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Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Witten’s Laplacian and the Morse Inequalities

Gianmarco Molino

University of Connecticut

December 1, 2017

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

1 Background 2 Morse Inequalities 3 Witten’s Idea 4 Local Approximation 5 Weak Morse Inequalities 6 Strong and Polynomial Morse Inequalities

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Morse Theory

Morse Theory is the study of critical points of a smooth function f : M → R. A smooth manifold M is a topological manifold with compatible smooth atlas (in the following all manifolds are assumed to be n-dimensional, smooth, oriented, closed, and without boundary.) A critical point q ∈ M of a smooth function f : M → R is a zero of the differential df . The Hessian Hf (q) of f at a critical point q ∈ M is the matrix of second derivatives. (Independent of coordinate system at critical points.)

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Morse Functions

A smooth function f : M → R is called Morse if its critical points are isolated and nondegenerate (that is, the Hessian

• f f is nonsingular.)

Remark: Nondegenerate critical points are necessarily isolated.

The Morse index mq of a critical point q is the dimension

• f the negative eigenspace of Hf (q).

The i-th Morse number Mi is the number of critical points with Morse index i.

Remark: The Morse numbers are invariant under diffeomorphism.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Betti Numbers

Associated to every smooth manifold is the sequence of Betti numbers βi, 0 ≤ i ≤ n defined as βi = dim Hi

dR(M) = dim {α ∈ Ωi : dα = 0}

{dβ : β ∈ Ωi−1} where Ωi is the space of differential i-forms. This sequence is a topological invariant, and notably χ(M) =

n

• i=0

(−1)iβi

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Morse Inequalities

The Weak Morse Inequalities are a classical result, proved using geometric techniques by Milnor [1]. Theorem (Weak Morse Inequalities) Let f : M → R be Morse. Then for any 0 ≤ i ≤ n βi ≤ Mi and moreover χ(M) =

n

• i=0

(−1)iMi

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Morse Inequalities

Theorem (Polynomial Morse Inequalities) Let f : M → R be Morse. Then for any t ∈ R there exists a sequence of nonnegative integers Qi such that Mt − Pt :=

n

• i=0

Miti −

n

• i=0

βiti = (1 + t)

n−1

• i=0

Qiti Theorem (Strong Morse Inequalities) Let f : M → R be Morse. Then for any 0 ≤ k ≤ n

k

• i=0

(−1)i+kβi ≤

k

• i=0

(−1)i+kMi

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Edward Witten

In 1982, Edward Witten published a proof [2] of the Morse Inequalities, essentially using the idea of the flow generated by a Morse functions, with an intuition deriving from Quantum

• Mechanics. He was awarded a Fields Medal in 1990, partially

for this work.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Supersymmetry

A Hilbert space H is called supersymmetric if there exists a decomposition H = H+ ⊕ H− and maps Q1 : H+ → H− Q2 : H− → H+ H, (−1)F : H → H that obey certain symmetry rules.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Supersymmetry

Notice that the space of differential forms is supersymmetric, splitting into even and odd forms, with Ω∗ =

n/2

• i=0

Ω2p ⊕

n/2−1

• i=0

Ω2p+1 Q1 = d + δ Q2 = i(d − δ) H = ∆ = dδ + δd

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Witten Deformation

Witten generalizes this idea, conjugating d with the flow etf for t ≥ 0, f Morse. dt = e−tf detf δt = etf δe−tf ∆t = dtδt + δtdt It is easily verifiable that Ω∗ is still a supersymmetric space using these deformed operators.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Hodge Theory

To understand the motivation for the Witten Laplacian, we need to look to Hodge Theory. Theorem (Hodge Theorem) For 0 ≤ i ≤ n, the maps hi : ker ∆i → Hi

dR(M)

ω → [ω] are isomorphisms. Corollary For 0 ≤ i ≤ n, βi = dim ker ∆i

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Hodge Theory

The corollary is the starting point for analytic approaches to the Betti numbers. We will prove the Hodge Theorem using a heat flow argument, the following lemma will be necessary. Lemma For all smooth differential forms ω, ∆e−t∆ω = e−t∆∆ω and de−t∆ω = e−t∆dω The first claim follows from self-adjointness of ∆, while the second can be proved using the uniqueness of solutions to the heat equation in L2.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Hodge Theory

Proof (Hodge Theorem). Let {ωi}i∈N be an orthonormal basis for Ωp with ∆ωi = λiωi. This can be done since M is compact, and follows from the Spectral Theorem for compact, self-adjoint operators applied to the heat operator e−t∆. Then lim

t→∞ e−t∆ω = lim t→∞

• i

aie−tλiωi =

N

• i=0

aiωi where {ωi, . . . , ωN} is an orthonormal basis for ker ∆p. Thus as t → ∞, ω flows to its harmonic component.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Hodge Theory

Proof (Hodge Theorem). Then for any closed differential form ω, e−t∆ω − ω = t ∂t(e−t∆ω)dt = d

• −

t e−t∆δωdt

• so

e−t∆ω = ω + d

• −

t e−t∆δωdt

• ∈ [ω]

which implies that heat flow preserves the cohomology class of a form.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Hodge Theory

Proof (Hodge Theorem). We see that each cohomology class contains a harmonic form, lim

t→∞ e−t∆ω = ω − lim t→∞ d

t e−t∆δω dt = ω − d∆−1δω (which is well-defined, after showing δω is independent of ker ∆) , now finally we show that the form is unique. Assume there exist harmonic forms η1 = η2 with [η1] = [η2] Then η1 = η2 + dθ δη1 = δη2 + δdθ 0 = δdθ so 0 = θ, δdθ = dθ, dθ = dθ2

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Witten Laplacian

Lemma For any t ≥ 0, βi = dim ker ∆i

t

Proof. Observe that dte−tf = (e−tf detf )e−tf = e−tf d, which implies that e−tf : Ωi → Ωi is an isomorphism making the following diagram commute, · · · Ωi Ωi+1 · · · · · · Ωi Ωi+1 · · ·

d d e−tf d e−tf dt dt dt

so βi = dim ker ∆i = dim ker ∆i

t.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Witten Laplacian

Now we will see Witten’s key insight: that the kernel of ∆i

t is

much simpler to understand as t → ∞. It is necessary to expand the Witten Laplacian directly.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Witten Laplacian

For ω1, ω2 ∈ Ωi dtω1 = e−tf detf ω1 = e−tf (etf dω1 + tetf df ∧ ω1) = (d + tdf ∧)ω1 and δtω1, ω2 = ω1, dtω2 = ω1, (d + tdf ∧)ω2 = ω1, dω2 + ω1, tdf ∧ ω2 = δω1, ω2 + tι∇f ω1, ω2 = (δ + tι∇f )ω1, ω2

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Witten Laplacian

Expanding ∆t, ∆t = dtδt + δtdt = (d + tdf ∧)(δ + tι∇f ) + (δ + tι∇f )(d + tdf ∧) = dδ + tdf ∧ δ + tdι∇f + t2df ∧ ι∇f + δd + tι∇f d + tδdf ∧ +t2ι∇f df ∧ = ∆ + t2df 2 + th where h = L∇f + L∗

∇f

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Witten Laplacian

Then as t becomes large, ω ∈ ker ∆t implies that ω can be nonzero only on small neighborhoods of the critical points of f . We will now consider a neighborhood Uq of a critical point q of f , and compute ∆t in a local coordinate system on Uq.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Morse Lemma

We will use the Morse Lemma to provide a coordinate system. Theorem (Morse Lemma) Let q be an isolated, nondegenerate critical point for f ∈ C ∞(M, R). Then there exists a coordinate system {x1, . . . , xn} on a neighborhood Uq of q such that for x = (x1, . . . , xn) ∈ Uq, f (x) = f (q) −

mq

• i=1

x2

i + n

• i=mq+1

x2

i

where mq is the Morse index of q.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Morse Lemma

To prove the Morse Lemma, we follow the following outline:

1 Prove Hadamard’s Lemma (1st order Taylor expansion). 2 Show that if f (x) = f (q) − λ i=1 x2 i + n i=λ+1 x2 i , then

λ = mq.

3 Choose a coordinate system x1(q) = · · · = xn(q) = 0, and

apply Hadamard’s Lemma twice (using the fact that df (q) = 0.)

4 Argue inductively on the coordinates xi.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Local Coordinate Expansion

In the coordinate system provided by the Morse Lemma, df 2 = 4

n

• i=1

x2

i ,

∇f = −

mq

• i=1

2xi ∂ ∂xi +

n

• i=mq+1

2xi ∂ ∂xi , and by choosing the metric in local coordinates to be g = n

i=1(dxi)2 at q,

∆ = −

n

• i=1

∂2 ∂x2

i

.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Local Coordinate Expansion

A (long) computation for h = L∇f + L∗

∇f gives

h = 2

n

• i=1

ηi[dxi, ι ∂

∂xi

] where ηi =

• −1

i ≤ mq, 1, i > mq.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Local Coordinate Expansion

Thus on Uq we can approximate ∆t by, Hq,t =

n

• i=1

− ∂2 ∂x2

i

+ 4t2x2

i + 2ηit[dxi∧, ι ∂

∂xi

] =

n

• i=1

Ji + 2tKi Where Ji := − ∂2

∂x2

i + 4t2x2

i and Ki := ηi[dxi, ι ∂

∂xi

].

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Local Coordinate Expansion

Notice, for a differential form ω = fIdxI (with I a multiindex,) Kiω =

• −ω

(i ≤ mq and i ∈ I) or (i > mq and i / ∈ I) ω

• therwise

so Ki = ±1 = ⇒ [Ji, Ki] = 0 which implies that they can be simultaneously diagonalized.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Local Coordinate Approximation

Proposition (Kernel of Hq,t) For any q ∈ Cr(f ), the map Hq,t : Uq → R defined in the local coordinates {xi} given by the Morse lemma on the neighborhood Uq of q by Hq,t =

n

• i=1

− ∂2 ∂x2

i

+ 4t2x2

i + 2ηit[dxi∧, ι ∂

∂xi

] has kernel of dimension one, and is generated by the eigenform e−t|x|2dx1 ∧ · · · ∧ dxmq. and moreover all of the nonzero eigenvalues of Hq,t are greater than Ct for some fixed C > 0.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Eigenvalues of the Quantum Harmonic Oscillator

The operator Ji is a scaling of the simple quantum harmonic

• scillator from physics. It’s spectrum is well-known.

Proposition The eigenvalues of Ji are precisely 2t(1 + 2j) for non-negative integers j. Moreover, the 2t-eigenfunction of Ji is e−tx2

i

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Eigenvalues of the Quantum Harmonic Oscillator I

To determine the eigenvalues of the quantum harmonic

• scillator, argue using the ‘Dirac Ladder Operator’ method:

1 Define p = −i ∂ ∂xi and a = √t

• xi + i

2t p

• so that

Ji = 4t2x2

i + p2 2 Then

2t(1 + 2a†a) = Ji so an eigenvalue of N = a†a is an eigenvalue of Ji.

3 Show that

Nafλ = (λ − 1)afλ Na†fλ = (λ + 1)a†fλ.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Eigenvalues of the Quantum Harmonic Oscillator II

4 Argue that eigenvalues must be nonnegative since

λfλ, fλ = fλ, a†afλ = afλ2 ≥ 0

5 Argue that if λ is not a nonnegative integer, applying a

sufficiently many times to fλ would result in a function with negative eigenvalue, a contradiction.

6 Conclude the first claim using 2t(1 + 2N) = Ji. 7 Finally, show that e−tx2

i is the 2t-eigenfunction of Ji

directly, solving Naf0 = 0.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Local Coordinate Approximation

Now we can prove the proposition. For an eigenform ω of Hq,t, Hq,tω = t

• 2

n

• i=1

(1 + 2j + Ki)

• ω

so if ω = g(x)dxI ∈ ker Hq,t is nontrivial, then it must be that j = 0 which forces g(x) = e−t|x|2 and also i ∈ I if and only if i ≤ mq. In conclusion, ω = e−t|x|2dx1 ∧ · · · ∧ dxmq generates the kernel of Hq,t.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Physical Intuition

There is now a good intuition as to why the Weak Morse Inequalities should hold. For each q ∈ Cr(f ) with mq = p, the kernel of Hq,t is generated by a single p-form locally. There will be precisely Mp critical points of f with p-forms generating the kernel of Hq,t|p

Ω locally.

Globally it seems reasonable that for ω ∈ ker ∆p

t it must

be that ω ∈ Hq,t so Mp ≥ dim ker ∆p

t = βp.

Witten argues along these lines. Here is presented a justification by global analysis of the low-lying eigenvalues

• f ∆p

t , adapted from [3].

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of the Weak Morse Inequalities

Denote by E p

t (c) the eigenspace of ∆p t with eigenvalues in the

interval [0, c]. The following key theorem will give the Weak Morse Inequalities as a corollary. Theorem (Key Theorem) For any c > 0, there exists a t0 > 0 such that for any t > t0, dim E p

t (c) = Mp

where Mp is the p-th Morse number, 0 ≤ p ≤ n. Corollary (Weak Morse Inequalities) For 0 ≤ p ≤ n, βp ≤ Mp

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of the Weak Morse Inequalities

To prove the key theorem, we will need estimates on Sobolev

• norms. Without loss of generality assume Uq is an open ball

centered at critical point q with radius 4a, and choose γ ∈ C ∞(R, [0, 1]) to be such that γ(x) =

• 1

|x| < a |x| > 2a

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of the Weak Morse Inequalities

Define αq,t =

• γ(|x|)e−t|x|2
• 2

0 =

• Uq

γ(|x|)2e−2t|x|2 dx1 ∧ · · · ∧ dxn ρq,t = γ(|x|) √αq,t e−t|x|2dx1 ∧ · · · ∧ dxmq The ρq,t will have unit length, and are motivated by the local generators for the kernel of Hq,t. Denote by Et the vector space genersated by the ρq,t for q ∈ Cr(f ).

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Projection Lemma

The following lemma estimating the orthogonal projections Pt(c): H0(M) → Et(c) will allow us to prove the key theorem. Lemma There exist constants C, t3 > 0 such that for any t ≥ t3 and any σ ∈ Et, Pt(c)σ − σ0 ≤ C t σ0

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Deformed Witten Operator

Denote by Dt the ‘deformed Witten operator.’ Dt = dt + δt and observe that ∆t = dtδt + δtdt = (dt + δt)(dt + δt) = D2

t

since d2

t = δ2 t = 0.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Deformed Witten Operator

Since the ρq,t are compactly supported, Et is a finite dimensional subspace of H0(M), and so there is an orthogonal decomposition H0(M) = Et ⊕ E ⊥

t

with projections pt : H0(M) → Et p⊥

t : H0(M) → E ⊥ t

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of Key Theorem

Proof (Key Theorem). By the lemma, when t is large enough the Pt(c)ρq,t will be linearly independent, so dim Et(c) ≥ dim Et. Assume for contradiction that dim Et(c) > dim Et. Then there must be a nonzero s ∈ Et(c) orthogonal to Pt(c)Et. That is s, Pt(c)ρq,t0 = 0 for all q ∈ Cr(f ).

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of Key Theorem

Proof (Key Theorem). Then we can write the projection pts =

• q∈Cr(f )

s, ρq,t0ρq,t =

• q∈Cr(f )

s, ρq,t0(ρq,t − Pt(c)ρq,t) +

• q∈Cr(f )

s, ρq,t − Pt(c)ρq,t0Pt(c)ρq,t so by the lemma pts0 ≤ C t s0.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of Key Theorem

Proof (Key Theorem). Then p⊥

t s0 ≥ s0 − pts0 ≥ C ′s0

and then using the proposition, CC ′√ ts0 ≤ Dtp⊥

t s0 ≤ Dts0 + Dtpts0

≤ Dts0 + 1 t s0. Rearranging, Dts0 ≥ CC ′t3/2 − 1 t s0 which as t → ∞ contradicts s ∈ Et(c).

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of Key Theorem

Proof (Key Theorem). Now dim Et(c) = dim Et =

n

• i=0

Mi and Et(c) is generated by Pt(c)ρq,t. Let Qi denote the projection H0(M) → L2Ωi. We have that ∆tQis = Qi∆ts = µ2Qis so that Qis is a µ2-eigenform of ∆t. We wish to show that for t large enough, dim QiEt(c) = Mi.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Proof of Key Theorem

Proof (Key Theorem). By the lemma, for t large enough QmqPt(c)ρq,t − ρq,t0 ≤ C t thus the forms QmqPt(c)ρq,t are linearly independent and dim QiEt(c) ≥ Mi. However,

n

• i=0

dim QiEt(c) ≤ dim Et(c) =

n

• i=0

Mi forcing dim QiEt(c) = Mi, and completing the proof.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Polynomial Morse Inequalities I

Witten proves the Polynomial Morse Inequalities. The Strong Morse Inequalities follow by equivalence. We outline the proof

• f the Polynomial Morse Inequalties:

1 Let Cp(f ) be the free abelian group generated by the

critical points q ∈ M with Morse index mq = p. Denote by dp

t the map Cp(f ) → Cp+1(f ) determined by dt,

identifying critical points with the associated eigenforms of ∆t. Consider the Morse-Smale-Witten chain complex 0 → C1(f ) → · · · → Cn(f ) → 0.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Polynomial Morse Inequalities II

2 The sequence

0 → ker dp

t → Cp(f ) dp

t

− → im dp

t → 0

is exact, so Mp = rank Cp(f ) = rank ker dp

t + rank im dp t 3 The sequence

0 → im dp−1

t

→ ker dp

t → Hk(C∗(f ), d∗ t ) → 0

is also exact, so βp = rank Hk(C∗(f ), d∗

t ) = rank ker dp t − rank im dp−1 t

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Polynomial Morse Inequalities III

4 Then letting Qp = Mp − rank ker dp t ≥ 0,

Mt − Pt =

n

• p=0

(rank ker dp

t + rank im dp t )tp

−

n

• p=0

(rank ker dp

t − rank im dp−1 t

)tp = (1 + t)

n−1

• p=0

(Mp − rank ker dp

t )tp

= (1 + t)

n−1

• p=0

Qptp

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Strong Morse Inequalities I

Theorem The Strong and Polynomial Morse Inequalities are equivalent. This can be proved as follows:

1 Assume the Strong Inequalities. Then

M−1 =

n

• i=0

(−1)iMi =

n

• i=0

(−1)iβi = P−1 implies that Mt − Pt is divisible by (1 + t).

2 Then for some Qi ∈ Z,

Mt − Pt = (1 + t)

n−1

• i=0

Qiti

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

Strong Morse Inequalities II

3 Arguing by induction using the Strong Inequalities, we can

show that the Qi must be nonnegative, proving the Polynomial Inequalities.

4 Assume the Polynomial Inequalities. Then by induction for

k ∈ {0, 1, . . . , n − 1},

k

• i=0

(−1)i+kMi =

k

• i=0

(−1)i+kβi + Qk

5 Letting t = −1, n

• i=0

(−1)iMi =

n

• i=0

(−1)iβi completing the equivalence.

Witten’s Laplacian and the Morse Inequalities Gianmarco Molino Background Morse Inequalities Witten’s Idea Local Approximation Weak Morse Inequalities Strong and Polynomial Morse Inequalities References

John W. Milnor. Morse Theory. Princeton University Press, 1973. isbn: 978-0691080086. Edward Witten. “Supersymmetry and Morse Theory”. In:

• J. Differential Geom. 17.4 (1982), pp. 661–692. doi:

10.4310/jdg/1214437492. Weiping Zhang. Lectures on Chern-Weil Theory and Witten Deformations. Singapore: World Scientific Publishing, 2001. isbn: 981-02-4685-4.