)$. For $2 \leq i \leq n$, we have $m_ i( \alpha , m_{i-1}(\beta , \gamma ) ) = m_{i-1}( \beta , m_ i(\alpha , \gamma ) )$.

Proof. Applying Lemma 3.2.3.7 to the tuple $( [e], \ldots , [e], \beta , m_{i-1}(\beta , \gamma ), \gamma , [e], \ldots , [e] )$, we deduce that the tuple $( [e], \ldots , [e], \beta , m_ i( \alpha , m_{i-1}(\beta , \gamma ) ), m_ i( \alpha , \gamma ), [e], \ldots , [e] )$ bounds, which is equivalent to the asserted identity. $\square$

Lemma 3.2.3.9. Let $\alpha $ and $\beta $ be elements of $\pi _{n}(X,x)$. For $2 \leq i \leq n$, we have $m_{i}(\alpha , \beta ) = m_{i-1}( \beta , \alpha )$.

Proof. Combining Lemma 3.2.3.8 with Example 3.2.3.6, we obtain

\[ m_ i( \alpha , \beta ) = m_ i( \alpha , m_{i-1}(\beta , [e] ) ) = m_{i-1}( \beta , m_ i( \alpha , [e] ) ) = m_{i-1}( \beta , \alpha ). \]
$\square$

Proof of Theorem 3.2.2.10. For every pair of elements $\alpha , \beta \in \pi _{n}(X,x)$, let $\alpha \beta $ denote the homotopy class $m_1(\alpha ,\beta )$, where $m_1: \pi _{n}(X,x) \times \pi _{n}(X,x) \rightarrow \pi _{n}(X,x)$ is the multiplication map of Lemma 3.2.3.5. We first note that this multiplication is associative: for every triple of elements $\alpha , \beta , \gamma \in \pi _{n}(X,x)$, Lemmas 3.2.3.9 and 3.2.3.8 yield identities

\begin{eqnarray*} \alpha (\beta \gamma ) & = & m_1( \alpha , m_1(\beta , \gamma )) \\ & = & m_1( \alpha , m_2( \gamma , \beta ) ) \\ & = & m_2( \gamma , m_1( \alpha , \beta ) ) \\ & = & m_1( m_1(\alpha , \beta ), \gamma ) \\ & = & (\alpha \beta ) \gamma . \end{eqnarray*}

Example 3.2.3.6 shows that $[e]$ is a two-sided identity with respect to multiplication. For every element $\alpha \in \pi _{n}(X,x)$, Lemma 3.2.3.4 implies that we can choose an element $\beta \in \pi _{n}(X,x)$ for which the tuple $( \alpha , [e], \beta , [e], [e], \ldots , [e])$ bounds, so that $\alpha \beta = m_1( \alpha , \beta ) = [e]$. This shows that $\alpha $ has a right inverse, and a similar argument shows that $\alpha $ has a left inverse. It follows that multiplication determines a group structure on the set $\pi _{n}(X,x)$, having $[e]$ as the identity element.

We now verify that the multiplication on $\pi _{n}(X,x)$ satisfies condition $(b)$ of Theorem 3.2.2.10. Suppose we are given an $(n+1)$-tuple $\vec{\eta } = ( \eta _0, \eta _1, \ldots , \eta _{n+1} )$ of elements of $\pi _{n}(X,x)$. We wish to show that $\vec{\eta }$ bounds if and only if the product $\eta _0^{-1} \eta _1 \eta _2^{-1} \cdots \eta _{n+1}^{(-1)^{n}}$ is equal to the identity element of $\pi _{n}(X,x)$. If $\vec{\eta } = ( [e], [e], \ldots , [e] )$, there is nothing to prove. Otherwise, there exists some smallest positive integer $i$ such that $\eta _{i-1} \neq [e]$. We proceed by descending induction on $i$. If $i > n$, we must show that $([e], [e], \ldots , [e], \eta _{n}, \eta _{n+1} )$ bounds if and only if $\eta _ n = \eta _{n+1}$, which follows from Example 3.2.3.6. Let us therefore assume that $1 \leq i \leq n$. Define $\vec{\eta }' = ( \eta '_0, \eta '_1, \ldots , \eta '_{n+1} )$ by the formula

\[ \eta '_{j} = \begin{cases} m_ i( \eta _{i-1}^{-1}, \eta _{j} ) & \text{ if $j = i-1$ or $j=i$ } \\ \eta _ j & \text{ otherwise. } \end{cases} \]

Invoking Lemma 3.2.3.9 repeatedly, we obtain

\[ \eta '_{i-1} = m_ i( \eta _{i-1}^{-1}, \eta _{i-1} ) = \begin{cases} \eta _{i-1}^{-1} \eta _{i-1} & \text{ if $i$ is odd } \\ \eta _{i-1} \eta _{i-1}^{-1} & \text{ if $i$ is even } \end{cases} = [e] \]

\[ \eta '_{i} = m_ i( \eta _{i-1}^{-1}, \eta _{i} ) = \begin{cases} \eta _{i-1}^{-1} \eta _{i} & \text{ if $i$ is odd } \\ \eta _{i} \eta _{i-1}^{-1} & \text{ if $i$ is even }\end{cases}. \]

We therefore have an equality

\[ \eta _0^{-1} \eta _1 \eta _{2}^{-1} \cdots \eta _{n+1}^{(-1)^{n} } = \eta '^{-1}_{0} \eta '_{1} \eta '^{-1}_{2} \cdots \eta '^{(-1)^{n}}_{n+1}. \]

Invoking our inductive hypothesis, we conclude that this product vanishes if and only if the tuple $\vec{\eta }'$ bounds. By virtue of Lemma 3.2.3.7, this is equivalent to the assertion that $\vec{\eta }$ bounds.

We now complete the proof of Theorem 3.2.2.10 by showing that the multiplication on $\pi _{n}(X,x)$ is commutative. Fix a pair of elements $\sigma , \sigma ' \in \Sigma $. Then the tuples of $n$-simplices $( \sigma , e, \sigma ', \bullet , e, e, \ldots , e)$ and $(\sigma ', e, \sigma , \bullet , e, e, \ldots , e)$ determine maps of simplicial sets $f, f': \Lambda ^{n+1}_{3} \rightarrow X$ (Exercise 1.1.2.14). Since $X$ is a Kan complex, we can extend $f$ and $f'$ to $(n+1)$-simplices of $X$, which we will denote by $\tau $ and $\tau '$, respectively. It follows from the preceding arguments that the faces $d_3(\tau )$ and $d_3(\tau ')$ are representatives of the products $[\sigma '] [\sigma ]$ and $[\sigma ] [\sigma ']$ in $\pi _ n(X,x)$, respectively. Let $\overline{e}: \Delta ^{n+1} \rightarrow X$ denote the constant map taking the value $x$. Then the tuple of $(n+1)$-simplices $(\tau , s_0(\sigma ), s_1(\sigma ), \tau ', \bullet , \overline{e}, \overline{e}, \ldots , \overline{e} )$ determines a map of simplicial sets $g: \Lambda ^{n+2}_{4} \rightarrow X$ (Exercise 1.1.2.14). Since $X$ is a Kan complex, we can extend $g$ to an $(n+2)$-simplex of $X$. Then the fourth face of this extension witnesses that the tuple of $n$-simplices $( d_3( \tau ), e, e, d_3(\tau '), e, \ldots , e)$ bounds, so that we have an equality $[\sigma '] [\sigma ] = [ d_3(\tau ) ] = [ d_3(\tau ') ] = [ \sigma ] [\sigma ']$ in the homotopy group $\pi _{n}(X,x)$. $\square$