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?


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?


You also need the second inclusion-induced homomorphism to be injective, in order to reach the desired conclusion. That is: If $Y$, $Z$, and $Y\cap Z$ are connected, aspherical CW-complexes, and the inclusion-induced homomorphisms $\pi_1(Y\cap Z)\to \pi_1(Y)$ and $\pi_1(Y\cap Z)\to \pi_1(Z)$ are both injective, then the CW-complex $X=Y\cup Z$ is also aspherical. This is an old result of J. H. C. Whitehead: On the asphericity of regions in a 3-sphere, Fundamenta Mathematicae 32 (1939), no. 1, 149-166.


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