A unique decomposition of involutions of handlebodies

Abstract:

We consider a PL involution of an orientable, $3$-dimensional handlebody for which each component of the fixed point set is $2$-dimensional. The handlebody is uniquely equivariantly decomposed as a disk sum of handlebodies ${M_i}$ such that if ${M_i} \approx {A_i} \times I$, then $h|{M_i}$ is equivalent to (i) $\alpha \times {\text {i}}{{\text {d}}_I}$, where $\alpha$ is an involution of $A$, or to (ii) ${\text {i}}{{\text {d}}_a} \times r$, where $r(t) = - t$ for all $t \in I = [ - 1,1]$.