Linking of tree manifolds (Q1813677)

Let \((B,w)\) denote a weighted tree, where \(w: E(B)\to\mathbb{Z}\) is a weight function on the set of vertices \(E(B)\). By ``plumbing'' [see \textit{F. Hirzebruch} and \textit{K. H. Mayer}, \(O(n)\)-Mannigfaltigkeiten, exotische Sphären und Singularitäten. Lecture Notes Math. 57 (1968; Zbl 0172.25304)] together a collection of \(2m\)-dimensional vector bundles \(\{V_b\to S^{2m}\mid b\in E(B), e(V_b)=-w(b)\}\), one obtains as the boundary a closed \((4m-1)\)-dimensional manifold \(M(B,w)\). This ``tree-manifold'' \(M(B,w)\) has a collection of disjoint embedded \(S^{2m-1}\) spheres, namely the collection of fibers \(S(b)\) of the sphere bundles \(S(V_b)\), \(b\in E(B)\). The aim of this paper is to determine the linking numbers of these spheres. For \(e,f\in E(B)\), let \(L(e,f)\) denote the shortest path in \(B\) from \(e\) to \(f\). The complement \(B\setminus L(e,f)\) is a disjoint union of trees \(B_1\cup\dots \cup B_r\), where \(E(B_1)\cup\dots\cup E(B_r)=EB\setminus EL(e,f)\) and the new edge set consists of all edges in \(B\) which do not meet \(L(e,f)\). The determinant \(d(B)\) of a weighted tree is (by definition) the determinant of the matrix \(\{b_{ij}\mid i,j\in E(B)\}\), where \(b_{ii}=w(i)\), \(b_{ij}=1\) if \(i\) and \(j\) are joined by an edge \((i\neq j)\) and \(b_{ij}=0\) otherwise. Theorem 1: The linking number \(v(e,f)\) of two fibers \(S(e)\) and \(S(f)\) in \(M(B,w)\) is given by \(-d(B)^{-1}\prod^r_{i=1}d(B_ i)\in\mathbb{Q}\), assuming that \(d(B)\) and the \(d(B_i)\) are nonzero.