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A continuous map $f:X\to Y$ induces homomorphisms $f_{\ast }:H_n(X)\to H_n(Y)$ satisfying $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $({\mathrm{id}}_X{)}_{\ast }={\mathrm{id}}_{H_n(X)}$.

A singular $n$-simplex in a space $X$ is a continuous map $\sigma :{\Delta }^n\to X$ from the standard $n$-simplex. Its alternating restrictions to the faces form its boundary.