Proposition 3

Let $(X, e)$ be a Hopf space and $m: X\times X\rightarrow X$ its composition law. For any loops $c,c'$ of X at $e$, one has:

where $\sim$ denotes the relation of strict homotopy. The law of composition of the group $\pi_1(X, e)$ is the homomorphism obtained by composing the canonical isomorphism $\pi_1(X, e)\times \pi_1(X, e)\rightarrow \pi_1(X\times X,(e, e))$ (III, p. 297, corollary) and the homomorphism $\pi_1(m,(e, e))$ of $\pi_1(X\times X,(e, e))$ into $\pi_1(X, e)$. The group $\pi_1(X, e)$ is commutative.

Let $\mu:\pi_1(X, e)\times \pi_1(X, e)\rightarrow \pi_1(X, e)$ be the group homomorphism obtained by composing the canonical isomorphism of $\pi_1(X, e)\times \pi_1(X, e)$ onto $\pi_1(X\times$ $X,(e, e))$ and the homomorphism $\pi_1(m,(e, e))$. It is to be proved that one has

for all $\gamma ,\gamma '\in \pi_1(X, e)$. Taking account of Remark 2 of III, p. 297, one has:

Let $c\in \Omega_e(X)$ be a loop of class $\gamma$; let us denote by $m_1: X\rightarrow X$ the mapping defined by $m_1(x) =m(x, e)$; then $\mu(\gamma , \varepsilon_e)$ is the class of $m_1\circ c$. By definition of a Hopf space, the mappings pointed at $e$, $m_1$ and Id$_X$, have the same pointed homotopy class at $e$. Hence the loops at $e$, $m_1\circ c$ and $c$, are strictly homotopic (III, p. 296, Cor. 1 of Prop. 3) and one has $\mu(\gamma , \varepsilon_e) =\gamma$. One proves analogously that one has $\mu(\varepsilon_e, \gamma ') =\gamma '$. Relation (1) then follows from relation (2).