Exercise 26, § 5

In the numerical plane $\mathbf{R}^2$, denote by $D_0$ the disk with center $(0,0)$ and radius 1 and by $C_0$ its boundary; put $a_0= (-1,0)$. Let $g$ be a natural number $\geqslant 1$, let $r$ be a strictly positive real number and let $(v_1, . . . , v_g)$ be a sequence of real numbers such that $r <$ inf($v_1+ 1,(v_2-v_1)/2, . . . ,(v_g-v_{g-1})/2,1-v_g$). For every $j\in \{1, . . . , g\}$, denote by $D_j$ the closed disk with center $(0, v_j)$ and radius $r$ and by $C_j$ the circle with center $(0, v_j)$ and radius $r$; also put $a_j= (-r, v_j)$. Let X be the space D $-\bigcup^g_{j=1}\mathring{D}_j$.

For every $j\in \{1, . . . , j\}$, denote by $u_j$ the class in $\varpi (X)$ of a path with origin $a_0$ and term $a_j$ whose image is the segment $[a_0, a_j]$; let us also denote by $c_j\in \pi_1(X, a_j)$ the class of the loop given by $t\mapsto (-r$ cos(2$\pi t$)$,-r$ sin(2$\pi t$)$+v_j)$ and $\gamma_j=u_jc_ju^{-1}_j$.

a) Let $\gamma_0$ be the class of the loop in X given by $t\mapsto (-$ cos(2$\pi t$)$,-$ sin(2$\pi t$)). Prove that one has $\gamma_0=\gamma_1. . . \gamma_g$.

b) Prove that the unique homomorphism of the free group with $g$ generators $x_1, . . . , x_j$ into $\pi_1(X, a_0)$ which applies $x_j$ to $\gamma_j$ is a group isomorphism. (Apply the van Kampen theorem to the covering of X defined by its intersections with the half-planes $\mathbf{R}_+\times \mathbf{R}$ and $\mathbf{R}_-\times \mathbf{R}.$)

c) Let $f: X\rightarrow \mathbf{R}_+$ be a continuous function such that $\overset{-1}{f}(0) =\bigcup^g_{j=0}C_j$ and let Y be the subspace of $\mathbf{R}^3$ formed by the points $(x, y, z)$ such that $z^2=f(x, y)$. Denote by $i_+$ (resp. $i_-$) the continuous mapping from X into Y given by $(x, y)\mapsto$ $(x, y,\surd f(x, y))$ (resp. $(x, y)\mapsto (x, y,-\surd f(x, y))$). For every class of path $\gamma$ in X, define classes of paths in $\varpi (Y)$ by $\gamma^+= (i_+)_*(\gamma )$ and $\gamma^-= (i_-)_*(\gamma )$.

For every $j\in \{1, . . . , g\}$, we finally define an element of $\pi_1(Y, a_0)$ by $\delta_j=u^+_ju^-_j$. Prove that one has the relations $\gamma^+_j\delta_j=\delta_j\gamma_j^-$ for $1\leqslant j\leqslant g$, and $\gamma_1^+. . . \gamma_g^+=\gamma_1^-. . . \gamma^-_g$.

d) Let $\varphi$ be the unique homomorphism of a free group F with $2g$ generators $x_1, . . . , x_g, y_1, . . . , y_g$ into $\pi_1(Y, a_0)$ which maps $x_j$ onto $\gamma_j^+$ and $y_j$ onto $\delta_j$. Prove that the homomorphism $\varphi$ is surjective and that its kernel is the smallest normal subgroup of F which contains the element

(Apply the theorem of van Kampen to the covering of Y defined by its intersections with the half-spaces $\mathbf{R}^2\times \mathbf{R}_+$ and $\mathbf{R}^2\times \mathbf{R}_-.$)

e) Let $\varphi '$ be the unique homomorphism of a free group F with $2g$ generators $x_1, . . . , x_g, y_1, . . . , y_g$ into $\pi_1(Y, a_0)$ which, for every $j$, maps $x_j$ onto

and $y_j$ onto $\gamma_j^+$. Prove that the homomorphism $\varphi '$ is surjective and that its kernel is the smallest normal subgroup of F which contains the element