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 $$ d_n:\langle e_1, \ldots, e_n\rangle \mapsto \sum_{k=1}^n \langle e_1, \ldots, e_{k-1}, e_{k+1}, \ldots, e_n\rangle $$

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 $$C_n(X)=\langle \sigma : \Delta_n \to X \rangle $$ 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 $$ H_n = \ker d_n/ \operatorname{Im} d_{n+1} $$ (Exercise: prove that this is well-defined.)