Definition of singular cubical homology As we know , a continuous function $T : [0,1]^{n} \to X$ is called a singular $n$ cube in $X$ . Denote $Q_{n}$ is free abelian group with basis is the set of all singular $n$ cube in $X$ . Let $D_{n},C_{n}$ are free abelian group generated by degenerate cubes and nondegenerate cubes , respectively . Is $Q_{n}$ the direct sum of $D_{n}$ and $C_{n}$ ?

Apologies: I originally misunderstood your question.

In singular _cubical_ homology, degenerate cubes are cubes that are constant in one of the coordinates.

The group $C_n$ is defined to be the _quotient_ , $$ C_n = \frac{Q_n}{D_n}.$$ This is the appropriate relationship between $Q_n$, $D_n$ and $C_n$.

From the way your question was asked, I would assume that you were thinking of the elements in $C_n$ as cube chains. They are not. The elements in $C_n$ are _equivalence classes_ of cube chains, modulo some degenerate cubes that you ignore.

Note that the boundary map $\delta_n$ sends $D_n$ into $D_{n-1}$, which ensures that $\delta_n$ descends to a well-defined map from the quotient $C_n$ to the quotient $C_{n - 1}$.