# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## On the monodromy at isolated singularities of weighted homogeneous polynomialsHTML articles powered by AMS MathViewer

by Benjamin G. Cooper
Trans. Amer. Math. Soc. 269 (1982), 149-166 Request permission

## Abstract:

Assume $f:{{\mathbf {C}}^m} \to {\mathbf {C}}$ is a weighted homogeneous polynomial with isolated singularity, and define $\phi :{S^{2m - 1}} - {f^{ - 1}}(0) \to {S^1}$ by $\phi (\overrightarrow z ) = f(\overrightarrow z ) / |f(\overrightarrow z )|$. If the monomials of $f$ are algebraically independent, then the closure ${\overline F _0}$ of ${\phi ^{ - 1}}(1)$ in ${S^{2m - 1}}$ admits a deformation into the subset $G$ where each monomial of $f$ has nonnegative real values. For the polynomial $f({z_1}, \ldots ,{z_m}) = z_1^{{a_1}}{z_2} + \cdots + z_{m - 1}^{{a_{m - 1}}}{z_m} + z_m^{{a_m}}{z_1}$, $G$ is a cell complex of dimension $m - 1$, invariant under a characteristic map $h$ of the fibration $\phi$, and the inclusion $G \to {F_0}$ induces isomorphisms in homology. To compute the homology of the link $K = {f^{ - 1}}(0) \cap {S^{2m - 1}}$ it thus suffices to calculate the action of ${h_{\ast }}$ on ${H_{m - 1}}(G)$. Let $d = {a_1}{a_2} \cdots {a_m} + {( - 1)^{m - 1}}$. Let ${w_1}, {w_2}, \ldots ,{w_m}$ be the weights associated with $f$, satisfying ${a_j} / {w_j} + 1 / {w_{j + 1}} = 1$ for $j = 1, 2, \ldots , m - 1$ and ${a_m}/{w_m} + 1/{w_1} = 1$. Let $n = d/{w_1}$, $q = \gcd (n, d)$, $r = q + {( - 1)^m}$. Then ${H_{m - 2}}(K) = {Z^r} \oplus {z_{d/q}}$ and ${H_{m - 1}}(K) = {Z^r}$.
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