I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105:
Hatcher defines $\Delta_n(X)$ to be the free abelian group with basis the open n simplices ${e_\alpha}^n$ of X where elements are the n-chains in the form $\sum_\alpha n_\alpha {e_\alpha}^n$ where $n_\alpha \in \mathbb{Z}$.
Hatcher then goes on to say that this is equivalent to $\sum_\alpha n_\alpha \sigma_\alpha $ where $\sigma_\alpha:\Delta^n \longrightarrow X$ is the characteristic map of ${e_\alpha}^n$.
What is the characteristic map in this definition and how does it fit into the context of $\Delta$-Complexes? the only character i have seen in other modules is that of characters for representations but am unsure if this is relevant.
thanks in advance for the help
As defined in Section 2.1 of Hatcher's book, a $\Delta$-complex structure on a space $X$ is a collection of maps $\sigma_\alpha : \Delta^n \to X$, satisfying a list of properties.
About a page after the beginning of that definition, Hatcher gives the notation $e^n_\alpha = \sigma_\alpha(\text{interior}(\Delta^n))$, and the terminology that $\sigma_\alpha$ is the characteristic map of the open cell $e^n_\alpha$. (He writes a little circle atop $\Delta^n$ instead of $\text{interior}(\Delta^n)$ but I do not know how to reproduce that here.)