# Spaces homotopy dominated by a Moore space

‎A Moore space of degree $n\geq 2$ is a simply connected $CW$-complex X with a single non-vanishing homology group of degree $n$‎, ‎that is $\tilde{H}_{i}(X,\mathbb{Z})=0$ for $i\neq n$‎. ‎We write $X=M(A,n)$, where $A\cong \tilde{H}_{n} (X,\mathbb{Z})$‎. Recall that a topological space $X$ is homotopy dominated by a space $Y$ if there exist maps $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$ so that $g\circ f\simeq id_X$.

My question is that:

If $B$ is direct summand of $A$, then is $M(B,n)$ homotopy dominated by $M(A,n)$?

Yes. This follows, for instance, from the fact that Moore spaces are unique up to homotopy equivalence. Now, if $A=B\oplus C$ and $Z$ is an $M(C,n)$, then $Y\vee Z$ is an $M(A,n)$ and hence is homotopy equivalent to $X$. Since $Y$ is a retract of $Y\vee Z$, it is therefore homotopy dominated by $X$.