Thom–Sebastiani theorem

Remarks

Sebastiani and Thom proved the theorem in 1971. The idea is that $F_{f \oplus g}$ has the homotopy type of the join $F_f * F_g$. The reduced homology of a join is the tensor product of the reduced homologies, shifted by one degree. The monodromy of $f \oplus g$ is the join of the two monodromies.

Taking ranks gives $\mu(f \oplus g) = \mu(f) \, \mu(g)$ for the Milnor numbers. This also follows from algebra, since the local algebra of $f \oplus g$ is the tensor product of the local algebras of $f$ and $g$.

Example: $f(x) = x^p$ and $g(y) = y^q$. The Milnor fibre of $x^p$ is $p$ points, cyclically permuted by the monodromy, whose eigenvalues on reduced $H_0$ are the $p$-th roots of unity other than $1$. So the monodromy of $x^p + y^q$ on $H_1$ has as eigenvalues the products $\zeta \eta$ with $\zeta^p = 1 \ne \zeta$ and $\eta^q = 1 \ne \eta$, and $\mu(x^p + y^q) = (p-1)(q-1)$.

Going deeper

For a student

The theorem reduces many computations to one-variable functions. Iterating it, for a Brieskorn–Pham polynomial $z_0^{a_0} + \cdots + z_n^{a_n}$, the monodromy is diagonalizable with eigenvalues $\zeta_0 \cdots \zeta_n$, where $\zeta_j^{a_j} = 1 \ne \zeta_j$, and $\mu = \prod_j (a_j - 1)$. This is the computation behind the topology of Brieskorn manifolds. Adding a square $z^2$ in a new variable does not change $\mu$, and it changes the monodromy only by a sign. This explains the stable behaviour of singularities under adding squares.