1
$\begingroup$

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?

Thanks

  • 2
    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. http://en.wikipedia.org/wiki/Suspension_%28topology%292012-10-16

0 Answers 0