The statement of the index theorem
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Atiyah-Singer Index Theorem (1968). Let D : C ∞ (E) → C ∞ (F ) be an
elliptic differential operator. Then
Index D = t-Index D
Remark.
1. Index D is called the analytic index.
2. Index D := dim ker D − dim cok D, where cok D = C ∞ F/D(C ∞ E).
3. t-Index D is called the topological index. There are 3 points of view:
(a) t-Index D := (−1)n (ch(σD) Td(X))[T X] “cohomology form of index theorem”
(b) t-Index D = i! (σD) ∈ K 0 (pt) “K-theory form of the index theorem”
R
(c) t-Index D = X KAS where KAS is an explicitly defined poly in
the curvatures of X, E, and F . “geometric form of the index
theorem”
Glossary
• E, F are smooth complex vector bundles over a closed smooth
manifold X.
• C ∞ E and C ∞ F : vector spaces of smooth sections.
• σD is the symbol of D. This is a bundle σD ↓ TC∗ X.
• ch and Td are the chern character and the the Todd class; they
are characteristic classes
Four classical elliptic operators
DeRham operator, Index D = χ(X)
Signature operator, defined for oriented manifolds, Index D = sign(X). Implies the Hirzebruch signature theorem
Dolbeault operator, defined for complex manifolds. Implies the RiemannRoch Theorem
Dirac operator, defined for Spin manifolds
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Local differential operators
For α = (α1 , . . . , αn ), let
∂1α
∂α
. . . n : C ∞ (Rn , R) → C ∞ (Rn , R)
∂xα1
∂xαn
Dα =
|α| = α1 + · · · + αn is the order of this differential operator.
A (linear) differential operator
D : C ∞ (Rn , RN ) = C ∞ (Rn , R)N → C ∞ (Rn , RM ) = C ∞ (Rn , R)M
is a linear combination
D=
X
Aα (x)Dα
α
α
where A (x) is an M × N matrix of real-valued functions of n-variables.
For D of order m, for x, ξ ∈ Rn , the (local) symbol is
σD (x, ξ) ∈ MM ×N (R)
where σD (x, ξ) =
P
|α|=m
Aα (x)ξ α with ξ α = ξ1α1 · · · ξnαn
Definition. D is elliptic if ∀x, ∀ξ 6= 0, σD (x, ξ) is an isomorphism. (Hence
M = N ).
Note: For f (x) = u(x) + iv(x) ∈ C ∞ (Rn , C),
∂f
∂u
∂v
=
+i
∂xi
∂xi
∂xi
Example.
Del ∇ =
∂
∂
∂
i+
j + k = (∂/∂x1 , ∂/∂x2 , ∂/∂x3 )
∂x
∂y
∂z
• Grad ∇ : C ∞ (R3 ; R) → C ∞ (R3 ; R)3
• Div ∇· : C ∞ (R3 ; R)3 → C ∞ (R3 ; R)
• Curl ∇× : C ∞ (R3 ; R)3 → C ∞ (R3 ; R)
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• Laplacian ∆ = ∇ · ∇ : C ∞ (R3 ; R) → C ∞ (R3 ; R)
• Exterior derivative
d : C ∞ (Λ1 (T ∗ (R3 )) → C ∞ (Λ2 (T ∗ (R3 ))
1-forms 7→ 2-forms
ω = f1 dx1 + f2 dx2 + f3 dx3 7→ dω = df1 ∧ dx1 + df2 ∧ dx2 + df3 ∧ dx3
where dfi = (∂fi /∂x1 )dx1 + (∂fi /∂x2 )dx2 + (∂fi /∂x3 )dx3
Question: Does there exist an order 1 elliptic operator with X = R3 ?
Topological Manifolds
Definition. A topological manifold X of dim n is a Hausdorff, 2nd countable
space so that ∀x ∈ X, ∃ nbdh U of x homeo to Rn
Remark. Any subset of RN is Hausdorff, 2nd countable.
Theorem. Any n-manifold is homeo to a subset of R2n+1 .
Definition. For an n-dim’l manifold X, a chart h : U → V is a homeo with
U open in M and V open in Rn . h−1 : V → U is a local parameterization.
An atlas is a collection of charts A = {hi : Ui → Vi } so that {Ui } cover X.
Smooth Manifolds
Definition. Two charts h1 : U1 → V1 and h2 : U2 → V2 are smoothly related
if h2 h−1
1 : h1 (U1 ∩ U2 ) → h2 (U1 ∩ U2 ) is a diffeomorphism.
Definition. An atlas on M is smooth if all its charts are smoothly related.
Proposition. Every smooth atlas A is contained in a unique max’l smooth
atlas D(A).
In fact D(A) = all charts smoothly related to A.
Definition. A smooth manifold is a pair (X, D) consisting of a topological
manifold and a maximal smooth atlas.
Remark. Any smooth atlas (X, A) determines an unique smooth manifold.
From now on, manifold = smooth manifold, chart = smooth chart, atlas
= smooth atlas
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eg. X = Rn and A = {Id : Rn → Rn }
e.g. X = S n A = {hN : (S n − N ) → Rn , hS : (S n − S) → Rn }
Smooth atlas determines manifold:
given smooth atlas A = {hi : Ui → Vi }i∈Λ on X,
a
X∼
Vi / ∼
=
i∈Λ
−1
where x ∈ Vi ∼ y ∈ Vj ⇔ h−1
i (x) = hj (y)
Definition. A k-dim’l submanifold is a subset Y ⊂ X n so that for every
y ∈ Y , there is chart h : U → V with y ∈ U so that h(U ∩ Y ) = Rk × 0.
Definition. A function f : X → Y is smooth if it is “locally smooth” if
∀x ∈ X, ∃ charts (h : U → V ), (k : W → Z) with x ∈ U, f (x) ∈ W s.t.
kf h−1 is smooth.
Tangent bundles and derivatives
derivatives need tangent bundles
X n = manifold
Tangent bundle π : T X → X where Tx X = π −1 x is a n-dim’s v.s.
Tangent bundle of a submanifold X of of RN
T X = {(x, v) ∈ X × RN | ∃γ : [−1, 1] → X, γ(0) = x, γ 0 (0) = v}
Abstract tangent bundle
If X has atlas {hi : Ui → Vi }, then
`
Vi × Rn
TX =
∼
−1
−1
where (x, u)i ∼ (y, v)j if h−1
i (x) = hj (y) and d(hj hi )x u = v.
If f : X → Y is smooth, the derivative df : T X → T Y
df
T X −−−→


y
f
TY


y
X −−−→ Y
hence dfx : Tx X → Tf (x) Y .
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Transversality
transversality produces manifolds
Why is sphere a manifold?
Definition. smooth f : X → Y is transverse to submanifold L ⊂ Y , if
∀x ∈ f −1 L
dfx Tx X + Tf (x) L = Tf (x) Y
Write f t L
Implicit function theorem implies
Theorem (Transversality theorem 1). f t L ⇐⇒ f −1 L ⊂ X is a submanifold and
∼
=
ν(f −1 L ,→ X) −
→ ν(L ,→ Y )
Transversality is generic
Theorem (Transversality theorem 2). Let f : X → Y be cont. and L ⊂ Y
be a submanifold. Then f ' g with g t L.
Smooth vector bundles and elliptic operators
Two definitions of rank n vector bundle
Definition (Milnor-Stasheff). (p : E → X, ∀x ∈ X v.s. structure on p−1 x),
so that
• ∀x ∈ X, ∃ nbhd U and homeo φ
φ
p−1 U
U × Rn
U
so that p−1 x → x × Rn is a v.s. iso.
Definition (Steenrod). (p : E → X, homeos B = {φi : p−1 Ui → Ui × Rn })
so that
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1.
φi
p−1 Ui
Ui × Rn
Ui
commutes
2. {Ui } open cover of X
3. ∀i, j, ∃ cont θij : Ui ∩ Uj → GLn (Rn )
−1
φ−1
i (x, v) = φj (x, (θij (x)v)
4. B is max’l with respect to the above three properties.
Definition (Milnor-Stasheff). smooth vector bundle: require E and X are
manifolds, and φis are diffeos.
Definition (Steenrod). smooth vector bundle: require X is manifold and θij
are smooth.
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Definition. smooth section is smooth map s : X → E s.t. p ◦ s = IdX .
C ∞ (E) is the v.s. of smooth sections.
e.g. E = X × RM , C ∞ (E) = C ∞ (X, R)M
Exercise. Make precise sense of the following remark.
Remark. A smooth section of an M -plane bundle over an n-manifold is
locally an element of C ∞ (Rn , RM ) with V open in Rn . (Explicitly use charts
h and φ)
C ∞ (E) is a module over C ∞ (X, R)
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(f, s) 7→ f s = (x 7→ f (x)s(x)).
Definition of differential operators on manifolds
Let E → X and F → X be smooth vector bundle.
Definition. Let Op(E, F ) = Hom(C ∞ E, C ∞ F ) (all linear maps). Define
Op(E, F ) × C ∞ (X, R) → Op(E, F )
(D, f ) 7→ [D, f ] = Df − f D
Thus [D, f ]s = D(f s) − f D(s).
Inductively define
DOm (E, F ) =⊂ Op(E, F )
DO0 (E, F ) = {D ∈ Op(E, F ) | [D, f ] = 0}
DOm (E, F ) = {D ∈ Op(E, F ) | [D, f ] ∈ DOm−1 (E, F )}
[
DOm (E, F )
DO(E, F ) =
m
A differential operator is an element of DOm (E, F ) for some m. The minimal
m is called the order of D.
Thus DOm (E, F ) is the differential operators of order less than or equal
to m.
Definition. For s ∈ C ∞ E, support s = s−1 (E − 0)
Definition. D ∈ OP (E, F ) is local if ∀s, support Ds ⊂ support s.
Lemma. A differential operator is local.
Proof. By induction on m. True for m = −1. Assume true for m − 1. Let
D ∈ DOm (E, F ) and let s be a section. ∀ open U ⊃ support s, ∃ smooth
f : X n → [0, 1] with f ≡ 0 outside U and f ≡ 1 in support s. Then f s = s.
D(s) = D(f s) = [D, f ]s + f D(s)
So support Ds ⊂ support s ∪ support f ⊂ U . Choose Ui so that ∩ U i =
support s.
Corollary. If U is a neighborhood of x ∈ X, and s1 and s2 are sections that
agree on U , then D(s1 )x = D(s2 )x.
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Consider the support of s1 − s2 .
Corollary. For an open set U ⊂ X, then is a map DOm (E, F ) → DOm (E|U , F |U )
Apply cutoff functions which are 1 in a neighborhood of x ∈ U and 0
outside of U .
In fact, the map U 7→ DOm (E|U , F |U ) is a sheaf and
Theorem (Peetre, 1959). D is local if and only if D is a differential operator.
Local analysis:
Serre-Swan theorem
symbol as map
clutching
metric
symbol
The four classic elliptic operators
Complex of elliptic operators
DeRham operator
Dolbeault operator
Signature operator
Dirac operator
Algebraic top. of vector bundles
1. Thom isomorphism
2. Euler Class and euler characteric
3. splitting principle and characteristic classes
4. topological K-theory
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The Index Theorem applied to the four classic
elliptic operators
Analysis
Key question - why are the kernel and cokernel of an elliptic operator finite
dimensional. Hodge theory . . .
Other topics
Here are possibilities:
1. The proof of the K-theoretic version (involves pseudo-differential operators)
2. Derivation of cohomological version from K-theoretic version
3. The heat kernel proof
4. Chern-Weil theory
5. APS theory
6. L2 -index theorem
7. applications to positive scalar curvature
8. G-signature theorem
9. Fixed point theorems
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