Proposition 2

Let A be a connected topological space locally connected by arcs, let B be an unravelable topological space, and let $f: A\rightarrow B$ be a continuous mapping. Let $a$ be a point of A and $b=$ $f(a)$. The following assertions are equivalent:

(i) The homomorphism $\pi_1(f, a):\pi_1(A, a)\rightarrow \pi_1(B, b)$ is surjective.

(ii) For every pair $(E,E')$ of coverings of B, the mapping

which associates to a B-morphism $g: E\rightarrow E'$ the A-morphism

is bijective.

By means of the homomorphism $\pi_1(f, a)$, every $\pi_1(B, b)$-set is endowed with a structure of $\pi_1(A, a)$-set. Condition (i) is then equivalent to the following condition (i$'$):

(i$'$) For every pair $(F,F')$ of $\pi_1(B, b)$-sets, every $\pi_1(A, a)$-morphism of F into $F'$ is a $\pi_1(B, b)$-morphism.

In fact, if the homomorphism $\pi_1(f, a)$ is surjective, every $\pi_1(A, a)$-morphism of $\pi_1(B, b)$-sets is a $\pi_1(B, b)$-morphism. Conversely, let us take a $\pi_1(B, b)$-set F reduced to a point and let $F'=\pi_1(B, b)/f_*(\pi_1(A, a))$. The mapping of F into $F'$ whose image is $f_*(\pi_1(A, a))$ is a $\pi_1(A, a)$-morphism but is not a $\pi_1(B, b)$-morphism if $F'$ is not reduced to a point, that is to say if $\pi_1(f, a)$ is not surjective.

Since the space B is unrollable, every $\pi_1(B, b)$-set is isomorphic to the $\pi_1(B, b)$-set $E_b$, where E is a covering of B (IV, p. 344, remark 1). The equivalence of (i$'$) and (ii) therefore follows from prop. 2 of III, p. 310.