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.)