Topology of the Milnor fibers: the case of an
isolated point
Ronan Herry
April 21, 2016
The setting
n ≥ 1 integer.
f ∈ C[z1 , . . . , zn+1 ].
f (0) = 0 and 0 isolated critical point.
V = f −1 (0).
D = {|z| ≤ } and S = {|z| = }.
K = V ∩ S .
The Milnor fibration: φ : z ∈ S \ K 7→
The Milnor fibers: Fθ = φ−1 (eiθ ).
f (z)
|f (z)|
∈ S1 .
Main goal
Understand the topology of the fibers Fθ .
Theorem (6.5)
Fθ has the homotopy type of a bouquet of n-sphere.
Theorem (6.6)
For n 6= 2, F¯θ is diffeomorphic to the handlebody: D2n + n-handles
(and probably for n = 2 too).
From the previous chapters
Lemma (Curve selection)
If A algebraic set such that 0 ∈ closure(A ∩ (Cn \ 0)) then there
exist a smooth path p such that p0 = 0 and for all t > 0, pt ∈ A
and pt 6= 0.
Lemma (5.1)
Fθ has the homotopy type of a n CW-complex.
Lemma (5.8)
There exists a Morse function sθ : Fθ → R+ with Morse index ≥ n
so that sθ (z) = |f (z)| when it is close enough to 0.
Structure of the fibers
For > 0 small enough, F¯θ , the closure of Fθ in S :
Lemma (6.1)
is a smooth 2n manifold with boundary ∂ F¯θ = K and interior Fθ .
Lemma (6.2)
has the same homotopy type as its complement S \ F¯θ .
Lemma (6.3)
satisfies Hi = 0 for i < n.
Lemma (6.4)
is (n − 1)-connected.
Homotopy type of the fibers
Theorem (6.5)
Fθ has the homotopy type of a bouquet of n-sphere.
1. In view of Lemmas 5.1 and 6.3, the homology groups are the
ones of a bouquet:
Hn = ⊕Zn and Hi = 0 (i < n).
2. By the Hurewicz theorem (n > 1), πn ' ⊕Zn and πn admits a
finite basis fi : (Sn , v ) → (Fθ , z).
3. ∨fi : ∨ Sn → Fθ induces an isomorphism of homology groups
so it is an homotopy equivalence (Whitehead theorem).
Form of the fibers
Theorem (6.6)
For n 6= 2, F¯θ is diffeomorphic to the handlebody: D2n +
n-handles.
Proved by invoking a big theorem of Smale that rely on the
Poincaré conjecture in dimension 2n − 1. Since it has been proved
in dimension = 3 it is probably true for n = 2.
Boundary of the fibers
Lemma (6.1)
F¯θ is a smooth 2n manifold with boundary ∂ F¯θ = K and interior
Fθ .
1. f S has no critical points on K (will contradict the Curve
selection Lemma).
2. Choose real local coordinates u1 , . . . , u2n such that
f = u1 + iu2 .
3. Locally,
F0 = φ−1 (1) = {u1 > 0, u2 = 0} and F¯0 = {u1 ≥ 0, u2 = 0}
which is a smooth 2n manifold with interior F0 and boundary
K.
Homotopy of the complement
Lemma (6.2)
F¯θ has the same homotopy type as its complement S \ F¯θ .
1. S \ F¯θ = φ−1 (S1 \ eiθ ).
2. Since deformation retract is an homotopy equivalence S \ F¯θ
is equivalent to Fθ0 hence to Fθ (they are diffeomorphic).
Trivial homology groups
Lemma (6.3)
F¯θ has Hi = 0 for i < n.
1. F¯θ ⊂ S ' S2n+1 is compact and locally contractile (because
S1 \ eiθ is).
2. By duality for reduced (co)homology Hi ' H 2n−i = 0 for
2n − i > n by Lemma 5.1.
(n − 1)-connectedness
Lemma (6.4)
Fθ is (n − 1)-connected.
1. If n = 1 ok.
2. Otherwise just need to prove π1 = 0 (Whitehead theorem).
3. Prove that S \ F¯θ is simply connected and use the Lemma 6.2.
4. F¯θ ' union of handles of index ≤ n attached to a disk (−sθ
Morse function).
5. A sphere minus a disk is a disk and is simply connected.
6. Adding handles of index ≤ n ≤ dim(S ) − 3 = 2n − 2 does not
change the fundamental group.