Proposition 4

Let $(X, e)$ be a Hopf space (IV, p. 373, definition 2) and let $m: X\times X\rightarrow X$ be its law of composition. Assume that the space X is connected and locally arcwise connected. Let $X'$ be a connected covering of X; let $p$ denote its projection and let $e'$ be a point of the fibre $X'_e$.

a) The covering $X'$ is Galois and the group of its automorphisms is commutative.

b) There exists a continuous law of composition $m': X'\times X'\rightarrow X'$ and only one such that one has $p\circ m'=m\circ (p, p)$ and $m'(e', e') =e'$. Endowed with this law of composition, the pointed space $(X', e')$ is a Hopf space.

c) Endowed with the law of composition induced by $m'$, the fibre $X'_e$ is a group with identity element $e'$. The mapping of $\pi_1(X, x)$ into $X'_e$ given by $\gamma \mapsto e'\cdot \gamma$ is a surjective group homomorphism with kernel $p_*(\pi_1(X', e'))$. The mapping $g\mapsto g(e')$ is an isomorphism of the group Aut$_X(X')$ onto the group $X'_e$.

a) The group $\pi_1(X, e)$ is commutative (prop. 3). Since the covering $X'$ of X is connected, it is then a Galois covering and the group Aut$_X(X')$ is isomorphic to the quotient group $\pi_1(X, e)/p_*(\pi_1(X', e'))$ (III, p. 312, corollary 4 to proposition 2); this group is commutative.

b) Let us denote by $q: X'\times X'\rightarrow X$ the mapping $m\circ (p, p)$. The required mapping $m'$ is a continuous lifting of $q$ to $X'$ such that $m'(e', e') =e'$. The space $X'\times X'$ is locally arcwise connected; it is therefore enough, in order to prove the existence of such a continuous lifting, to verify that $q_*(\pi_1(X'\times X',(e', e')))$ is contained in $p_*(\pi_1(X', e'))$ (III, p. 308, prop. 1). By prop. 3, the mapping $\pi_1(m,(e, e))$ identifies with the law of composition of the group $\pi_1(X, e)$ when one identifies $\pi_1(X\times X,(e, e))$ with $\pi_1(X, e)\times \pi_1(X, e)$. The group $q_*(\pi_1(X'\times X',(e', e')))$ is therefore equal to the group $p_*(\pi_1(X', e'))$, whence the existence of $m'$. Since $X'\times X'$ is connected, the uniqueness of $m'$ follows from I, p. 34, corollary 1 of prop. 11.

Let us prove that the mapping $m'$ endows the pointed space $(X', e')$ with a structure of Hopf space. Let $m_1: X\rightarrow X$ be the mapping $g\mapsto m(g, e)$ and let $\sigma_1: X\times \mathbf{I}\rightarrow X$ be a homotopy pointed at $x$ joining $m_1$ to Id$_X$. Put $\tau =\sigma_1\circ (p$, Id$_{\mathbf{I}}): X'\times \mathbf{I}\rightarrow X$. The mapping $m'_1: X'\rightarrow X'$ defined by $h\mapsto m'(h, e')$ lifts the mapping $\tau (\cdot ,0)$; let $\tau ': X'\times \mathbf{I}\rightarrow X'$ be the lifting of $\tau$ which is a homotopy with origin $m'_1($III, p. 301, prop. 2). The mapping $t\mapsto \tau '(e', t)$ is a lifting to $X'$ of the constant mapping $t\mapsto e$; since $\tau '(e',0) =e'$, we therefore have $\tau '(e', t) =e'$ for every $t\in \mathbf{I}$. The mapping $\tau '(\cdot ,1)$ is then a lifting to $X'$ of the mapping $p$ which maps $e'$ to $e'$. Since $X'$ is connected, Id$_{X'}$ is the unique X-morphism of $X'$ into itself which fixes the point $e'$; we therefore have $\tau '(\cdot ,1) =$ Id$_{X'}$. This shows that $\tau '$ is a homotopy pointed at $e'$ which joins the mapping $m'_1$ to the mapping Id$_{X'}$. Analogously, there exists a homotopy pointed at $e'$ joining the mapping $m'_2:h\mapsto m'(e', h)$ to the mapping Id$_{X'}$. This proves that the pointed space $(X', e')$, endowed with the law of composition $m'$, is a Hopf space.

c) Since $X'$ is a connected covering of X, the orbital mapping of $e'$ induced by the operation of $\pi_1(X, e)$ on $X'_e$ is surjective and induces a bijection of $\pi_1(X, e)/p_*(\pi_1(X', e'))$ onto $X'_e($III, p. 305, Theorem 1).

Let $c$ and $d$ be loops in X at $e$, and let $\gamma$ and $\delta \in \pi_1(X, e)$ be their classes. Let $c'$ and $d'$ be the paths with origin $e'$ in $X'$ which lift $c$ and $d$. We have $e'\cdot \gamma =c'(1)$ and $e'\cdot \delta =d'(1)$ (III, p. 304). By Proposition 3 of IV, p. 374, the loop $m\circ (c, d)$ is strictly homotopic to the loop $c*d$; hence we have $e'\cdot (\gamma \delta ) =e'\cdot (m\circ (c, d))$. Now, the path $m'\circ (c', d')$ is a lift with origin $e'$ of the path $m\circ (c, d)$; hence we have

The mapping $\gamma \mapsto e'\cdot \gamma$ is therefore a group homomorphism of $\pi_1(X, e)$ into the set $X'_e$ endowed with the law of composition induced by $m'$. Consequently, $X'_e$ is a group for the law of composition induced by $m'$, and the orbital mapping of $e'$ is an isomorphism of the quotient group $\pi_1(X, e)/p_*(\pi_1(X', e'))$ onto $X'_e$.

The last part of assertion c) then results from Corollary 3 of III, p. 311.