Definition 29.3 (Boundary of a singular simplex):

Given a singular $n-$simplex $\sigma:\Delta_n\longrightarrow X$, its $j-$th singular boundary is the singular $(n-1)$ simplex $\sigma\circ \Phi_j^n$ and the boundary $\partial _n\sigma$ of $\sigma$ is then the $(n-1)$ chain given by the algebraic sum of its singular faces:

$$\partial _n\sigma = \sum_{j=0}^{n}(-1)^j(\sigma\circ \Phi_j^n). \eqno(29.4)$$

The map $\partial _n$ then extends as a group homomorphism $\sigma_n:S_n(X)\longrightarrow S_{n-1}(X)$. When $n = 0$ we define the boundary map $\partial _0$ to be the zero map.

The most important property of the maps $\partial _n$ is the vanishing of $\partial _{n-1}\circ\partial _n$ which we now prove.