Proposition 3

Suppose that X is unlaceable. The homomorphism $\mathsf{N}$ is then surjective, and its kernel is the smallest distinguished subgroup of $\pi_1(X, o)*F(G)$ containing the elements

($R_2$)$r_2(k, v) = [k]^{-1}v[k](\beta_k^{-1}k^*(v)^{-1}\beta_k)$

for $k\in G$ and $v\in \pi_1(X, o)$;

($R'_3$)$r'_3(k, j) = [k](\beta_k^{-1}k^*(\gamma_j)^{-1}\gamma_j)$

for $k\in G$ and $j\in \pi_0(X^k)$;

($R''_3$)$r''_3(k, h) = [kh]^{-1}[k][h](\beta_h^{-1}h^*(\beta_k^{-1})\beta_{kh})$

for $k$ and $h\in G$.

The groupoid morphism $\varpi (f)$ induces a morphism

which is an isomorphism by Theorem 3 (IV, p. 403), since the space X is supposed unlaceable. The proposition then results from II, p. 211, prop. 6.

The three corollaries below follow from the immediately corresponding corollaries of prop. 6 of II, p. 211.