IMPLEMENTACIÓN DE MANUFACTURA ESBELTA

Having understood_V>M _{for sufficiently large M, we begin revealing the rest of the variety}

by decreasing the value of M. We use Morse theory to examine how the topology changes as we do so.

There are two important structural theorems that will allow us to accomplish our task, which are as follows.

Theorem 3.3.1 (First Structure Theorem). Let m1 < m2 ∈ R be such that h has no critical values in the interval [m1, m2]. Then V>m1 is topologically equivalent to V>m2; in fact _V>m2 _{is a deformation retract of V}>m1_.

For a discussion, see [Mil63]. The only special feature of hthat is used in the preceding theorem is the fact thath is proper on_VQ. This is guaranteed by Assumption 3.2.4, which implies that the inverse images under h

VQ of bounded sets are again bounded.

follows:

Theorem 3.3.2(Second Structure Theorem). Letcbe a critical value of the height function

h on _VQ, and define the set Σ(c)⊆Σby

Σ(c) ={σ ∈Σ :h(σ) =c},

necessarily finite by Assumption 3.1.2. Letm1, m2∈Rbe values such thatm1 < c < m2and such that c is the onlycritical value ofhin the interval[m1, m2]. Then, topologically, V>m1 can be obtained from V>m2 _{by the attachment of |Σ(c)| disks. Each disk is attached along}

k disjoint subarcs of its boundary, wherek is the degree of degeneracy of the corresponding critical point.

Note that in decreasing the value of M, no new components of _V>M are ever formed. ThusV>M _{is never empty unless V}

Q is empty itself.

Before continuing with the proof of this theorem, we note that it could be proved from standard Morse theorems by a perturbation argument, modifying h slightly to assure that the critical points are non-degenerate and occur at different heights. The reason we present a full proof, however, is that the structure of the proof will be useful examining the homology class of the intersection cycle on_VQ.

Proof. We first prove the theorem assuming that Σ(c) is but a single point σ0 = (x0, y0). Then we will explain how the proof may be modified to allow for more critical points.

Our goal is to understand how the topology of_V>m changes asm decreases fromm2 to

m1. The idea is to modify the height function h to obtain a new proper height function ˜h that has all the same critical points, but such that the height atσ0will have been lowered to a new value ˜csuch thatm1<˜c < c. We shall denote by ˜V the varietyV endowed with this

auxiliary height function. We will then use ˜_V as a tool in understanding _V>m _{for various}

values ofm. The reasoning is detailed in the following steps:

1. The construction of ˜h will be such that ˜h_≤h everywhere and ˜h=h on _V>m2_{. Thus} we have _V>m2 _{= ˜V}>m2_.

2. Now let m0 be any value such that ˜c < m0 < c. By the first structure theorem of Morse Theory, as there are no critical values of ˜h in [m0, m2], we have that ˜V>m2 is topologically equivalent to ˜_V>m0_.

3. We now return to our original height functionh. We will be able to understand exactly what happens topologically as we shift from ˜_V>m0 _toV>m0_.

4. Finally, again by the first structure theorem of Morse Theory and due to the fact that there are no critical values of h in [m1, m0], we have that V>m0 is topologically equivalent to V>m1_.

Thus by understanding the topological changes in the third step, we will completely understand how V>m2 _{and V}>m1 _{differ. With that outline in mind, we turn toward an} examination of the height function.

We begin by examining the height function near the critical point σ0. By smoothness of the variety atσ0,VQmay be parameterized local toσ0 in terms of eitherxory. Assume without loss of generality that we can parameterize in terms of x. Then because h has a critical point at σ0, and because h is the real part of the complex analytic functionH, we must have that dH_dx vanishes at x0 (when H is parameterized by x). Thus locally H takes

the form: H(x) =c+id+ ∞ X j=k cj(x−x0)j,

wherek≥2 and ck6= 0. Denoting C=c+idand g(x) =P∞j=kcj(x−x0)j−k we get H(x) =C+ (x−x0)kg(x),

whereg(x0)6= 0.

Now as g(x0)6= 0, we have in a small neighborhood ofx0 thatg(x)1/k, the kthprincipal root of g(x), is an analytic function. Thus locally the function

w= (x−x0)g(x)1/k

is analytic, and by inspection we see that the derivative dw_dx does not vanish at x=x0. So by the inverse function theorem we can parameterize H locally by w. Note that, in terms of the variablew,H takes on the relatively nice form:

H(w) =C+wk.

Next we move our attention fromH to the actual height functionh. Representing win polar coordinates asw=ρeiθ,we get

H(ρ, θ) =C+ρkeikθ,

soh= ReH takes the form

h(ρ, θ) =c+ρkcos(kθ).

under this parameterization. Now let δ > 0 be sufficiently small so that the ball Bδ(0)

about w = 0 lies within the region on which this parameterization holds, and such that

δ <(m2−c)1/k. The second condition guarantees thath(w)< m2 on Bδ(0), which we will

We now begin the construction of the modified height function ˜h. To that end we must first construct a bump function for lowering the height ofσ0. So let b: [0, δ]→ Rbe some smooth function such that

b(ρ) =        1 forρ_≤δ/3 0 forρ_≥2δ/3

and 0 ≤ b(ρ) ≤ 1 for δ/3 ≤ ρ ≤ 2δ/3. Then ˜h is constructed as follows: on Bδ(0), in

polar coordinates, we have ˜h(ρ, θ) =h(ρ, θ)₋εb(ρ) (where εis a sufficiently small positive constant). And outside of Bδ(0) we have that ˜h ≡ h. Thus ˜h is obtained from h by

subtracting a very small bump function in a neighborhood of the critical point σ0. Note that ˜h is still proper.

Assuming that ε is sufficiently small, ˜h will have no new critical points beyond those of the original height function (∇h 6= 0 inside the modified annulus), and this will be the sought-after modified height function. Note that the height of σ0 has been lowered: ˜

c = ˜h(σ0) =c−ε (which is between m1 and c for sufficiently smallε). Note further that because ˜h < m2 onBδ(0), we indeed have ˜h=h on V>m2 as required.

Now let m0 =c−ε/2, a value between ˜c and c. We wish to understand what the set ˜

V>m0 _{looks like inside the neighborhood B}

δ(0). To do that we first examine the constant

height curves ˜h=m0 in this neighborhood. By the definition of ˜h, these curves consist of 1. The constant height curve h=m0 on points w=ρeiθ such thatρ≥2δ/3.

2. The constant height curveh=m0+ε=c+ε/2 on pointsw=ρeiθ such thatρ≤δ/3. 3. Curves that interpolate between these segments for points w=ρeiθ such that δ/3<

Figure 3.1: The region VQ local to σ0, where ˜V>m0 is shaded and the degeneracyk= 3.

Figure 3.2: The region VQ local to σ0, where V>m0 is shaded and the degeneracyk= 3.

Then we see that the set ˜V>m0_∩B

δ(0) is composed ofkconnected components bound inside Bδ(0) by the curves described above. See Figure 3.1 for an example with k= 3.

Finally, we wish to return to our original height function, and see how the topology changes when we go from ˜V>m0 _toV>m0_{. Becauseh and ˜h only differ on the set B}

2δ/3(0), we see that the change is only local toσ0. And by inspection, we see thatV>m0 is obtained from ˜_V>m0 _{by adding a “star-shaped” region that joins thekcomponents of ˜V}>m0_∩B

δ(0).

See Figure 3.2 for a picture. This proves the theorem in the case that _|Σ(c)_|= 1.

If _|Σ(c)_|>1 we proceed as before, however the height function ˜h is now obtained from

h by subtracting non-overlapping bump functions in a neighborhood of each of the points in Σ(c).

Thus we have our piecewise understanding of the topology of VQ. Before proceeding

structure theorem.

Corollary 3.3.3. Let M _∈R be such that

M >max_{h(σ) :σ _∈Σ_}.

Then _V>M _{is diffeomorphic to a finite set of disjoint, punctured open disks, each of which}

is an x-component or a y-component, but not both.

Proof. By Theorem 3.2.6, _V>M0 _{takes the desired form for sufficiently largeM}

0> M. But ash has no critical values in the interval [M, M0], the first structure theorem tells us that the topologies of V>M _andV>M0 _{are identical.}

Furthermore, we noted that the components of X>M0 _{and Y}>M0 _{are disjoint for suf-} ficiently large M0. As V>M deformation retracts onto V>M0, the same is true of their individual connected components. HenceX>M _andY>M _{are likewise disjoint.}