Subcomplexes

Suppose we are given a cubical complex $K$. Suppose also that all cells of another cubical complex $L$ belong to $K$, i.e., $$L \subset K.$$

Now, suppose $$\partial \colon C(K) \rightarrow C(K)$$

is the boundary operator of $K$ defined on the chain complex $C(K)$ of $K$. Then the boundary operator $$d \colon C(L) → C(L)$$

of L is simply the restriction $\partial$ to $C(L)$: $$d = \partial |_{C(L)}.$$

The subcomplex construction produces the inclusion function: $$i_L \colon C(L) \rightarrow C(K) {\rm \hspace{3pt} given \hspace{3pt} by \hspace{3pt}} i_L(x) = x.$$

This function is a chain map.