27 De Rham Theorem and Double Theorem

By an “isomorphism” map between two spaces, sometimes something will not be changed. For example, a linear isomorphism between two vector spaces X and Y , their dimension dim X = dim Y . In this case, we may say that dimension is an invariant of vector spaces.

For a real differentiable manifold, there are several ways to define its “invariants”, either by topological way (homology), or by differentiable way (cohomology). The famous de Rham theorem, which will be presented below, will give a link to show that two differently defined invariants are actually identical. The complex version of the similar result is the Dolbeault theorem, which will also be presented in this section.

Homology group theory For a topological space X, it can associates some invariant groups called “homology groups” Hp(X) in the sense that if f : X → Y is a homeomorphism, it induces a group isomorphism f∗ : Hp(X) → Hp(Y ), ∀p.

Let X be a topological space. A chain complex C(X) is a sequence of abelian groups or modules with homomorphisms ∂n : Cn → Cn−1 which we call boundary operators. That is,

∂n+1 ∂n ∂n−1 ∂2 ∂1 ∂0 ... −−−→ Cn −→ Cn−1 −−−→ ... −→ C1 −→ C0 −→ 0 where 0 denotes the trivial group and Cj = 0 for j < 0. We also require the composition of any two consecutive boundary operators to be zero. That is, for all n,

∂n ◦ ∂n+1 =0.

This means im(∂n+1) ⊆ ker(∂n).

Now since each Cn is abelian, im(Cn) is a normal subgroup of ker(Cn). We deﬁne the n-th homology group of X with respec to the chain complex C(X) to be the factor group (or quotient module) Hn(X)= ker(∂n)/im(∂n+1)

We also use the notation Zn(X) := ker(∂n) and Bn(X) := im(∂n+1), so

Hn(X)= Zn(X)/Bn(X).

Clearly the deﬁnition of homology group depends on which abelian groups used in the construction. Next we shall introduce two ways.

153 Simplicial homology 49 First we define a specific homology group: simplicial homology. The simplicial homology groups Hn(X) are defined by using the simplicial chain complex C(X), with C(X)n the free abelian group generated by the n-simplices of X. Here an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices.

If σn = [p0, ..., pn], then n k ∂nσn := (−1) [p0, ..., pk−1,pk+1, ...pn]. Xk=0

We can verify that ∂n+1 ◦ ∂n = 0. For example, if σ = [p0,p1,p2] is a 2-simplex. ∂2(σ) = [p1,p2] − [p0,p2] + [p0,p1] and ∂1 ◦ ∂2(σ)= ∂1([p1,p2] − [p0,p2] + [p0,p1]) = [p2] − [p1] − [p2]+ [p0] + [p1] − [p0] = 0. Singular homology 50 There is another way to deﬁne homology group: singular homology. The singular homology groups Hn,sing(X) are deﬁned by using the singular simplicial chain complex C(X)sing, with C(X)n,sing the free abelian group generated by the singular n-simplices of X. A singular n-simplex is a continuous mapping σn from the standard n-simplex ∆n to a topological space X: σn : ∆n → X.

If we designate the range of σn by its vertices

[p0,p1, ··· ,pn] = [σn(e0), σn(e1), ··· , σn(en)], n n where ek are the vertices of the standard n-simplex ∆ , then the boundary of σn(∆ ), n denoted as ∂nσn(∆ ), is deﬁned to be n n k ∂nσn(∆ )= (−1) [p0, ..., pk−1,pk+1, .., pn]. Xk=0

We can verify that ∂n+1 ◦ ∂n = 0. By a fundamental result from algebraic topology, simplicial homology and singular homology are isomorphic: Hp(X) ≃ Hp,sing(X). Cohomology theory 51 Cohomology is fundamental to modern algebraic topology, i.e., p C (X,G) := Hom(Cp(X),G) where G is an abelian group. Then we have the cochain complex which has opposite direction by comparing with homology chain complex: δ δ ... ← Cn+1(X,G) ←− Cn(X,G) ←− ... ← C0(X,G) ← 0

49see wikipedia: simplicial homology 50see wikipedia: Singular homology. 51see wikipedia: Cohomology.

154 and we can similarly deﬁne cohomology group: Hp(X)= ker(∂p)/im(∂p−1). Its importance was not seen for some 40 years after the development of homology. The concept of dual cell structure, which Henri Poincar´eused in his proof of his Poincar’e Duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.

Historic remarks • In 1930, Alexander defined a first cochain notion. • In 1931, Georges de Rham related homology and exterior differential forms, prov- ing De Rham’s theorem. In fact, De Rham’s original work was to define relative de Rham groups and to prove that the resulting homology theory satisfied the axioms of Eilenverg and Steenrod. This result is now understood to be more naturally interpreted in terms of cohomology. • In 1934, L. Pontryagin proved the Pontryagin duality theorem; a result on topological groups. This (in rather special cases) provided an interpretation of Poincaréduality and Alexander duality in terms of group characters. • In 1935, A. Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure. • In 1936 Norman Steenrod published a paper constructing Cechˇ cohomology by dual- izing Cechˇ homology. • From 1936 to 1938, Hassler Whitney and Eduard Cechˇ developed the cup product (making cohomology into a graded ring) and cap product, and realized that Poincaré duality can be stated in terms of the cap product. Their theory was still limited to finite cell complexes. • In 1944, Samuel Eilenberg overcame the technical limitations, and gave the modern definition of singular homology and cohomology

De Rham cohomology groups The de Rham complex is the cochain complex of exterior diﬀerential forms on some smooth manifold M, with the exterior derivative d as the diﬀerential. d d d 0 →A0(M) −→A1(M) −→A2(M) −→A3(M) → ...

155 where A0(M) is the space of smooth functions on M, A1(M) is the space of 1-forms, and so forth. The de Rham cohomology group of M in degree p is Ker( d : Ap(M) →Ap+1(M)) Hp (M, R) := . DR Im( d : Ap−1(M) →Ap+1(M)) p R The Betti number of M is bp := dimR HDR(M, ). Stokes theorem E. Cartan developed the theory of exterior diﬀerentials in 1920s. In 1936-1937 in Paris, E. Cartan discovered the general Stokes theorem [?] for any dimension, which gives a link between analysis and geometry

dω = ω ZD Z∂D where ω is a smooth p-form, and D is a (p + 1)-dimensional orientable submanifold.

In classical complex, Cauchy integral theorem, Cauchy integral formulas, Residue theorem are consequence of Stokes’ theorem.

We claim that the map k R R R HDR(M, ) × Hk(M, ) → 7→ (ω, σ) σ ω R is well - deﬁned. In fact, if ω,ω′ are in the same cohomology class, i.e., ω − ω′ = dδ for some ∈ k−1 − ′ ′ δ Ω (M), then σ(ω ω ) = σ dδ = dσ δ = {0} δ = 0, so that σ ω = σ ω . Here we have used Stokes’ theorem.R R R R R R

If σ, σ′ are in the same holomology class, i.e., σ − σ = dψ for some (k + 1)-chain ψ. Then − σ ω σ′ ω = dψ ω = ψ dω = 0, so that σ ω = σ′ ω. Here we used Stokes’ theorem. Our Rclaim isR proved.R R R R

In other words, Stokes’ theorem is an expression of duality between de Rham cohomology and the homology of chains. The above pairing of diﬀerential forms and chains, via integra- k tion, gives a homomorphism from de Rham cohomology HDR(M, R) to singular cohomology k R R R 52 groups Hsing(M; ) := HomR(Hk(M, ), ): k R k R HDR(M, ) → Hsing(M, ) 7→ 7→ ω (σ σ ω) 52 R HomR(X, R) means the set of all homomorphisms from X to R, i.e., the set of all linear frunctionals.

156 De Rham’s theorem, proved by Georges de Rham in 1931, states that for a smooth manifold M, this map is in fact an isomorphism.

The wedge product endows the direct sum of these groups with a ring structure: Hk (M, R) × Hr (M, R) → Hk+r(M, R) DR DR DR (100) (ω, ω) 7→ ω ∧ ω

A further result of the theorem is thate the two cohomology ringse are isomorphic (as graded rings), where the analogous product on singular cohomology is the cup product.

The de Rham theorem says p R p R HDR(M, ) ≃ Hsing(M, ) p R R where Hsing(M, ) is the simplicial cohomology group of M with respect to . Also, p R ˇ p R HDR(M, ) ≃ H (M, ) where Hˇ p(M, R) is the Cechˇ cohomology group of M for the constant sheat R. (cf., [?], p.44).

If we consider complex-valued forms, we take the complex Kp := C∞(M, C ⊗ ∧pT ∗M) p C p C p C and can similarly deﬁne HDR(M, ) := Z (M, )/B (M, ). We have p C C p R HDR(M, )= ⊗ HDR(M, ).

Dolbeault cohomology groups The complex analygue of the de Rhan theorem was discovered by Dolbeault in 1953.

Let X be a complex manifold. The Dolbeault cohomology is the vector space Ker(∂ : Ap,q(X) →Ap,q+1(X)) Hp,q(X) := Hq(Ap,•(X), ∂) := . Im(∂ : Ap,q−1(X) →Ap,q(X)) Dolbeault isomorphism theorem says that p,q C q p H (X, ) ≃ H (X, ΩX ). (101)

p p where ΩX is the sheaf Ω of holomorphic p-forms over X.

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