Lecture
The topology of the nonsingular fiber near a singular point of a complex analytic function is a fundamental topic in singularity theory. It describes how the space of solutions or level sets of a function is arranged in a neighborhood of a point where the function loses regularity.
Here is a brief and structured explanation:
Let f:Cn→C be a holomorphic function with a singular point at zero: f(0)=0, and ∇f(0)=0.
For small ε≠0, the nonsingular fiber — is a smooth complex manifold of dimension n−1.
It describes what a level set of the function looks like near the singular point, but outside the singularity itself.
The most important result: Milnor's theorem states that for sufficiently small δ and ε, the nonsingular fiber f−1(ε)∩Bδ is smooth and has a topology independent of the choice of ε.
Moreover, there exists the so-called Milnor fiber — it describes the local topology of the function near the singularity.
The boundary of the fiber is a connected manifold, called the Milnor link, and it plays a key role in the study of vanishing cycles and monodromy.
As one goes around the singular point along a small circle in the space of values of the function, the nonsingular fiber is transformed — this is described by monodromy.
Inside the fiber there arise vanishing cycles — homology classes which vanish as one approaches the singularity.
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