Intuition for homology of $S^n$ We know that the sphere $S^n$ has $n$-th singular homology $H_n(S^n)= \mathbb{Z}$. A generator is given by a fundamental class, which is nothing else than the sum of the simplices in some triangulation. Thus my question is: Is there any intuition behind the fact that such a sum of simplices generates the whole homology? Is there any picture one can have in mind to see every element of the singular homology is a power of this fundamental class?
 A: The idea is simply this.  Suppose you have (a representative of) a nonzero top homology class.  Then it has $k\not= 0$ copies of some n-simplex.  Since this is a manifold, each boundary face is attached to the boundary face of exactly one other n-simplex, so in order for our chain to be a cycle, we must have $k$ copies of those simplices too.  These, in turn, force their neighbors to be in our chain with multiplicity $k$, and so everything propagates around the entire (connected component of) the manifold until we see that this is just $k$ copies of the fundamental class...
Or we get a contradiction.  Precisely in the case that we're working with a nonorientable manifold, we'll be able to come back around to our original simplex and we'll want to say that it has multiplicity $-k$, which is impossible unless $k=0$.  This illustrates that the top integral homology of a nonorientable manifold is 0, and also suggests why you get a fundamental class for any manifold when you use $\mathbb{Z}/2$ coefficients.
