Combinatorial moves on ambient isotopic submanifolds in a manifold (Q5939220)

In classical knot theory, ambient isotopy of links is well known to be the same as equivalence by a finite number of Reidemeister moves which are combinatorial moves over (and supported by a neighborhood of) a 2-simplex each. The principal theorem of the present paper implies that, for compact proper locally flat \(n\)-submanifolds of codimension at least one in an arbitrary PL-\(q\)-manifold, the following equivalence relations are the same: ambient isotopy, transformation by proper moves (supported by a \(q\)-disk), by cellular moves (essentially over an \((n+1)\)-disk), and by \((n+1)\)-simplex moves. Moreover, it does no harm to require a \((q-1)\)-submanifold in the boundary of the ambient manifold to be kept fixed and, with respect to ambient isotopy and transformability by \((n+1)\)-simplex moves, to allow for non-flatness along a subset of dimension 1 or 0.