Help identifying two connecting homomorphisms

Question

I'm running into a little trouble with the proof of Proposition 11.1 on page 64 of the first chapter of Goerss and Jardine's book Simplicial Homotopy Theory.

Proposition 11.1. Suppose $X$ is a Kan complex, then the unit $$\eta_X:X\to S|X|$$ arising from the adjunction $|-|:\mathrm{sSet}\leftrightarrows\mathrm{Top}:S$ is a weak equivalence of simplicial sets, in the sense that it induces isomorphisms in all possible homotopy groups.

The proof is by induction. One shows that the path components are in one to one correspondence, i.e. one proves the statement for the $0$-th homotopy groups, and one hopes to apply the long exact homotopy sequence of the path fibration to get to higher homotopy groups by shifting the dimension using the loop space. However, they draw a diagram with one map I don't know how to interpret properly: $$\begin{matrix} \pi_{n+1}(X,x) & \xrightarrow{(\eta_X)_*} & \pi_{n+1}(S|X|,x) \\ \partial\downarrow && \downarrow{\color{red}\partial}\\ \pi_{n}(\Omega X,x) & \xrightarrow{(\eta_{\Omega_x X})_*} & \pi_{n}(S|\Omega_x X|,x) \end{matrix}$$ (I have written abusively $x$ everywhere, when it should be something like $\eta_X(x)$, or the constant loop at $x$) The problem I have is with the red ${\color{red}\partial}$. In the paragraph below I explain how I believe this diagram arises. However, once this is established I still don't see how to finish:

We want to perform a dimension shift argument using the long exact sequence of the path fibration of $S|X|$, but this would require we replace the red map $$\pi_{n+1}(S|X|,x)\xrightarrow{{\color{red}\partial}}\pi_{n}(S|\Omega_x X|,x)$$ with the connecting homomorphism arising from the path fibration of $S|X|$, i.e. with $$\pi_{n+1}(S|X|,x)\xrightarrow{{\color{blue}\partial}}\pi_{n}(\Omega_xS| X|,x)$$ it would have to land in $\pi_{n}(\Omega_x S| X|,x)$, but then the arrow $(\eta_{\Omega_X})_*$ would have the wrong target.

I doubt that there should be an isomorphism of simplicial sets $\Omega_x S|X|\simeq S|\Omega_x X|$. What follows is presumably how they got the diagram.

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Theorem 1.10 tells us that the geometric realization $|P_xX|\to |X|$ of the path space fibration $P_xX\to X$ is a Serre fibration, and it easily follows that $S|P_xX|\to S|X|$ is a Kan fibration. The loop space at $x$ is defined by pullback (a finite limit): $$\begin{matrix} \Omega_x X & \rightarrow & P_xX\\ \downarrow & &\downarrow\\ \Delta^0 &\rightarrow & X \end{matrix}$$ Also, since $|-|$ preserves finite limits by Proposition 2.4, we get a pullback diagram upon applying geomtric realization to the previous diagram: $$\begin{matrix} |\Omega_x X| & \rightarrow & |P_xX|\\ \downarrow & &\downarrow\\ \lbrace\mathrm{pt}\rbrace &\rightarrow & |X| \end{matrix}$$ $S$ is a right adjoint, hence preserves limits, and we get a further pullbakc diagram $$\begin{matrix} S|\Omega_x X| & \rightarrow & S|P_xX|\\ \downarrow & &\downarrow\\ \Delta^0 &\rightarrow & S|X| \end{matrix}$$ As was noted earlier, the vertical map on the right is a fibration, and all simplicial sets involved are Kan complexes, so this gives us the natural setting for the long exact sequence of homotopy groups of a fibration. This diagram has a natural map coming from the original diagram $$\begin{matrix} \Omega_x X & \rightarrow & P_xX\\ \downarrow & &\downarrow\\ \Delta^0 &\rightarrow & X \end{matrix}$$ induced by the unit $\eta$ of the adjunction, and the naturality of the long exact sequence of a fibration gives a ladder diagram $$ \begin{matrix} \cdots\rightarrow & \pi_{n+1}(X,x) & \xrightarrow{\partial} & \pi_{n}(\Omega_xX,x) & \rightarrow & \pi_{n}(P_xX,x) & \rightarrow & \pi_{n}(X,x) & \xrightarrow{\partial} & \pi_{n-1}(\Omega_xX,x) &\rightarrow\cdots \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ \cdots\rightarrow & \pi_{n+1}(S|X|,x) & \xrightarrow{{\color{red}\partial}} & \pi_{n}(S|\Omega_xX|,x) & \rightarrow & \pi_{n}(S|P_xX|,x) & \rightarrow & \pi_{n}(S|X|,x) & \xrightarrow{{\color{red}\partial}} & \pi_{n-1}(S|\Omega_xX|,x) &\rightarrow\cdots \end{matrix}$$

This at least clears up what the original red ${\color{red}\partial}$ meant, and how the diagram was conceived.

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Naturality of the path fibration applied to the map $\eta_X:X\to S|X|$ affords one with a similar diagram contining the second connecting homomorphism. $$ \begin{matrix} \cdots\rightarrow & \pi_{n+1}(X,x) & \xrightarrow{\partial} & \pi_{n}(\Omega_xX,x) & \rightarrow & \pi_{n}(P_xX,x) & \rightarrow & \pi_{n}(X,x) & \xrightarrow{\partial} & \pi_{n-1}(\Omega_xX,x) &\rightarrow\cdots \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ \cdots\rightarrow & \pi_{n+1}(S|X|,x) & \xrightarrow{{\color{blue}\partial}} & \pi_{n}(\Omega_xS|X|,x) & \rightarrow & \pi_{n}(P_xS|X|,x) & \rightarrow & \pi_{n}(S|X|,x) & \xrightarrow{{\color{blue}\partial}} & \pi_{n-1}(\Omega_xS|X|,x) &\rightarrow\cdots \end{matrix} $$

Presumably one has to link the two diagrams and do a little diagram chase, but I don't see how.

Answer 1

It seems the proof involves several results. $\mathrm{Top}$ stands for the category of compactly generated Hausdorff spaces.

Proposition 2.4. Geometric realization $|\cdot|:\mathrm{sSet}\to\mathrm{Top}$ preserves finite limits.

Theorem 7.10. Let $X,Y$ be two Kan complexes and $p:X\to Y$ a map of simplicial sets. Then $f$ is a fibration and a weak equivalence iff all lifting problems $$\begin{matrix} \partial\Delta^n & \to & X \\ \downarrow & \nearrow & \phantom{p}\downarrow p\\ \Delta^n & \rightarrow & Y\end{matrix}$$ have solutions

Theorem 10.10. The geometric realization of a Kan fibration is a Serre fibration.

So we start with the diagram that defines the simplicial loop space at $x$ as a pullback: $$\begin{matrix} \Omega_xX & \to & P_xX\\ \downarrow & & \downarrow\\ \Delta^0&\xrightarrow{x}&X \end{matrix}$$ By Proposition 2.4., the geometric realization maps this to yet another pullback diagram, this time in $\mathrm{Top}$: $$\begin{matrix} |\Omega_xX| & \to & |P_xX|\\ \downarrow & & \phantom{|p|}\downarrow |p|\\ \mathrm{pt}&\xrightarrow{x}&|X| \end{matrix} $$ since $S$ is a right adjoint, we get a third pullback diagram in $\mathrm{sSet}$: $$\begin{matrix} S|\Omega_xX| & \to & S|P_xX|\\ \downarrow & & \phantom{S|p|}\downarrow S|p|\\ \Delta^0&\xrightarrow{x}&S|X| \end{matrix}$$

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By Theorem 10.10. , $|p|$ is a Serre fibration. Using the usual properties of adjoint functors, and given a continuous map $f:\mathcal X\to \mathcal Y$ we see that the existence of a lift in either diagram below is equivalent to the existence of a lift in the other $$\begin{matrix} \Lambda^n_k & \to & S\mathcal X\\ \downarrow & \nearrow & \phantom{Sf}\downarrow Sf\\ \Delta^n & \xrightarrow{x}&S\mathcal Y \end{matrix} \quad\Longleftrightarrow\quad \begin{matrix} |\Lambda^n_k| & \to & \mathcal X\\ \downarrow & \nearrow & \phantom{f}\downarrow f\\ |\Delta^n| & \xrightarrow{x}&\mathcal Y \end{matrix}$$ Hence $f:\mathcal X\to \mathcal Y$ is a Serre fibration iff $Sf:S\mathcal X\to S\mathcal Y$ is a Kan fibration. Thus, $S|p|$ is a Kan fibration. We have a natural transformation of pointed fibrations $$ \begin{matrix} S|\Omega_xX| & \to & S|P_xX| & \xrightarrow{S|p|} & S|X|\\ \phantom{\eta_{\Omega_xX}}\uparrow\eta_{\Omega_xX}&& \phantom{\eta_{P_xX}}\uparrow\eta_{P_xX}&& \phantom{\eta_{X}}\uparrow\eta_{X}\\ \Omega_xX & \to & P_xX & \to & X \end{matrix} $$ So by naturality of the long exact sequence of a fibration, we get a commutative diagram of long exact sequences of homotopy groups: $$ \begin{matrix} \cdots\rightarrow & \pi_{n+1}(X,x) & \xrightarrow{\partial} & \pi_{n}(\Omega_xX,x) & \rightarrow & \pi_{n}(P_xX,x) & \rightarrow & \pi_{n}(X,x) & \xrightarrow{\partial} & \pi_{n-1}(\Omega_xX,x) &\rightarrow\cdots \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ \cdots\rightarrow & \pi_{n+1}(S|X|,x) & \xrightarrow{{\color{blue}\partial}} & \pi_{n}(\Omega_xS|X|,x) & \rightarrow & \pi_{n}(P_xS|X|,x) & \rightarrow & \pi_{n}(S|X|,x) & \xrightarrow{{\color{blue}\partial}} & \pi_{n-1}(\Omega_xS|X|,x) &\rightarrow\cdots \end{matrix} $$

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The Kan complex $P_xX$ is shown to be contractible (it strongly deformation retracts onto the constant path at $x$), so all its homotopy groups are zero (and $\pi_0=\mathrm{pt}$). Let's consider such a strong deformation retraction $$h:P_xX\times \Delta^1\to P_xX$$ Using Proposition 2.4, there is a canonical isomorphism $|P_xX\times \Delta^1|\simeq|P_xX|\times|\Delta^1|$, and we get a strong deformation retraction of topological spaces $$|h|:|P_xX|\times |\Delta^1|\to |P_xX|$$ contracting $|P_xX|$ to a point. Hence the map $$\begin{matrix} |P_xX|\\\downarrow\\\mathrm{pt}\end{matrix}$$ is a fibration and a weak equivalence, and thus solves all lifting problems $$ \begin{matrix} |\partial\Delta^n| & \to &|P_xX|\\ \downarrow & \nearrow & \downarrow\\ |\Delta^n| & \to & \mathrm{pt} \end{matrix} $$ which, by adjunction, are equivalent to lifting problems in $\mathrm{sSet}$ $$ \begin{matrix} \partial\Delta^n & \to &S|P_xX|\\ \downarrow & \nearrow & \downarrow\\ \Delta^n & \to & \Delta^0 \end{matrix} $$ Since both $S|P_xX|$ and $\Delta^0$ are Kan complexes, Theorem 7.10. kicks in, and we get that the map $S|P_xX|\to\Delta^0$ is a weak equivalence (and a fibration) of simplicial sets, and thus all homotopy groups of $S|P_xX|$ are trivial. We then get a diagram like so $$ \begin{matrix} 0 & \rightarrow & \pi_{n+1}(X,x) & \xrightarrow{\partial} & \pi_{n}(\Omega_xX,x) & \rightarrow & 0\\ && (\eta_{X})_*\downarrow\phantom{(\eta_{X})_*} & & \phantom{(\eta_{\Omega_xX})_*}\downarrow(\eta_{\Omega_xX})_* & \\ 0 & \rightarrow & \pi_{n+1}(S|X|,x) & \xrightarrow{\partial} & \pi_{n}(\Omega_xS|X|,x) & \rightarrow & 0 \end{matrix} $$ The two connecting homomorphisms are necessarily isomorphisms by exactness, and so is the vertical map on the right, by the induction hypothesis. This completes the proof.