Topology/Singular Homology

First we define the standard simplices as the convex span of the standard basis vectors. We then take as a boundary map $d_n:⟨e_1,\dots ,e_n⟩\mapsto \sum _{k=1}^n⟨e_1,\dots ,e_{k-1},e_{k+1},\dots ,e_n⟩$

Next we transport this structure to a topological space X: A simplex s in X is the image of a continous map from some standard simplex.

Now let $C_n(X)=⟨\sigma :{\mathrm{\Delta }}_n\to X⟩$ be the free groups on the simplices in X. The maps $d_n$ now induce a new chain map on the complex $C_{\bullet }$

Now using the definition of homology as in the previous section we define $H_n=\ker d_n/\mathrm{Im}{d}_{n+1}$ (Exercise: prove that this is well-defined.)