Free abelian groups in Algebraic Topolgogy

Question

In the context of algebraic topology, free abelian groups are frequently used like the parameter of functions $\textbf{X}$ in $f(\textbf{X})$

Since i am lack of knowledge about abstract algebra, i'd like to know what is free abelian group and how it is used in algebraic topology in some sense.

Answer 1

See Wolfram Mathworld - "A free Abelian group is a group $G$ with a subset which generates the group $G$ with the only relation being $ab=ba$."

"For instance, in algebraic topology, chains are formal sums of simplices, and the chain group is the free abelian group whose elements are chains." (Wikipedia).