The homology groups of the links of quasi-ordinary singularities
A \(d\)-dimensional complex analytic hypersurface germ \((X,x) \subset (\mathbb{C}^{d+1},0)\) is called a quasi-ordinary singularity if there is a branched covering \(\pi:(X,x) \to (\mathbb{C}^ d,0)\), which is unbranched outside the union of the coordinate hyperplanes in \(\mathbb{C}^ d\). The local ring \({\mathcal O}_{X,x}\) of germs of holomorphic functions is \(\mathbb{C}\{t_ 1,\dots,t_ d\} [\zeta]\), where \(\mathbb{C}\{t_ 1,\dots,t_ d\}\) is the convergent power series in \(d\) variables. \(\zeta\) is called a parametrization of \((X,x)\). Set \[ \begin{aligned} D_ i & = \{t \in \mathbb{C} \mid | t | \leq \varepsilon_ i\},\;\varepsilon_ i>0, \\ \partial {\mathbf D}^{d+1} & = \partial (D_ 1 \times \dots \times D_{d+1}), \\ L & = X \cap \partial {\mathbf D}^{d+1}. \end{aligned} \] The local topology of \((X,x) \subset (\mathbb{C}^{d+1},0)\) is determined by the pair \((\partial {\mathbf D}^{d+1},L)\) and \(L\) is called the link of the singularity \((X,x)\). The collection \({\mathcal U} = \{U_ i=\pi^{-1}(V_ i)\}\), where \(V_ i=(D_ 1 \times \dots \times D_{i-1} \times D^*_ i \times D_{i+1} \times \dots \times D_ d) \cap \partial {\mathbf D}^ d\), forms an open covering of \(L\). The Čech resolution of singular chain complexes of \(L\) with respect to the covering \({\mathcal U}\) gives rise to a spectral sequence: \[ E^ 2_{p,q} (L,{\mathcal U})=H^ \delta_ p \bigl( \bigoplus_{i_ 0< \cdots < i_ p} H_ q(U_{i_ 0 \dots i_ p},\mathbb{Z}) \bigr) \Rightarrow H_{p+q} (L,\mathbb{Z}), \] where \(U_{i_ 0 \dots i_ p}=U_{i_ 0} \cap \cdots \cap U_{i_ p}\). Now the map \(\pi\) induces a morphism \(\pi_ *\) between this spectral sequence and the following spectral sequence: \[ E^ 2_{p,q} (\partial {\mathbf D}^ d,{\mathcal V})=H^ \delta_ p \bigl( \bigoplus_{i_ 0<\cdots<i_ p} H_ q(V_{i_ 0\dots i_ p}, \mathbb{Z}) \bigr) \Rightarrow H_{p+q} (\partial {\mathbf D}^ d,\mathbb{Z}), \] where \({\mathcal V}=\{V_ i\}\) and \(V_{i_ 0 \dots i_ p}=V_{i_ 0} \cap \cdots \cap V_{i_ p}\). Our particular choice of the covering \({\mathcal V}\) forces that the spectral sequence \(E^ 2_{p,q} (\partial {\mathbf D}^ d,{\mathcal V})\) degenerates at \(E^ 2\). In fact, \(E^ 2_{p,q} (\partial {\mathbf D}^ d,{\mathcal V})=0\) unless \((p,q)=(0,0)\) or \((d-1,d)\). Hence \(E^ 2_{p,q} (L,{\mathcal U})\) can be calculated in terms of the quotient groups \[ \wedge^ qG_{i_ 0 \dots i_ p}=H_ q(V_{i_ 0 \dots i_ p},\mathbb{Z})/ \pi_ *\bigl( H_ q(U_{i_ 0 \dots i_ p}, \mathbb{Z}) \bigr). \] We show these quotient groups are presented by matrices involving the exponents of the terms of the parametrization \(\zeta\) and get various results on the homology groups of the link. Among them, we show our computation of the homology group \(H_{2d-3} (L,\mathbb{Z})\) via the spectral sequences agrees with \textit{J. Lipman}'s in Mem. Am. Math. Soc. 388, 1-106 (1988; Zbl 0658.14003).
scientific article; zbMATH DE number 561309
scientific article