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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 4 • 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 ). 6 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 7 • 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}