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Notation (The $n$-Simplex). For each integer $n \geq 0$, we let $| \Delta ^{n} |$ denote the set of $(n+1)$-tuples of nonnegative real numbers $(t_0, t_1, \cdots , t_ n )$ which satisfy the equation $t_0 + t_1 + \cdots + t_ n = 1$. We regard $| \Delta ^{n} |$ as a topological space (with the topology inherited from standard topology on Euclidean space $\mathbf{R}^{n+1}$). If $X$ is a topological space, we will refer to a continuous function $\sigma : | \Delta ^ n | \rightarrow X$ as a singular $n$-simplex in $X$.