Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum
$$
\Sigma^t X\cong \bigvee \_{k=1}^\infty Y\_k.
$$


(1). Does this imply 
$$
X\to \Sigma^tX\to \bigvee \_{k=1}^\infty Y\_k\to Y\_k
$$
induce an epimorphism on homology
$$
H\_\*(X)\to H\_\*(Y\_k)?
$$


(2). Can we construct a map
$$
Y\_k\to \bigvee \_{k=1}^\infty Y\_k\to \Sigma^tX \to X
$$
such that the map induces a monomorphism on homology?



stable splitting into a wedge sum
Rather than thinking about maps $X\to \Sigma^tX$ or $\Sigma^tX\to X$ you should just pre- or post-compose with the suspension isomorphism. 


For instance,
$$
H\_\*(X) \cong \tilde{H}\_{\*+t}(\Sigma^t X) \cong \tilde{H}\_{\*+t}\left(\bigvee\_{k=1}^\infty Y\_k\right) \twoheadrightarrow \tilde{H}\_{\*+t}(Y\_k)
$$
is an epimorphism (the last map is an epimorphism because it's induced by a retraction). So we get an epimorphism $H\_\*(X)\to H\_\*(Y\_k)$ up to a degree shift.


Similarly you'll get a monomorphism $H\_\*(Y\_k)\to H\_\*(X)$ up to a degree shift.


Note that the inclusion $X\to \Sigma^t X$ is always null-homotopic.


