In the usual diagram of inclusion homomorphisms, if the upper two maps are injective, then so are the other two.
More formally, consider a space 
which is expressible as the union
of pathwise-connected open
sets 
,
each containing the basepoint 
such that each intersection 
is pathwise-connected.
Then, the homomorphism induced by the inclusion
map from the free product of the fundamental
groups of the 
s
to the fundamental group of 
, i.e.,
 |
(1)
|
is surjective (Hatcher 2001, p. 43). In addition, if each intersection 
is pathwise-connected, then the kernel of 
is the normal subgroup 
generated by all elements of the form
 |
(2)
|
where 
is the homomorphism induced by the inclusion 
, and so 
induces an isomorphism
 |
(3)
|
