Transfer maps for triples of manifolds
If \(Y^{n-q}\subset X^{n}\) is a codimension \(q\) submanifold with normal bundle \(\xi\) in an \(n\)-manifold \(X\), and \(\partial U_{\xi}\) is the boundary of a tubular neighbourhood of \(Y\) in \(X\) one can define transfer maps of Wall groups \[ \begin{aligned} \text{tr}_{\xi} &: L_{n-q}(\pi_{1}(Y)) \to L_{n}(\pi_{1}(\partial U_{\xi}) \to \pi_{1}(U_{\xi})),\\ \text{tr}'_{\xi} &: L_{n-q}(\pi_{1}(Y)) \to L_{n}(\pi_{1}(\partial U_{\xi})).\end{aligned} \] This paper describes relations between the transfer maps associated to a triple of embeddings \(Z \subset Y\subset X\). This is done by introducing a relative transfer map \[ \text{tr}^{\text{rel}}: L_{n-q}(\pi_{1}(Y \setminus Z) \to \pi_{1}(Y)) \to L_{n} (\pi_{1}(X \setminus Z) \to \pi_{1}(X)) . \]
scientific article; zbMATH DE number 5155464
scientific article