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For n=1. We consider two copies of $\Delta^1$, $1$ simplicies and identifying their boundaries we get a loop, that is $S^1$.

For n=2, identifying boundaries of two copies of $\Delta^2$ via identify map, we get a compact convex subset of $R^3$, hence it is $S^2$.

These give $\Delta$ complex structure of $S^1$ and $S^2$.

How do we visualize this standard $\Delta$ complex structure for $S^n$ in general?


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It's not clear what you mean by "visualize" in $n$-dimension, but viewing $\Delta^2$ as a disk, taking two disks and gluing them together along their borders yields a sphere. A more advanced way to show it is to first note that the boundary of $\Delta^n$ is $S^{n-1}$ and the gluing you describe yields the suspension of that boundary. – Thomas Andrews Oct 16 '12 at 13:50
thank you very much. – user38764 Oct 16 '12 at 15:13

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