Product involutions with 0 or 1-dimensional fixed point sets on orientable handlebodies (Q1335492)

The author determines involutions (periodic self-homeomorphisms with period 2) on orientable handlebodies with 0- or 1-dimensional fixed point sets. We say that a subset \(S\) of a handlebody \(M\) is \(A\)-horizontal in \(M\) if \(M \cong A \times I\) \((I = [ - 1,1] )\) and \(S\) can be isotoped into \(A \times \{0\}\). If a subset of \(M\) is isomorphic to the fixed point set of some involution on \(A\) with \(M \cong A \times I\), we say it is \(A\)-fixable. Then the theorem is: Let \(M\) be an orientable handlebody with involution \(h\) having 0- or one-dimensional fixed point set \(\text{Fix} h\). Suppose also that \(\text{Fix} h\) is \(B\)-horizontal and \(B\)-fixable where \(M \cong B \times I\). Then there exists a compact, bordered surface \(A\) such that \(M \cong A \times I\) and \(h\) is equivalent to \(\tau \times r\), where \(\tau\) is a nonidentity involution of \(A\) and \(r(t) = - t\) for all \(t \in I = [ - 1,1]\). By finding \(h\)-equivariant disks and cutting along these disks, the theorem is proved by induction over the genera of handlebodies.