Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell.edu/~hatcher/AT/AT.pdf, section 4.F, last lines) it's stated that the natural map $$\underset{\rightarrow}{\lim} \ \Omega Z_n \rightarrow \Omega \underset{\rightarrow}{\lim} \ Z_n$$ is a weak homotopy equivalence (the map is given by the universal property of the direct limit); I also recall that the direct limit of a sequence of CW-complexes is the mapping telescope. I'm trying to prove this fact but I don't know how to proceed; in particular, I don't know how to relate the homotopy groups of the mapping telescope to the homotopy groups of the spaces $Z_n$. There is a relation for the homology groups, namely $$H_i(\underset{\rightarrow}{\lim}\ Z_n)\simeq\underset{\rightarrow}{\lim}H_i(Z_n)$$ but I don't know how this can help.

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    $\begingroup$ I see you posted this on MO as well. It is generally preferable that you wait a significant period of time before doing so, but more importantly, that when you do so, you add links to both questions to avoid duplication of effort. $\endgroup$ – user98602 Oct 13 '18 at 19:45
  • $\begingroup$ @MikeMiller Nice catch. $\endgroup$ – Kevin Carlson Oct 13 '18 at 20:05
  • $\begingroup$ @MikeMiller Ok, I didn't know what was the usual procedure in this cases; I also asked myself if this question was more suitable for MO or for MSE. I'll consider this in the future. $\endgroup$ – Giuseppe Bargagnati Oct 14 '18 at 10:24
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    $\begingroup$ @GiuseppeBargagnati For future reference, my opinion is that this was probably a better fit here-an exercise in an introductory graduate textbook is usually not "research-level" math, MO's job, although it's close. $\endgroup$ – Kevin Carlson Oct 14 '18 at 18:43

The key intuition is that the loop space is the space of maps from $S^1$, and that such mapping spaces commute with direct limits up to weak equivalence because $S^1$ is a finite complex.

More formally, the claim follows from cellularity for relative complexes: any map $(X,A)\to Y$ of relative complexes which is cellular on $A$ is homotopic rel $A$ to a map which is cellular on $X$. Thus every map from a finite relative complex $(K,K')$ into $\varinjlim X_i$ factors up to homotopy rel $K'$ through a finite subcomplex, and thus (perhaps up to another relative homotopy) through some $X_i$.

In particular, this applies when $(K,K')=(S^k,*)$, implying surjectivity of the desired map on $\pi_k$. For injectivity, we use $(K,K')=(S^k\wedge I_+,S^k\vee S^k)$. The factorization result here implies that if maps $f,g:S^k\to X_i$ become homotopic in $\varinjlim X_i$, they become homotopic already in some $X_j$, which gives the result.

  • $\begingroup$ Could you just point out where you show that the weak homotopy equivalence is induced by the natural map? I can see that these two spaces have the same homotopy groups but here I have difficult showing that the natural map induce the isomorphisms (I think injectivity is my problem) $\endgroup$ – Giuseppe Bargagnati Oct 23 '18 at 20:50
  • $\begingroup$ @GiuseppeBargagnati For injectivity, it suffices to show that any homotopy in the direct limit between two spheres factoring through $X_i$ (via the natural map) factors through some $X_j$ (via the natural map,) as I’ve done here. $\endgroup$ – Kevin Carlson Oct 23 '18 at 22:24

We have $\Omega = \operatorname{Map}_*(S^1, -)$ and $S^1$ is a compact topological space. Therefore, $\Omega$ commutes with directed colimits given by closed $T_1$-inclusions (such as a mapping telescope of CW complexes).

In more words, wlog we may assume $\{Z_n\}$ is a sequence of CW inclusions by passing to the mapping telescope. Since $S^1$ is compact, its image under a map $S^1 \to \varinjlim Z_n$ can't escape more than finitely many $Z_n$, and so every map $S^1 \to \varinjlim Z_n$ factors through some initial part $Z_m$, thus determining an element in $\Omega Z_m$ and in $\varinjlim \Omega Z_m$. This provides an inverse to the natural map $\varinjlim \Omega Z_n \to \Omega \varinjlim Z_n$.

  • $\begingroup$ Do you know how to prove that this inverse can be chosen to be continuous without appealing to Whitehead's theorem or another combinatorialization? This would be interesting since it's nontrivial to prove that loop spaces of CW complexes are CW complexes. $\endgroup$ – Kevin Carlson Oct 13 '18 at 19:53

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