Download 1 Theorems

1
Theorems
Theorem 1 (Classification of Covering Spaces) Let X be a locally 1-connected topological space.
The the following hold:
1. For any covering p : E → X the map p∗ : π1 (E) → π1 (X) is a monomorphism
2. There is a one-to-one correspondence between the set p−1 (x0 ) and the cosets π1 (X)/p∗ π1 (E).
3. For any subgroup G ⊂ π1 (X) there is a covering p : E → X such that p∗ (E) = G.
4. Consider coverings of punctured spaces, i.e. fix points e0 ∈ E and e00 ∈ E 0 in the preimages of the
base point x0 and consider two coverings p : E → X and p0 : E 0 → X of punctured spaces to be
equivalent if the homeomorphism f : E → E 0 maps f (e0 ) = e00 . Then two coverings are equivalent
in this punctured classification if and only if the following two subgroups of of π1 (X, x0 ) coincide:
p∗ π1 (E, e0 ) = p0∗ π1 (E 0 , e00 )
5. Two coverings are equivalent in the sense of the usual definition of equivalence if and only if the
groups p∗ π1 (E, e0 ) and p0∗ π1 (E 0 , e00 ) are conjugate as subgroups of π1 (X, x0 ), for some e0 ∈ p−1 (x0 )
and e00 ∈ (p0 )−1 (x0 ).
Theorem 2 (Eilenberg-Zilber) For topological spaces X and Y , there is a natural chain equivalence
ζ : S• (X × Y ) → S• (X) ⊗ S• (Y ), unique up to chain homotopy, Hence
Hn (X × Y ) = Hn (S• (X) ⊗ S• (Y ))
for all n ≥ 0
Theorem 3 (Exact Sequence of a Pair) If A is a subset of X there is a exact sequence
···
/ Hn (A)
/ Hn (X)
/ Hn (X, A)
/ Hn−1 (A)
/ ···
Moreover if f : (X, A) → (Y, B) then there is a commutative diagram
···
/ Hn (A)
/ Hn (X)
/ Hn (X, A)
/ Hn−1 (A)
/ ···
···
/ Hn (B)
/ Hn (Y )
/ Hn (Y, B)
/ Hn−1 (B)
/ ···
where the vertical maps are induced by f .
Theorem 4 (Exact Sequence of a Triple) if A ⊂ B ⊂ X are subspaces, there is an exact sequence
···
/ Hn (B, A)
/ Hn (X, B)
/ Hn (X, A)
Moreover if there is a commutative diagram of pairs of spaces
1
/ Hn−1 (A, B)
/ ···
(A, B)
/ (X, B)
/ (X, A)
(A0 , B 0 )
/ (X 0 , B 0 )
/ (X 0 , A0 )
then there is a commutative diagram with exact rows
···
/ Hn (B, A)
/ Hn (X, B)
/ Hn (X, A)
/ Hn−1 (A, B)
/ ···
···
/ Hn (B 0 , A0 )
/ Hn (X 0 , B 0 )
/ Hn (X 0 , A0 )
/ Hn−1 (A, B)
/ ···
Theorem 5 (Excision I) Assume that U ⊂ A ⊂ X are subspaces with Ū ⊂ A◦ . Then the inclusion
i : (X − U, A − U ) → (X, A) induces isomorphisms
i∗ : Hn (X − U, A − U ) → Hn (X, A)
for all n.
Theorem 6 (Excision II) Let X1 and X2 be subspaces of X = X1◦ ∪ X2◦ . Then the inclusion j :
(X1 , X1 ∩ X2 ) → (X1 ∪ X2 , X2 ) = (X, X2 ) induces isomorphisms
j∗ : Hn (X1 , X1 ∩ X2 ) → Hn (X, X2 )
for all n.
Theorem 7 (Frobenius Theorem) A subbundle E of T M is involutive if and only if it is integrable.
Theorem 8 (Gauss Bonnet) If X is a compact, even dimensional hypersurface in Rn+1 , then
Z
1
κ = γk χ(X)
2
X
Where χ(X) is the Euler characteristic of X and γk is the volume of the unit k−sphere S k
Theorem 9 (Homotopy Lifting) Given a covering space p : X̃ → X, a homotopy ft : Y → X and a
map f˜0 : Y → X̃ lifting f0 , then there exists a unique homotopy f˜t : Y → X̃ of f˜0 that lifts ft .
Theorem 10 (Hopf Theorem) Suppose that M is a connected, oriented smooth manifold without
boundary of dimension m and suppose that f, g : M → S m are smooth maps to the m-sphere. Then
f and g are smoothly homotopic if and only if f and g have the same brower degree (degree of maps).
Theorem 11 (Implicit Function Theorem) Let U ⊂ Rc−d × Rd be open and let f : U → Rd be C ∞ .
We denote the canonical coordinate system on Rc−d × Rd by (r1 , . . . , rc−d , s1 , . . . , sd ). Suppose that at
the point (r0 , s0 ) ∈ U
f (r0 , s0 ) = 0
2
and that the matrix
∂fi
|i,j=1,...,d
∂sj
is non-singular. Then there exists an open neighborhood V of r0 in Rc−d and an open neighborhood
W of s0 in Rd such that V × W ⊂ U , and there exists a C ∞ map G : V → W such that for each
(p, q) ∈ V × W
f (p, q) = 0 ⇐⇒ q = g(p)
Theorem 12 (Invariance of Domain) If a subspace X of Rn is homeomorphic to an open set in Rn ,
then X itself is open in Rn .
Theorem 13 (Inverse Function Theorem) Suppose U and V are open subsets of Rn , and F : U → V
is a smooth map. If DF (p) is nonsingular at some point p ∈ U , then there exists a connected neighborhoods
U0 ⊂ U of p and V0 ⊂ V of F (p) such that F |U0 : U0 → V0 is a diffeomorphism.
Theorem 14 (Künneth Forumla) For every pair of topological spaces X and Y and for every integer
n ≥ 0, there is a split exact sequence
/
0
P
i+j=n
Hi (X) ⊗ Hj (Y )
/ Hn (X × Y )
/
P
i+j=n−1
/0
T or(Hi (X), Hj (Y ))
In particular this gives rise to the isomorphism

 

M
M
Hp (X) ⊗ Hn−p (Y ) ⊕ 
Tor(Hp (X), Hn−1−p (Y ))
Hn (X × Y ) ∼
=
p∈Z
p∈Z
Theorem 15 (Mayer Vietoris) If X1 and X2 are subspaces of X with X = X1◦ ∪ X2◦ , then there is
an exact sequence
···
/ Hn (X1 ∩ X2 )
/ Hn (X1 ) ⊕ Hn (X2 )
/ Hn (X)Hn−1 (X1 ∩ X2 )
/ ···
Theorem 16 (Preimage Theorem) If y is a regular value of f : X → Y then the preimage f −1 (y) is
a submanifold of X with dim f −1 (y) = dim X − dim Y
Theorem 17 (Sard) Let f : U → Rn be a smooth map defined on an open set U ⊂ Rm and let
C = {x ∈ U |rank dfx < n}
Then the image f (C) ⊂ Rn has Lebesgue measure zero.
Theorem 18 (Stokes) Let M be a smooth, oriented n−dimensional manifold with boundary, and let ω
be a compactly supported smooth n − 1 form on M , then
Z
Z
dω = ω
c
∂c
3
Theorem 19 (Van Kampen) If X is the union of path connected open sets Aα each containing the
base point x0 ∈ X and if each intersection Aα ∩ Aβ is path connected, then the homomorphism Φ :
∗α π1 (Aα ) → π1 (X) is surjective. In addition if each intersection Aα ∩ Aβ ∩ Aγ is path connected, then
the kernel of Φ is a normal subgroup N generated by all elements of the form iαβ (ω)iβγ (ω)−1 , and so Φ
induces an isomorphism
π1 (X) ∼
= ∗α π(Aα )/N
Theorem 20 (Universal Coefficients for Cohomology) If a chain complex C• of free abelian groups
has homology groups Hn (C• ) then the cohomology groups H n (C• ; G) of the cochain complex Hom(Cn , G)
are determined by split exact sequences
0
/ Ext(Hn−1 (C• ), G)
/ H n (C; G)
/ Hom(Hn (C, G))
/0
Theorem 21 (Universal Coefficents Theorem for Homology) For every space X and every abelian
group G, there are exact sequences for all n ≥ 0
0Hn (X) ⊗ G
α
/ Hn (X; G)
/ Tor(Hn−1 (X), G)
/0
In particular
Hn (X; G) ∼
= Hn (X) ⊗ G ⊕ Tor(Hn−1 (X), G)
2
Functors
2.1
Tensor
A⊗B
• A⊗Z=A
• A ⊗ Q is free.
• Z/m ⊗ Z/n ∼
= Z/ gcd(n, m)
2.2
Hom
(A,B)
• Contravariant Functor
2.3
Tor
Tor(A,B) Let
P• : Pn
/ Pn−1
/ ···
/ P0
/A
Be a projective resolution for A.
Compute C• ⊗ B and H• (C• ⊗ B). The nth homology Hn (C• ⊗ B) is the nth tor product
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• Tor(A, Z) = Tors(Z, A) = 0
• Tor(A, B) = Tors(B, A)
• Tor(Z/m, Z/n) ∼
= Z/ gcd(m, n)
• Tor(A, (B ⊕ C)) ∼
= Tors(A, B) ⊕ Tors(A, C)
2.4
Ext
Ext(A,B) Let
P• : Pn
/ Pn−1
/ ···
/ P0
/A
Be a projective resolution for A.
Compute Hom(C• , A) and H• (Hom(C• , B). The nth homology Hn (C• ⊗ B) is the nth ext product.
• Ext(A, B) 6= Ext(B, A)
• Ext(Z, A) = 0
• Ext(Z/m, Z/n) ∼
= Z/ gcd(m, n)
• Ext(A, (B ⊕ C)) ∼
= Ext(A, B) ⊕ Ext(A, C)
3
Definitions
Chain Complex: A sequence of Abelian groups and homomorphisms
/ Ci
···
∂i
/ Ci−1
∂i−1
/ C0
/ ···
/0
is a chain complex if ∂i−1 ◦ ∂i = 0 for all i. The ∂i are called boundary morphisms.
• Closed: ω ∈
Vk
M is closed if dω = 0
• Closed: A set is closed if it is the complement of an open set
Cohomology: Let C• be a co-chain complex
··· o
δ
C i+1 o
i+1
Ci o
δi
C i−1 o
···
The cohomology of this chain complex is
ker δ i+1
Imδ i
Covering Space: Let X be a topological space, a covering space is a map p : Y
any x0 ∈ X there is a neighborhood U of x0 such that p−1 (U ) = ∆ × U where ∆ is a
Degree of a Map: Let M, N be a closed, orientable, manifold of dimension m
Let f : M → N be a continuous map. The degree of f is Hn f (1).
Lie Derivative: Let V and W be smooth vector fields on a smooth manifold M .
H i (C) =
5
→ X such that for
discrete set.
and n respectively.
The Lie Bracket is
[V, W ] (f ) = V W (f ) − W V (f )
∂
∂
Let V = Vi ∂x
and W = Wi ∂x
be the coordinate expressions for V and W in terms of some smooth
i
i
local coordinates xi of M . Then the Lie Bracket [V, W ] has the following expression
∂
∂Wj
∂Vj
[V, W ] = Vi
−
∂xi
∂xi ∂xj
or more concisely
[V, W ] = (V Wi − W Vi )
∂
∂xi
Diffeomorphism: Let ϕ : M → N be C ∞ . ϕ is a diffeomorphism if ϕ is injective and surjective,
and ϕ−1 is also C ∞
Vk
Differential Form: Let M be a n−dimensional manifold. Denote by
M , the subset T k M . A
Vk
section of
M is called a differential k−form.
In any smooth char, a k− form ω can be written locally as
X
ω=
ωI dxI
I
Distribution: Let M be a d-dimensional manifold M . Let 1 ≤ c ≤ d. A c−dimensional distribution
D is a choice of a c−dimensional subspace D(m) of Mm for each m ∈ M . D is smooth if for each for
each m ∈ M there is a neighborhood U of m and there are c vector field X1 , . . . , Xc of class C ∞ on U
which span D at each point of U .
A vector field X on M is said to belong to (or lie in) the distribution D if Xm ∈ D(m) for each
m ∈ M.
Vk
• Exact: Of a differential form: A smooth differential form ω ∈
M is exact if there exists a
smooth (k − 1)-form η on M such that dη = ω.
• Exact: Of a sequence: The sequence
/B
α
A
β
/C
Is exact if ker β = imα.
Exterior
V Derivative: The exterior derivative of a differential k−form on a n− dimensional vector
space, ω ∈ V is
n
X
∂fj
X
n
k
1≤j≤(
)
i=1
∂xi
dxi
Fundamental Class: Let M be a compact, connected, orientable, d−dimensional manifold. A
fundamental class on M is a generator of Hd (M ).
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Fundamental Group: Let X be connected, locally path connected, topological space, and x0 ∈ X.
Then π1 (X, x0 ) is defined as the set of paths Pxx00 (X) in X, starting and ending at the point x0 , considered
up to homotopy.
Homeomorphism: map h : X → Y is a homeomorphism of h is a continuous bijection with
continuous inverse.
Homology: Let C• be a chain complex.
···
/ Ci
∂i
/ Ci−1
∂i−1
/ ···
/ C0
/0
the ith homology of C• is
ker ∂i
im∂i+1
• Let f, g : X → Y be continuous maps. f is Homotopic to g (written f g̃) if there exists a map
F : X × I → Y such that F (t, 0) = f and F (t, 1) = g
• Homotopy Equivalent (Homotopy Type): Two spaces X and Y are said to be homotopy
equivalent if there exist maps f : X → Y and g : Y → X with the property that f ◦ g 1̃Y and
g ◦ f 1̃X
Hurewicz Homomorphism: The Hurewicz map h : πn (X, A, x0 ) → Hn (X, A) is a homomorphism
assuming n ≥ 1 so that πn (X, A, x0 ) is a group.
In dimension 1 the hurewicz map is given as π1 (X, x0 ) → H1 (X) is given by π1 (X, x0 )/C where C is
the commutator. Involutive: Let E ⊂ T M be a distribution on M . We say that E is involutive if for
any two vector fields X and Y defined on open sets of M and which take values in E, [X, Y ] takes values
in E as well.
Integrable: et E ⊂ T M be a distribution on M . We say that E is integrable if for any m ∈ M there
is a local submanifold N ⊂ M of E at m containing m, whose tangent bundle is exactly E restricted to
N.
Manifold: A Second Countable Hausdorff topological space M with countable basis if for each
x ∈ M there exists a neighborhood Nx and homeomorphisms ϕ(x) : Nx → Rn .
Metric Space: A Metric Space (X, ρ) is a topological space X equipped with a function ρ with the
properties
• ρ(x, x) = 0 for all x ∈ X
• ρ(x, y) ≥ 0 for all x, y ∈ X
• ρ(x, y) = ρ(y, x) for all x, y ∈ X
• ρ(x, y) ≤ ρ(x, z) + ρ(z, y) for all x, y, z ∈ X
Smooth Manifold: A smooth manifold is a topological manifold M , in which every chart is smooth
(C ∞ ).
Smooth: map A → B is said to be smooth if it, and all the derivatives of it, are continuous with
continuous inverse.
Topological Space: A topological space is a set X endowed with a topological structure (a topology)
τ , τ ⊂ 2X , whose elements are called open. The family of open sets should satisfy
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• The union of any set of open sets is an open set.
• The intersection of a finite collection of open sets is an open set
• the empty set and the whole set X are both open.
Vector Field: A vector field X along a curve σ : [a, b] → M is a mapping X : [a, b] → T (M ) which
lifts σ; That is, π ◦ X = σ. A vector field X is called smooth (C ∞ ) if X is a C ∞ function.
Tangent Space: add something
Exterior Algebra: add something
Cup Product: add something
Cap Product: add something
4
Spaces
n-Sphere: The n-dimensional sphere is the of solutions to the equation
{x1 , . . . , xn+1 :
n
X
i=1
embedded in R
n+1
8
x2i = 1}