Homotopic PL $n$-balls are isotopic

Abstract:

The following theorem extends a result of Martin and Rolfsen [Proc. Amer. Math. Soc. 19 (1968), 1290-1292]. Theorem. Let ${B^n}$ be a PL n-ball, ${Q^q}$ a $(2n - q + 1)$-connected PL q-manifold, $q \geqq n + 2$. Suppose Q is either compact or open and that, for $i = 0,1,{H_i}:{B^n} \to Q - \dot Q$ is a locally unknotted PL embedding. If there exists a homotopy $H:{B^n} \times I \to Q$ between ${H_0}$ and ${H_1}$ such that ${H_t}$ is fixed on ${\dot B^n}$, then there exists a PL ambient isotopy ${h_t}:Q \to Q$, fixed on ${H_0}({\dot B^n}) \cup \dot Q$, such that ${h_1}{H_0} = {H_1}$. Locally unknotted is taken here to mean that there exists a triangulation (L, K) of $(Q,{H_i}({B^n}))$ with $\dot K$ full in K and $({\text {lk}}(\nu ,L)$, ${\text {lk}}(\nu ,K))$ an unknotted sphere pair for all vertices $\nu \in K - \dot K$.