So I'm reading the beginning of chapter 2 of Hatcher, and there Hatcher defines cycles as loops without a fixed basepoint, and obtains these by "abelianizing".
I'm confused however, how we can ensure that things remain loops, or even sensible paths. Say we have three paths $a,b,c$ between a pair of points $x,y$. Then we have that $ab^{-1}ca^{-1}$ is a loop based at $x$. But then after abelianizing, we can turn this into $acb^{-1}a^{-1}$ which isn't actually a coherent path.
Concretely, how does this abelianization business work?
They don't. These cycles are actually just formal sums (and differences) of paths. In other words, while the group operation in the fundamental group was concatenation of paths, the group operation in homology has no immediate geometric meaning: we just formally say we're adding up the paths, and the result is a formal expression that is not itself a path. (It turns out that if your space is path-connected, then every element of the homology group $H_1$ can actually be represented by a loop, but this is not obvious from the definitions.)
If this doesn't entirely make sense, my main advice would be to just keep reading. At this point in the text Hatcher is just trying to give some rough intuition; he will give completely precise definitions of everything later.