String links and Whitehead torsion. (Q2773548)

This paper uses a refinement of Whitehead torsion to obtain invariants for string links. Let \(\pi\) be a group and suppose NEWLINE\[NEWLINE C: \dots \longrightarrow C_1 \longrightarrow C_0 \longrightarrow 0 NEWLINE\]NEWLINE is a chain complex of free \(\mathbb Z \pi\)-modules such that all the \(C_i\) are finitely generated free with a given basis. If \(C\) is acyclic, then one can define the torsion \(\tau (C) \in\) Wh\((\pi)\). This paper deals with cases when \(C\) is not acyclic. Suppose \(B\) is a ring with the property that \(B^p \cong B^q\) as \(B\)-modules implies \(p=q\). The author defines the concept of an acyclic representation \(\rho:\mathbb Z \pi \to B\), and then sets \(K_{\rho}(B) = K_1(B)/\rho(\pm \pi)\), which he calls the Milnor group of \(\rho\). In this situation one has a well defined \(\rho\)-torsion \(\tau_{\rho}(C) \in K_{\rho}(B)\), even if \(C\) is not acyclic.NEWLINENEWLINE Let \(E\) be an \(n\)-component string link in \(D^2 \times I\) with exterior \(X\) and let \(X_0 = X \cap D^2 \times 0\), and let \(\pi = \pi_1(X)\). Then in general the cellular \(\mathbb Z \pi\)-chain complex \(\widetilde C\) associated to \((X,X_0)\) will not be acyclic. However \(\tau_{\rho}(\widetilde C)\) for many representations \(\rho: \pi \to \Lambda\) will still be well defined. Various interesting examples are considered using what the author calls ``universal representations'', in some cases with computer help.