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Let $n\ge 2$. Compute the Euler characteristic of the $n$-sphere $S^n$ using the standard triangulation of the $n+1$-simplex.

I know the union of the proper faces of the $(n+1)$-simplex is homeomorphic to $S^n$, then also I am stuck...

Any help would be appreciated

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  • $\begingroup$ This might help: Topological definition for $n$ dimensions $\endgroup$
    – ccorn
    Mar 31, 2014 at 22:34
  • $\begingroup$ Alternatively, consider a height function $f(x_{1},...,x_{n+1})=x_{n+1}$, compute it's critical points and indexes. Then use Morse theorem: $\chi(M)=\sum_{k=1}(-1)^{k} cr_{k}(f)$, where $cr_{k}(f)$ = # critical points with index k. $\endgroup$
    – TKM
    Mar 31, 2014 at 23:05
  • $\begingroup$ When n is odd then the Euler characteristic is 0 -- there is no need to compute anything. $\endgroup$
    – Wlod AA
    Sep 18, 2021 at 6:33

1 Answer 1

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The number of $i$-dimensional faces of the $(n+1)$-simplex is ${n+2}\choose{i+1}$. Now, compute the sum of the terms $(-1)^{i}{{n+2}\choose{i+1}}$ for $i=0,1,\ldots,n$ and see what you get. (Hint: the answer depends on the parity of $n$.)

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