How to compare the group operations in homotopy and homology groups? Consider the following Lemma from p.173, Chapter IV of Bredon's Topology and Geometry:

If $f$ and $g$ are paths in $X$ such that $f(1)=g(0)$ then the $1$-chain $f*g-f-g$ is a boundary... Note that this lemma implies that one can replace the $1$-simplex $f*g$ by the $1$-chain $f+g$ modulo boundaries.

$*$ is concatenation for $n=1$ in particular and the group operation in $\pi_n(X)$ for all $n \ge 1$ in general.
Question: Does the analogous statement for $\pi_n(X)$, $H_n(X)$, $n>1$, hold?

If $f,g: S^n \to X$, $[f],[g] \in \pi_n(X)$, $n>1$, are such that $f*g$ is defined, then the $n$-chain $f*g -f-g$ is an $n$-boundary. In particular, one can replace the $n$-simplex $f*g$ by the $n$-chain $f+g$ modulo boundaries.

I suppose there is a subtlety I am overlooking, namely how to identify continuous maps $f,g,f*g:S^n \to X$ with continuous maps $\Delta_n\to X$ -- if this subtlety makes the question unanswerable in any meaningful sense please let me know.
A reference will suffice for an answer; please don't forget the page number.
 A: It's been a while since I read all the definitions, but ...
An element $\alpha$ of $\pi_n(X_n, x_0)$ is represented by a map $f$ from $D^n$ to $X$ where $\partial D^n$ is all mapped to $x_0$. Writing $D^n$ as a union of $n$-simplices ($Q = [0, 1] \times [0, 1]$ is the union of the triangles in $Q$ defined by $x \ge y$ and $x \le y$; the analogous cube $Q$ is the union of 6 tetrahedra, defined by all possible orderings of $x,y,z$, etc.), every map such as $f$ gives rise to an $n$-chain $c = W(f)$ with $n!$ simplices, defined by restricting $f$ to each of the simplices that constitute $D^n$. And $c$ is a cycle, because its boundary is (with some work) the restriction of $f$ to $\partial D^n$ combined with a bunch of internal pieces (like the diagonal of the square) that cancel out because of ... careful checking on the signs of permutations that correspond to each of the $n!$ constitutive simplices. Getting this right is a pain in the neck (for me, at least). 
If $f$ and $g$ represent two homotopy classes in $\pi_n(X, x_0)$ that can be "starred" (which, as I recall, involves pasting together their domains along an $n-1$-face, and then scaling by a factor of two alone the orthogonal line in the domain), then it's pretty clear that $W(f*g) = W(f) + W(g)$ as chains, where the "=" here means "differ by a boundary". 
I guess the conclusion I draw is that $W: \pi_n -> H_n$ is homomorphism. 
Ah...I'm right. It's called the Hurewicz homomorphism, and the linked Wikipedia article (or Hatcher's book, I'm sure) will tell you all about it. The Hurewicz theorem says that the homomorphism is an isomorphism IF the lower homotopy groups are trivial. 
