When is the $i$-th homology group of the $p$-skeleton of a complex isomorphic to the $i$-th homology group of that complex? For what values of $i$ is it true that $H_i(K^{(p)})\simeq H_i(K)$?
My guess is that this is true for $i>dim K$. Otherwise, we can use $n$-simplex for a counterexample.
 A: If $X$ is a simplicial or more generally a CW-complex, then $H_k(X^{n}) \cong H_k(X^{n+1})$ for all $k < n$. There's a way to see this without the machinery of cellular homology. 
Consider the long exact homology sequence of the pair $(X^{n+1}, X^{n})$, which is 
$$\cdots \to H_{k+1}(X^{n+1}/X^{n}) \to H_k(X^{n}) \to H_k(X^{n+1}) \to H_k(X^{n+1}/X^{n}) \to \cdots$$
$X^{n+1}/X^n$ is a wedge sum of $(n+1)$-spheres, which has nontrivial homology only at dimension $n+1$. Thus $H_k(X^{n+1}/X^n) \cong H_{k+1}(X^{n+1}/X^n) \cong 0$, which leads to the result. 
Thus, by induction, $H_k(X^n) \cong H_k(X)$ for all simplicial or more generally CW-complex $X$ whenever $k < n$. 
(as pointed out in a comment below by @StefanHamcke, this argument works only if $X$ is finite dimensional. However, this is also true for infinite dimensional CW-complexes. For a proof, look at Hatcher's Algebraic Topology, page 138)
A: By looking at cellular homology, we can see that the $n$th homology group of a CW complex is unaffected by adding cells of dimension $n+2$ and higher. (It affects neither the kernel of $\partial_n$ nor the image of $\partial_{n+1}$.)
The same thing applies for simplicial homology, replacing "cell" with "simplex".
