Kerodon

Proof of Proposition 3.2.6.2. Fix an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map carrying $\operatorname{\partial \Delta }^{n}$ to the base point $x \in X$. We wish to show that the homotopy class $[\sigma ]$ belongs to the image of the map $\pi _{n}(X_ s, x) \rightarrow \pi _{n}(X,x)$ if and only if the image $[f(\sigma )]$ is equal to the base point of $\pi _{n}(S,s)$. The “only if” direction is clear, since the composite map $X_{s} \hookrightarrow X \xrightarrow {f} S$ is equal to the constant map taking the value $s$. For the converse, suppose that $[ f(\sigma ) ]$ is the base point of $\pi _{n}(S,s)$. Then there exists a homotopy $h: \Delta ^{1} \times \Delta ^{n} \rightarrow S$ from $f(\sigma )$ to the constant map $\sigma '_0: \Delta ^{n} \rightarrow \{ s\} \subseteq S$, which is constant when restricted to the boundary $\operatorname{\partial \Delta }^ n$. Since $f$ is a Kan fibration, we can lift $h$ to a homotopy $\widetilde{h}: \Delta ^{1} \times \Delta ^{n} \rightarrow X$ from $\sigma $ to another $n$-simplex $\sigma ': \Delta ^{n} \rightarrow X$, where $\widetilde{h}$ is constant along the boundary $\operatorname{\partial \Delta }^{n}$ and $f( \sigma ') = \sigma '_0$ (Remark 3.1.5.3). Then $\sigma '$ represents a homotopy class $[\sigma '] \in \pi _{n}(X_ s, x)$, and the homotopy $\widetilde{h}$ witnesses that $[\sigma ]$ is equal to the image of $[\sigma ']$ in $\pi _{n}(X,x)$. $\square$

Proof. Combine Variant 3.2.5.5 with Proposition 3.2.6.4. $\square$

Proof of Proposition 3.2.6.4. Fix an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_ s$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map carrying $\operatorname{\partial \Delta }^{n}$ to the base point $x \in X_ s$. By construction, the homotopy class $[ \sigma ] \in \pi _{n}(X_ s, x)$ belongs to the image of the connecting homomorphism $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ if and only if there exists an $(n+1)$-simplex $\tau : \Delta ^{n+1} \rightarrow S$ such that $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $s$ and $\sigma $ is incident to $\tau $, in the sense of Definition 3.2.5.1. This condition is equivalent to the existence of an $(n+1)$-simplex $\widetilde{\tau }: \Delta ^{n+1} \rightarrow X$ satisfying $d^{n+1}_0( \widetilde{\tau } ) = \sigma $ and $d^{n+1}_ i( \widetilde{\tau } )$ is equal to the constant map $e: \Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $1 \leq i \leq n+1$. In other words, it is equivalent to the assertion that the tuple of $n$-simplices of $X$ $( \sigma , e, e, \ldots , e)$ bounds, in the sense of Notation 3.2.3.1. For $n \geq 1$, this is equivalent to the vanishing of the image of $[\sigma ]$ in the homotopy group $\pi _{n}(X,x)$ (Theorem 3.2.2.10). When $n=0$, it is equivalent to the equality $[\sigma ] = [x]$ in $\pi _0(X)$ by virtue of Remark 1.4.6.13. $\square$

Proof. Combine Variant 3.2.5.5 with Proposition 3.2.6.6. $\square$

Proof of Proposition 3.2.6.6. Fix an $(n+1)$-simplex $\tau : \Delta ^{n+1} \rightarrow S$ for which $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $s$. By construction, the connecting homomorphism $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ carries $[ \tau ]$ to the base point of $\pi _{n}(X_ s,x)$ if and only if the constant map $e: \Delta ^{n} \rightarrow \{ x\} \hookrightarrow X_{s}$ is incident to $\tau $, in the sense of Definition 3.2.5.1. This is equivalent to the requirement that $\tau $ can be lifted to a map $\widetilde{\tau }: \Delta ^{n+1} \rightarrow X$ for which $\widetilde{\tau }|_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$, which clearly implies that that $[\tau ]$ belongs to the image of the map $\pi _{n+1}(f): \pi _{n+1}(X,x) \rightarrow \pi _{n+1}(S,s)$. To prove the reverse implication, suppose that $[\tau ]$ belongs to the image of $\pi _{n+1}(f)$, so that we can write $[ \tau ] = [ f( \widetilde{\tau }' ) ]$ for some map $\widetilde{\tau }': \Delta ^{n+1} \rightarrow X$ for which $\widetilde{\tau }'|_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$. It follows that there is a homotopy $h: \Delta ^{1} \times \Delta ^{n+1} \rightarrow S$ from $f( \widetilde{\tau }' )$ to $\tau $ which is constant along the boundary $\operatorname{\partial \Delta }^{n+1}$. Since $f$ is a Kan fibration, we can lift $h$ to a map $\widetilde{h}: \Delta ^{1} \times \Delta ^{n+1} \rightarrow X$ such that $h|_{ \{ 0\} \times \Delta ^{n+1} } = \widetilde{\tau }'$ and $h|_{ \Delta ^{1} \times \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$ (Remark 3.1.5.3). The restriction $\widetilde{\tau } = h|_{ \{ 1\} \times \Delta ^{n+1} }$ then satisfies $f( \widetilde{\tau } ) = \tau $ and $\widetilde{\tau }|_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$. $\square$