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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 $0<ϵ<1$. Define an orientation-preserving diffeomorphism $R$ on each collar by $R(s,t)=(s+ϵ\chi (t),t)$ and as the identity outside the collars. Scaling $ϵ$ 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 $s+ϵ=s$ modulo one, impossible for the chosen $ϵ$. Outside the collars, $F=\iota$ exchanges the interiors of the halves and has no fixed points. Therefore $\iota$ 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.