Given two singlar n-simplicies $\phi$, $\gamma$ $:\Delta_n\rightarrow X$, how do we define the sum $\phi + \gamma$ in the singular chain group $S_n(X)$?
I can't come up with a natural way to define this since $X$ is an arbitrary topological space without a binary operation between points and you cant compose the two maps since their domains and codomains are different.
I can't find any information on the subject, everyone seems to think that the group operation should "arise" from somewhere, but i don't know where from.
Cheers.
-A total newbie in algebraic topology
The sum is purely formal: $\phi+\gamma$ is not itself a singular $n$-simplex or anything like that, but is merely a new symbol we invent and call the "sum" of $\phi$ and $\gamma$. More rigorously, a singular $n$-chain is actually defined to a function from the set $S$ of singular $n$-simplices in $X$ to $\mathbb{Z}$ (which takes the value $0$ on all but finitely many elements of $S$). We think of such a function $f:S\to\mathbb{Z}$ as representing the "sum" $$\sum_{s\in S}f(s)s.$$ That is, it is the formal sum of elements of $S$ in which we have $f(s)$ copies of $s$ for each $s\in S$. So, if $\phi$ and $\gamma$ are two particular elements of $S$, their "sum" is really the function $f:S\to\mathbb{Z}$ such that $f(\phi)=1$, $f(\gamma)=1$, and $f(s)=0$ for all other $s$. (This is assuming $\gamma\neq \phi$; if they are equal then their sum is given by $f(\phi)=2$ and $f(s)=0$ for all other $s$.)