Timeline for Loops in CW-complexes and the 2-skeleton
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2012 at 9:23 | comment | added | Ralph | 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). | |
| Aug 15, 2012 at 23:58 | comment | added | Fernando Muro | 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 $\omega$. | |
| Aug 15, 2012 at 23:23 | comment | added | Lee Mosher | 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. | |
| Aug 15, 2012 at 22:06 | history | asked | Ralph | CC BY-SA 3.0 |