= Solution
Let $P$ be a sphere with $g+1$ disjoint open disks removed, and form its double $S$ by gluing two copies along the boundary. Then $\chi(S)=2\chi(P)=2(1-g)$, so $S$ is an oriented closed surface of genus $g$. The <involution> $\iota$ swapping the two halves fixes precisely the $g+1$ seam circles.
Use disjoint collars with coordinates $(s,t)\in\mathbb R/\mathbb Z\times(-\eta,\eta)$, where $\iota(s,t)=(s,-t)$. Choose a smooth even bump function $\chi(t)$ supported inside the collar, with $\chi(0)=1$, and choose $0<\epsilon<1$. Define an orientation-preserving diffeomorphism $R$ on each collar by $R(s,t)=(s+\epsilon\chi(t),t)$ and as the identity outside the collars. Scaling $\epsilon$ from zero gives an isotopy from the identity to $R$, so $F=R\circ\iota$ is homotopic to $\iota$.
A fixed point of $F$ in a collar would satisfy $-t=t$, hence $t=0$. Its other coordinate would then require $s+\epsilon=s$ modulo one, impossible for the chosen $\epsilon$. Outside the collars, $F=\iota$ exchanges the interiors of the halves and has no fixed points. Therefore \b[$\iota$ has fixed circles but is homotopic to the fixed-point-free map $F$]. This is the <fixed-point-free perturbation of a doubled-surface reflection>; the construction includes genus zero, where the double is a sphere.
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