Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Witten’s Laplacian and the Morse Inequalities
Gianmarco Molino
University of Connecticut
December 1, 2017
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
1 Background
2 Morse Inequalities
3 Witten’s Idea
4 Local Approximation
5 Weak Morse Inequalities
6 Strong and Polynomial Morse Inequalities
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Morse Theory
Morse Theory is the study of critical points of a smoothfunction f : M → R.
A smooth manifold M is a topological manifold withcompatible smooth atlas (in the following all manifolds areassumed to be n-dimensional, smooth, oriented, closed,and without boundary.)
A critical point q ∈ M of a smooth function f : M → R isa zero of the differential df .
The Hessian Hf (q) of f at a critical point q ∈ M is thematrix of second derivatives. (Independent of coordinatesystem at critical points.)
Witten’sLaplacian and
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Morse Functions
A smooth function f : M → R is called Morse if its criticalpoints are isolated and nondegenerate (that is, the Hessianof f is nonsingular.)
Remark: Nondegenerate critical points are necessarilyisolated.
The Morse index mq of a critical point q is the dimensionof the negative eigenspace of Hf (q).
The i-th Morse number Mi is the number of critical pointswith Morse index i .
Remark: The Morse numbers are invariant underdiffeomorphism.
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Background
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Betti Numbers
Associated to every smooth manifold is the sequence ofBetti numbers βi , 0 ≤ i ≤ n defined as
βi = dimH idR(M) = dim
α ∈ Ωi : dα = 0dβ : β ∈ Ωi−1
where Ωi is the space of differential i-forms. This sequence is atopological invariant, and notably
χ(M) =n∑
i=0
(−1)iβi
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Morse Inequalities
The Weak Morse Inequalities are a classical result, proved usinggeometric 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
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Morse Inequalities
Theorem (Polynomial Morse Inequalities)
Let f : M → R be Morse. Then for any t ∈ R there exists asequence of nonnegative integers Qi such that
Mt − Pt :=n∑
i=0
Mi ti −
n∑i=0
βi ti = (1 + t)
n−1∑i=0
Qi ti
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
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Edward Witten
In 1982, Edward Witten published a proof [2] of the MorseInequalities, essentially using the idea of the flow generated bya Morse functions, with an intuition deriving from QuantumMechanics. He was awarded a Fields Medal in 1990, partiallyfor this work.
Witten’sLaplacian and
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Supersymmetry
A Hilbert space H is called supersymmetric if there exists adecomposition H = H+ ⊕H− and maps
Q1 : H+ → H−
Q2 : H− → H+
H, (−1)F : H → H
that obey certain symmetry rules.
Witten’sLaplacian and
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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
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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 spaceusing these deformed operators.
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Hodge Theory
To understand the motivation for the Witten Laplacian, weneed to look to Hodge Theory.
Theorem (Hodge Theorem)
For 0 ≤ i ≤ n, the maps
hi : ker ∆i → H idR(M)
ω 7→ [ω]
are isomorphisms.
Corollary
For 0 ≤ i ≤ n,βi = dim ker ∆i
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Hodge Theory
The corollary is the starting point for analytic approaches tothe Betti numbers. We will prove the Hodge Theorem using aheat flow argument, the following lemma will be necessary.
Lemma
For all smooth differential forms ω,
∆e−t∆ω = e−t∆∆ω
andde−t∆ω = e−t∆dω
The first claim follows from self-adjointness of ∆, while thesecond can be proved using the uniqueness of solutions to theheat equation in L2.
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Hodge Theory
Proof (Hodge Theorem).
Let ωii∈N be an orthonormal basis for Ωp with ∆ωi = λiωi .This can be done since M is compact, and follows from theSpectral Theorem for compact, self-adjoint operators applied tothe heat operator e−t∆. Then
limt→∞
e−t∆ω = limt→∞
∑i
aie−tλiωi =
N∑i=0
aiωi
where ωi , . . . , ωN is an orthonormal basis for ker ∆p. Thusas t →∞, ω flows to its harmonic component.
Witten’sLaplacian and
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Hodge Theory
Proof (Hodge Theorem).
Then for any closed differential form ω,
e−t∆ω − ω =
∫ t
0∂t(e
−t∆ω)dt
= d
(−∫ t
0e−t∆δωdt
)so
e−t∆ω = ω + d
(−∫ t
0e−t∆δωdt
)∈ [ω]
which implies that heat flow preserves the cohomology class ofa form.
Witten’sLaplacian and
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Hodge Theory
Proof (Hodge Theorem).
We see that each cohomology class contains a harmonic form,
limt→∞
e−t∆ω = ω − limt→∞
d
∫ t
0e−t∆δω dt = ω − d∆−1δω
(which is well-defined, after showing δω is independent ofker ∆) , now finally we show that the form is unique. Assumethere exist harmonic forms η1 6= η2 with [η1] = [η2] Then
η1 = η2 + dθ
δη1 = δη2 + δdθ
0 = δdθ
so0 = 〈θ, δdθ〉 = 〈dθ, dθ〉 = ‖dθ‖2
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Background
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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 followingdiagram commute,
· · · Ωi Ωi+1 · · ·
· · · Ωi Ωi+1 · · ·
d d
e−tf
d
e−tf
dt dt dt
so βi = dim ker ∆i = dim ker ∆it .
Witten’sLaplacian and
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Background
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Witten Laplacian
Now we will see Witten’s key insight: that the kernel of ∆it is
much simpler to understand as t →∞. It is necessary toexpand the Witten Laplacian directly.
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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〉
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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 ∧
= ∆ + t2‖df ‖2 + th
whereh = L∇f + L∗∇f
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Witten Laplacian
Then as t becomes large, ω ∈ ker ∆t implies that ω can benonzero only on small neighborhoods of the critical points of f .We will now consider a neighborhood Uq of a critical point q off , and compute ∆t in a local coordinate system on Uq.
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Morse Lemma
We will use the Morse Lemma to provide a coordinate system.
Theorem (Morse Lemma)
Let q be an isolated, nondegenerate critical point forf ∈ C∞(M,R). Then there exists a coordinate systemx1, . . . , xn on a neighborhood Uq of q such that forx = (x1, . . . , xn) ∈ Uq,
f (x) = f (q)−mq∑i=1
x2i +
n∑i=mq+1
x2i
where mq is the Morse index of q.
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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 x2i +
∑ni=λ+1 x
2i , then
λ = mq.
3 Choose a coordinate system x1(q) = · · · = xn(q) = 0, andapply Hadamard’s Lemma twice (using the fact thatdf (q) = 0.)
4 Argue inductively on the coordinates xi .
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Local Coordinate Expansion
In the coordinate system provided by the Morse Lemma,
‖df ‖2 = 4n∑
i=1
x2i ,
∇f = −mq∑i=1
2xi∂
∂xi+
n∑i=mq+1
2xi∂
∂xi,
and by choosing the metric in local coordinates to beg =
∑ni=1(dx i )2 at q,
∆ = −n∑
i=1
∂2
∂x2i
.
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Local Coordinate Expansion
A (long) computation for h = L∇f + L∗∇f gives
h = 2n∑
i=1
ηi [dxi , ι ∂
∂xi
]
where
ηi =
−1 i ≤ mq,
1, i > mq.
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Local Coordinate Expansion
Thus on Uq we can approximate ∆t by,
Hq,t =n∑
i=1
− ∂2
∂x2i
+ 4t2x2i + 2ηi t[dx i∧, ι ∂
∂xi
]
=n∑
i=1
Ji + 2tKi
Where Ji := − ∂2
∂x2i
+ 4t2x2i and Ki := ηi [dx
i , ι ∂∂xi
].
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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 otherwise
so Ki = ±1 =⇒ [Ji ,Ki ] = 0 which implies that they can besimultaneously diagonalized.
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Local Coordinate Approximation
Proposition (Kernel of Hq,t)
For any q ∈ Cr(f ), the map Hq,t : Uq → R defined in the localcoordinates xi given by the Morse lemma on theneighborhood Uq of q by
Hq,t =n∑
i=1
− ∂2
∂x2i
+ 4t2x2i + 2ηi t[dx i∧, 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 greaterthan Ct for some fixed C > 0.
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Eigenvalues of the Quantum Harmonic Oscillator
The operator Ji is a scaling of the simple quantum harmonicoscillator from physics. It’s spectrum is well-known.
Proposition
The eigenvalues of Ji are precisely 2t(1 + 2j) for non-negativeintegers j . Moreover, the 2t-eigenfunction of Ji is
e−tx2i
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Eigenvalues of the Quantum Harmonic Oscillator I
To determine the eigenvalues of the quantum harmonicoscillator, argue using the ‘Dirac Ladder Operator’ method:
1 Define p = −i ∂∂xi and a =√t(xi + i
2t p)
so that
Ji = 4t2x2i + 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.
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Eigenvalues of the Quantum Harmonic Oscillator II
4 Argue that eigenvalues must be nonnegative since
λ〈fλ, fλ〉 = 〈fλ, a†afλ〉= ‖afλ‖2 ≥ 0
5 Argue that if λ is not a nonnegative integer, applying asufficiently many times to fλ would result in a functionwith negative eigenvalue, a contradiction.
6 Conclude the first claim using 2t(1 + 2N) = Ji .
7 Finally, show that e−tx2i is the 2t-eigenfunction of Ji
directly, solving Naf0 = 0.
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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)dx I ∈ kerHq,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 ifi ≤ mq. In conclusion,
ω = e−t|x |2dx1 ∧ · · · ∧ dxmq
generates the kernel of Hq,t .
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Physical Intuition
There is now a good intuition as to why the Weak MorseInequalities should hold.
For each q ∈ Cr(f ) with mq = p, the kernel of Hq,t isgenerated by a single p-form locally.
There will be precisely Mp critical points of f with p-formsgenerating the kernel of Hq,t |pΩ locally.
Globally it seems reasonable that for ω ∈ ker ∆pt it must
be that ω ∈ Hq,t so Mp ≥ dim ker ∆pt = βp.
Witten argues along these lines. Here is presented ajustification by global analysis of the low-lying eigenvaluesof ∆p
t , adapted from [3].
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Proof of the Weak Morse Inequalities
Denote by Ept (c) the eigenspace of ∆p
t with eigenvalues in theinterval [0, c]. The following key theorem will give the WeakMorse Inequalities as a corollary.
Theorem (Key Theorem)
For any c > 0, there exists a t0 > 0 such that for any t > t0,
dimEpt (c) = Mp
where Mp is the p-th Morse number, 0 ≤ p ≤ n.
Corollary (Weak Morse Inequalities)
For 0 ≤ p ≤ n,βp ≤ Mp
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Proof of the Weak Morse Inequalities
To prove the key theorem, we will need estimates on Sobolevnorms. Without loss of generality assume Uq is an open ballcentered at critical point q with radius 4a, and chooseγ ∈ C∞(R, [0, 1]) to be such that
γ(x) =
1 |x | < a
0 |x | > 2a
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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 localgenerators for the kernel of Hq,t . Denote by Et the vectorspace genersated by the ρq,t for q ∈ Cr(f ).
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Projection Lemma
The following lemma estimating the orthogonal projectionsPt(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 andany σ ∈ Et ,
‖Pt(c)σ − σ‖0 ≤C
t‖σ‖0
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Deformed Witten Operator
Denote by Dt the ‘deformed Witten operator.’
Dt = dt + δt
and observe that
∆t = dtδt + δtdt = (dt + δt)(dt + δt) = D2t
since d2t = δ2
t = 0.
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Deformed Witten Operator
Since the ρq,t are compactly supported, Et is a finitedimensional subspace of H0(M), and so there is an orthogonaldecomposition
H0(M) = Et ⊕ E⊥t
with projections
pt : H0(M)→ Et
p⊥t : H0(M)→ E⊥t
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Proof of Key Theorem
Proof (Key Theorem).
By the lemma, when t is large enough the Pt(c)ρq,t will belinearly independent, so
dimEt(c) ≥ dimEt .
Assume for contradiction that dimEt(c) > dimEt . Then theremust be a nonzero s ∈ Et(c) orthogonal to Pt(c)Et . That is
〈s,Pt(c)ρq,t〉0 = 0
for all q ∈ Cr(f ).
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Proof of Key Theorem
Proof (Key Theorem).
Then we can write the projection
pts =∑
q∈Cr(f )
〈s, ρq,t〉0ρq,t
=∑
q∈Cr(f )
〈s, ρq,t〉0(ρq,t − Pt(c)ρq,t)
+∑
q∈Cr(f )
〈s, ρq,t − Pt(c)ρq,t〉0Pt(c)ρq,t
so by the lemma
‖pts‖0 ≤C
t‖s‖0.
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Proof of Key Theorem
Proof (Key Theorem).
Then‖p⊥t s‖0 ≥ ‖s‖0 − ‖pts‖0 ≥ C ′‖s‖0
and then using the proposition,
CC ′√t‖s‖0 ≤ ‖Dtp
⊥t s‖0 ≤ ‖Dts‖0 + ‖Dtpts‖0
≤ ‖Dts‖0 +1
t‖s‖0.
Rearranging,
‖Dts‖0 ≥CC ′t3/2 − 1
t‖s‖0
which as t →∞ contradicts s ∈ Et(c).
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Proof of Key Theorem
Proof (Key Theorem).
Now
dimEt(c) = dimEt =n∑
i=0
Mi
and Et(c) is generated by Pt(c)ρq,t . Let Qi denote theprojection H0(M)→ L2Ωi . We have that
∆tQi s = Qi∆ts = µ2Qi s
so that Qi s is a µ2-eigenform of ∆t . We wish to show that fort large enough, dimQiEt(c) = Mi .
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Proof of Key Theorem
Proof (Key Theorem).
By the lemma, for t large enough
‖QmqPt(c)ρq,t − ρq,t‖0 ≤C
t
thus the forms QmqPt(c)ρq,t are linearly independent anddimQiEt(c) ≥ Mi . However,
n∑i=0
dimQiEt(c) ≤ dimEt(c) =n∑
i=0
Mi
forcing dimQiEt(c) = Mi , and completing the proof.
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Polynomial Morse Inequalities I
Witten proves the Polynomial Morse Inequalities. The StrongMorse Inequalities follow by equivalence. We outline the proofof the Polynomial Morse Inequalties:
1 Let Cp(f ) be the free abelian group generated by thecritical points q ∈ M with Morse index mq = p. Denoteby 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’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Polynomial Morse Inequalities II
2 The sequence
0→ ker dpt → Cp(f )
dpt−→ im dp
t → 0
is exact, so
Mp = rankCp(f ) = rank ker dpt + rank im dp
t
3 The sequence
0→ im dp−1t → ker dp
t → Hk(C∗(f ), d∗t )→ 0
is also exact, so
βp = rankHk(C∗(f ), d∗t ) = rank ker dpt − rank im dp−1
t
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Polynomial Morse Inequalities III
4 Then letting Qp = Mp − rank ker dpt ≥ 0,
Mt − Pt =n∑
p=0
(rank ker dpt + rank im dp
t )tp
−n∑
p=0
(rank ker dpt − rank im dp−1
t )tp
= (1 + t)n−1∑p=0
(Mp − rank ker dpt )tp
= (1 + t)n−1∑p=0
Qptp
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
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
Qi ti
Witten’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
Strong Morse Inequalities II
3 Arguing by induction using the Strong Inequalities, we canshow that the Qi must be nonnegative, proving thePolynomial Inequalities.
4 Assume the Polynomial Inequalities. Then by induction fork ∈ 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’sLaplacian and
the MorseInequalities
GianmarcoMolino
Background
MorseInequalities
Witten’s Idea
LocalApproximation
Weak MorseInequalities
Strong andPolynomialMorseInequalities
References
John W. Milnor. Morse Theory. Princeton UniversityPress, 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 andWitten Deformations. Singapore: World ScientificPublishing, 2001. isbn: 981-02-4685-4.