plumbing

《La science et l'hypothèse》 · 第 128 页 · Section 4.6，Example 4.6.2（四维圆盘丛版本）

Let \(F\) be a closed, possibly disconnected or nonorientable surface, and let \(\pi:X\longrightarrow F\) be a \(D^2\)-bundle over \(F\). If \(D_1\) and \(D_2\) are disjoint disks in \(F\), then each \(\pi^{-1}(D_i)\) is a trivial bundle \(D_i\times D^2\). We plumb \(X\) at \(D_1\) and \(D_2\) by identifying \(D_1\times D^2\) with \(D_2\times D^2\), using a map that preserves the product structures but interchanges the factors. (As usual, we smooth corners.) […] A plumbing is a manifold obtained by finitely many applications of this procedure.