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Let $X$ be a path-connected CW-complex. If $\omega: [0,1] \to X$ is a loop in $X$ and $\partial$ the boundary operator in simplicial homology, then $\partial(\omega)=\omega(1)-\omega(0)=0$, i.e. $\omega$ is a cycle and represents a homology class $[\omega] \in H_1(X)$. Moreover, each homology class in $H_1(X)$ is represented by a loop (as is shown in the usual proof of $H_1(X) \cong \Pi_1(X)_{ab})$.

Denote by $X^2$ the 2-skeleton of $X$. It's well-known that the inclusion $X^2 \hookrightarrow X$ induces an isomorphism $H_1(X^2) \xrightarrow{\cong} H_1(X)$. Hence, as noted above, $\omega$ is homologous to a loop $\gamma: [0,1] \to X^2$.

Question: Is it possible to describe/construct $\gamma$ in an explicit way ?

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    $\begingroup$ More than homologous, $\omega$ is homotopic through loops to a loop in $X^2$, in fact to a loop in $X^1$. The homotopy is explicit in terms of $\omega$. To start, $image(\omega)\subset X^k$ for some $k$ (by compactness). Now do a tiny homotopy so that $image(\omega)$ misses the midpoint of each $k$-cell; this works as long as $k\ge 2$. Then compose $\omega$ with a deformation retraction from $X^k−$( midpoints of $k$−cells) to $X^{k−1}$. Then proceed by induction. $\endgroup$
    – Lee Mosher
    Commented Aug 15, 2012 at 23:23
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    $\begingroup$ The inclusion of the 2-skeleton also induces an isomorphism on $\pi_1$, by the cellular approximation theorem. That well-known theorem produces $\gamma$ out of $\omega$. $\endgroup$
    – Fernando Muro
    Commented Aug 15, 2012 at 23:58
  • $\begingroup$ Lee, Fernando, thanks for your answers. You can also post them as answers - I would be glad to accept them (however, it seems that the question will be closed soon). $\endgroup$
    – Ralph
    Commented Aug 16, 2012 at 9:23

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