Timeline for Geometric intuition behind this chain homotopy
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 17, 2019 at 22:29 | comment | added | Akerbeltz | That is right, I see what you are talking about. Thanks a lot for your answer and your time, your help was unvaluable. | |
| Nov 17, 2019 at 13:45 | comment | added | Allen Hatcher | @Akerbeltz: I think you are understanding this correctly. The twelve missing tetrahedra are grouped into six pairs with the two tetrahedra of each pair interchanged by reflecting the $I$ factor of $\Delta_2\times I$ across its midpoint. This means that after projecting $\Delta_2\times I$ onto $\Delta_2$ the two tetrahedra in each pair cancel algebraically. The inductive construction implies that there is a similar cancelation in all dimensions. | |
| Nov 15, 2019 at 19:02 | comment | added | Akerbeltz | Indeed, there are some missing elements of a subdivision of $\Delta_2\times\partial I\cup\partial\Delta_2\times I$ in my graphic, but the simplices of the chain $h\sigma$ can be considered to be the projection of a subcomplex of $\Delta_2\times I$ onto $\Delta_2\times\{0\}$, not the result of projecting an actual subdivision of $\Delta_2\times I$ | |
| Nov 15, 2019 at 15:22 | comment | added | Allen Hatcher | @Akerbeltz: If you form the cone on a triangulation of $\Delta_p\times \partial I \cup \partial \Delta_p \times I$ then the result is a triangulation of $\Delta_p\times I$. In your picture with 16 tetrahedra the underlying topological space is not homeomorphic to $\Delta_2\times I$ so it cannot give a triangulation of $\Delta_2\times I$. (A horizontal plane halfway between the top and bottom of your figure intersects the figure in something one-dimensional instead of two-dimensional.) | |
| Nov 14, 2019 at 17:56 | comment | added | Allen Hatcher | @Akerbeltz: Nice graphic at the link you provided, but something seems to be missing. There are 16 tetrahedra shown but there should be 28. | |
| Nov 13, 2019 at 3:20 | comment | added | Akerbeltz | If anyone is wondering how the cone of such subdivision of $\Delta_p\times\partial I\cup\partial\Delta_p\times I$ looks like, here is a representation of the case $p=2$: geogebra.org/3d/k8nr2wzs | |
| Nov 11, 2019 at 21:22 | vote | accept | Akerbeltz | ||
| Nov 11, 2019 at 21:18 | comment | added | Akerbeltz | Yes, there was a small typo in the original question. This answer is quite fulfilling, thanks. | |
| Nov 11, 2019 at 21:17 | history | bounty ended | Akerbeltz | ||
| Nov 11, 2019 at 16:11 | history | edited | Allen Hatcher | CC BY-SA 4.0 |
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| Nov 11, 2019 at 0:36 | history | answered | Allen Hatcher | CC BY-SA 4.0 |