Let $X$ be a reasonably nice topological space (say, a connected CW complex), and let $Y$ and $Z$ be reasonably nice connected subspaces of $X$ such that $X = Y \cup Z$.


Suppose that $Y$, $Z$, and $Y\cap Z$ are aspherical, and the homomorphism $\pi\_1(Y\cap Z)\to \pi\_1(Y)$ induced by inclusion is injective. Does it follow that $X$ is also aspherical?


I think this is true, but I have been unable to construct a proof. If excision held for homotopy groups, then we would know that $\pi\_n(X,Z) \cong \pi\_n(Y,Y\cap Z) = 0$ for all $n \geq 2$, and it would follow from the long exact sequence for $(X,Z)$ that $\pi\_n(X) = 0$ for all $n\geq 2$. 


However, this does not appear to be a case to which excision applies. It seems that both $\pi\_1(Y,Y\cap Z)$ and $\pi\_2(Z,Y\cap Z)$ may be nonzero, so the pair $(Y,Y\cap Z)$ is potentially only $0$-connected, and the pair $(Z,Y\cap Z)$ is potentially only $1$-connected.


So, is it true that $X$ is always aspherical under these circumstances?



Must the union of these two aspherical spaces be aspherical?
This is false in general. Let $Y=S^1\times S^1$, and let $Z$ be a disk whose boundary is identified with $S^1\times\ast$. Then $Y\cup Z\simeq S^1\vee S^2$ is not aspherical. Are there any additional conditions in the situation you care about that fail for this example?


