On automorphisms of high-dimensional solid tori (Q6595798)

The authors analyze the homotopy groups of \(B\text{Diff}_{\partial}(S^1\times D^{2n-1})\), where \(\text{Diff}_{\partial}(S^1\times D^{2n-1})\) are self diffeomorphisms of \(S^1\times D^{2n-1}\) that are the identity near the boundary. The method consists of studying the homotopy groups of \(B\text{Diff}_{\partial}(X_g)\) and of \(B\text{Emb}_{\partial/2}(X_g)\), where \(X_g\) is defined as the manifold \(X_g=S^1\times D^{2n-1} \#(S^n\times S^n)^{\# g}\). This is a solid torus \(S^1\times D^{2n-1}\) connected sum \(g\) times with \((S^n\times S^n)\); and \(\text{Emb}_{\partial/2}(X_g)\) are self embeddings that fix a portion of the boundary. Then the Weiss fiber sequence is applied\N\[ B\text{Diff}_{\partial}(S^1\times D^{2n-1})\to B\text{Diff}_{\partial}(X_g)\to B\text{Emb}_{\partial/2}(X_g). \] \NThe first result is \NTheorem A. Let \(n\geq 3\), \(p\) a prime number and \(0<k<\text{ min }(2p-3,n-2)\). Then the groups \(\pi_k(B\text{Diff}_{\partial}(S^1\times D^{2n-1}))\) are finitely generated when localized at \(p\). Moreover, if \(2p-3<n-2\) then there is a map \[ \bigoplus_{a>0}\mathbb{Z}/p\{t^a-t^{-a}\}\to \pi_{2p-3}(B\text{Diff}_{\partial}(S^1 \times D^{2n-1})) \] which is injective and whose cokernel is finitely generated when localized at \(p\). For \(p=2\), no localization is needed.\N\NThe authors give a result on the rational homotopy groups of \(B\text{Diff}_{\partial}(S^1\times D^{2n-1})\) in a certain range.