# Computing the Euler Characteristic of the $n$-sphere

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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• This might help: Topological definition for $n$ dimensions Mar 31, 2014 at 22:34
• 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.
– TKM
Mar 31, 2014 at 23:05
• When n is odd then the Euler characteristic is 0 -- there is no need to compute anything. Sep 18, 2021 at 6:33

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 $n$.)