New complexes from old

The main examples are the following (given for cubical complexes, same for cell complexes):

• Subcomplexes: Given a cubical complex $K$ and another cubical complex $L$ all cell of which also belong to $K$. If $\partial \colon C(K) \rightarrow C(K)$ is the boundary operator of $K$ defined on the chain complex of $K$, then the boundary operator $d \colon C(L) \rightarrow C(L)$ of $L$ is the restriction $\partial$ to $C(L)$.

• Products of complexes: Given two cubical complexes $K$ and $L$. Then $K \times L$ is the set of all pairwise products $a \times b$ of cells in $K$ and $L$ respectively. The boundary operator $d \colon C(K \times L) \rightarrow C(K \times L)$ of $K \times L$ is $d(a \times b) = \partial a \times \partial b$.

• Quotients of complexes: Given a complex $K$ and an equivalence relation $\sim$ on the chain complex $C(K)$ satisfying the quotient function $q \colon C(K) → C(K)/ \sim$ given by $q(x) = [x]$ is a chain map. Then the quotient set $C(K)/ \sim$ is a chain complex with the boundary operator $d([x]) = \partial (x)$.

Each of the three produces a chain map: the inclusion $i_L \colon C(L) \rightarrow C(K)$ given by $i_L(x) = x$, the projection-like functions $p_K \colon C(K \times L) \rightarrow C(K)$ and $p_L \colon C(K \times L) \rightarrow C(L)$ given by $p_K(a \times b) = a, p_L(a \times b) = b$, the quotient function $q \colon C(K) \rightarrow C(K)/ \sim$ given by $q(x) = [x]$. See Examples of maps.

Compare these complexes to the corresponding topological spaces in New topological spaces from old (also New sets from old).

Also:

 cell collapses,
 suspensions.