Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 3.2.4.4. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes, and let $n \geq 1$ be a positive integer, and let $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ be as in Proposition 3.2.4.2. Then $\partial $ is a group homomorphism.

Proof. To avoid confusion in the case $n=1$, let us use multiplicative notation for the group structures on both $\pi _{n+1}(S,s)$ and $\pi _{n}(X_ s,x)$. It is easy to see that the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X_{s}$ is incident to the constant map $\Delta ^{n+1} \rightarrow \{ s\} \subseteq S$, so the map $\partial $ carries the identity element of $\pi _{n+1}(S,s)$ to the identity element of $\pi _{n}(X_{s}, x)$. To complete the proof, it will suffice to show that if $(\eta _0, \eta _1, \ldots , \eta _{n+1})$ is an $(n+2)$-tuple of elements of $\pi _{n+1}(S,s)$ for which the product $\eta _0^{-1} \eta _1 \eta _{2}^{-1} \cdots \eta _{n+1}^{(-1)^{n}}$ vanishes in $\pi _{n+1}(S,s)$, then the product $\partial ( \eta _0)^{-1} \partial ( \eta _1) \partial ( \eta _2)^{-1} \cdots \partial ( \eta _{n+1})^{ (-1)^{n}}$ vanishes in $\pi _{n}(X_ s, x)$. To prove this, choose simplices $\tau _{i}: \Delta ^{n+1} \rightarrow S$ for which each restriction $\tau _{i}|_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $s$ and $[ \tau _ i ] = \eta _ i$. Using our assumption that $f$ is a Kan fibration, we can lift each $\tau _ i$ to a simplex $\widetilde{\tau }_{i}: \Delta ^{n+1} \rightarrow X$ carrying the horn $\Lambda ^{n+1}_{0}$ to the vertex $x \in X$, so that $\partial ( \eta _ i ) = [ d_0( \widetilde{\tau }_{i} ) ]$. Since $\pi _{n+1}(S,s)$ is abelian, the vanishing of the product $\eta _0^{-1} \eta _1 \eta _{2}^{-1} \cdots \eta _{n+1}^{(-1)^{n}}$ guarantees that we can choose an $(n+2)$-simplex $\rho : \Delta ^{n+2} \rightarrow S$ such that $d_{0}(\rho )$ is the constant map taking the value $s$ and $d_{i}( \rho ) = \tau _{i-1}$ for $1 \leq i \leq n+2$. Let $\widetilde{\rho }_{0}: \Lambda ^{n+2}_{0} \rightarrow X$ be the map given by the tuple of $(n+1)$-simplices $(\bullet , \widetilde{\tau }_{0}, \widetilde{\tau }_{1}, \ldots , \widetilde{\tau }_{n+1} )$ (see Exercise 1.1.2.14). Since $f$ is a Kan fibration, the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+2}_{0} \ar [r]^-{ \widetilde{\rho }_{0} } \ar [d] & X \ar [d]^{f} \\ \Delta ^{n+2} \ar [r]^-{ \rho } \ar@ {-->}[ur]^{ \widetilde{\rho } } & S } \]

admits a solution. Then $\sigma = d_0( \rho )$ is an $(n+1)$-simplex of $X_ s$ satisfying $d_ i( \sigma ) = d_0( \tau _{i} )$ for $0 \leq i \leq n+1$, and therefore witnesses that the product

\[ [ d_0(\sigma ) ]^{-1} [ d_1(\sigma ) ] [ d_2(\sigma ) ]^{-1} \cdots [ d_{n+1}(\sigma ) ]^{(-1)^{n} } = \partial ( \eta _0)^{-1} \partial ( \eta _1) \partial ( \eta _2)^{-1} \cdots \partial ( \eta _{n+1})^{ (-1)^{n}} \]

vanishes in the homotopy group $\pi _{n}(X_ s, x)$. $\square$