de Rham Theorem May 10, 2016

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1 de Rham Theorem May 10, 2016

2 Stokes formula and the integration morphism: Let M = σ Σ σ be a smooth triangulated manifold. Fact: Stokes formula σ ω = σ dω holds, e.g. for simplices. It can be used to define linear map Int k. The map Int k 1 : Ω k (M) Σ k defines a homomorphism of complexes. Note: Stokes thm. implies commutativity of the diagram: d k 1... Ω k 1 (M) Ω k (M) Ω k+1 (M)... d k Int k 1 Φ k 1 Int k Φ k Int k+1 Φ k+1 k 1... Σ k 1 Σ k Σ k+1... k

3 Elementary Forms: If p 1, p 2,... p s are the vertices of complex K, the set {St(p k )} k, where St(p k ) := σ:σ p k σ, forms an open cover for M. The partition of unity theorem guarantees the existence of a C -partition of unity φ 1,..., φ s subordinate to {St(p k )} k. Below we denote the dual basis to simpleces by the same letters. Let σ = [p λ0... p λk ] be an oriented simplex. Corresponding to σ is the elementary differential form of order k Φ k (p λ0,..., p λk ) = k! k i=0 ( 1)i φ λi dφ λ0 dφ λi dφ λk.

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6 1. Right inverse to integration via elementary forms Proof: induction on k. Below ω σ is the pairing by integration. k = 0 = Int 0 (Φ 0 (q i )) q j := (Φ 0 (q i ))(q j ) = Φ i (q j ) = δ ij. Inductive step k > 0 : if σ τ then τ M \ St(σ) (Lemma 1). Then Int k (φ k [σ]) τ = 0. Let σ =: α + [other (k-1)-faces of] σ. Then (α) = τ α τ. So, σ Φk (σ) = σ Φk (α) = σ dφk 1 (α) = σ Φk 1 (α) = α Φk 1 (α) = 1. (Using Stokes, Lemma 2, and the inductive hypothesis)

7 Two lemmas needed for Step 1. Remark: Supp Φ(σ) a V (σ) St(a) := St(σ). Lemma 1. τ σ; τ, σ Σ k = τ M \ St(σ). Proof. Let b V (τ) \ V (σ). Either (b, σ) Σ k+1 = supp Φ(σ) St(b) = or (b, σ) / Σ k+1. In the latter case, if β Σ k+1, b V (β) = σ / F(β) = St(b) σ =. In either case, σ Φ(τ) = τ Φ(σ) = 0, or, [σ] τ = [τ] σ = 0. Lemma 2. Φ k = d Φ k 1. Observe that φ i = 1, so dφi = 0. Also dφ k (q λ0... q λk ) = (k + 1)!dλ 0 dλ k.

8 Let σ = q λ0... q λk, then 1 (k+1)! Φk+1 [σ] = 1 (k+1)! [q r σ] Σ k+1 Φ k+1 [q r σ] = [q r σ] Σ k+1 [φ qr dφ λ0 dφ λk + k i=0 ( 1)i+1 φ λi dφ qr dφ λ0 dφ λi... dφ λk ] = [q r σ] Σ k+1 φ qr dφ λ0... dφ λk + [q r σ]/ Σ k+1,q r / V (σ) dφ q r k i=0 ( 1)i φ λi dφ λ0 dφ λi... dφ λk ] + k j=0 dφ λ j k i=0 ( 1)i φ λi dφ λ0 dφ λi dφ λk = dφ λ0... dφ λk = 1 (k+1)! dφk [σ].

9 de Rham Complex... ker Int k 1 (M) ker Int k (M) ker Int k+1 (M)... i i i d k 1 d k... Ω k 1 (M) Ω k (M) Ω k+1 (M)... Int k 1 Φ k 1 Int k Φ k Int k+1 Φ k+1 k 1 k... Σ k 1 Σ k Σ k+1... Let k th de Rham cohomology group H k (M) := ker d k /im d k 1. Let k th cohomology group of Σ H k (Σ) := ker k /im k 1. Note: Int k : Ω k (M) Σ k induces an isomorphism Int k : H k (M) H k (Σ) of differential complexes.

10 Acyclicity of the kernel of Int map. Basic fact: Poincare Lemma: If U is a contractible open set in R n and α a k-smooth closed form on U, then α is exact, i.e there exists a form β such that α = dβ. Observe that St(σ) is contractible so Poincare lemma applies. Next fact that we will need is the extension of forms theorem.

11 Extension of forms theorem. (a k ) Let U( σ)) be ngbhd of of σ, σ a s simplex, ω Ω k (U( σ)) closed, k 0, s 1. If σ ω = 0 and s = k + 1, then ω Ωk (U(σ)) cl.s.t. ω U( σ) = ω, perhaps by shrinking U( σ). (b k ) If s 1, k 1, σ an s simplex, ω Ω k (U(σ)) closed and α Ω k 1 (U( σ)), U( σ) U(σ), s.t. dα = ω U( σ). When s = k assume σ ω = σ α. Then exists α Ωk 1 (U(σ)) s.t. α U( σ) = α and d α = ω, maybe shrinking U(σ) U( σ).

12 Proof of acyclicity, by induction on s n. Consider an s-dim. subcomplex L s := i σs i and ω ker(int k ) a closed form. Outline: Construct inductively nbhds U(L s ) of L s and forms α s Ω k 1 (U(L s )) s.t. α s U(Ls) U(L s 1 ) = α s 1, dα s = ω U(Ls) and Int k 1 (α k 1 ) = 0. Then α n ker(int k 1 ) and dα n = ω, proving that ker(int ) is acyclic.

13 Proof of acyclicity, by induction. Basis step: Choose disjoint, contractible nbds U(σi 0 ). By Poincare Lemma exists α 0 Ω0 (U(σi 0)) with dα 0 = ω U(σi 0). Set α 0 := α 0 for k > 1 and α 0 := α 0 α 0 (σ0 i ) for k = 1 so Int 0(α 0 ) = 0 as required for s = 0. Inductive Step: Given α s 1, for each σ s i we now construct nbds U(σi s ) s.t. overlaps of each two are subsets of U(L s 1 ) and also forms

14 Proof of acyclicity, by induction. α s,i Ω k 1 (U(σi s)) that coincide with α s 1 on overlaps. Inductive assumption includes dα s 1 = ω U(Ls 1 ) and α s 1 ker(int k 1 (U(L s 1 ))) for s = k. Then (b k ) gives α s,i Ω k 1 (U(σi s)) s.t. d α s,i = ω U(σ s i ) and α s,i U( σ s i ) = α s 1. Glue α s,i into α s on U(L s ) := i U(σi s). We set α s := α s for s k 1 and α s := α s Φ k 1 (Int k 1 ( α s )) for s = k 1.

15 Proof of acyclicity, by induction (concluded). Note that Φ and Int are homomorphisms of complexes and the former is the right inverse of the latter by [1] imply dα k 1 = ω φ k (Int k (ω)) = ω on U(L s ) and also that Int k 1 (α k 1 ) = Int k 1 ( α k 1 ) Int k 1 ( α k 1 ) = 0 concluding the proof.

16 Euler characteristic χ(t (M)) does not depend on the triangulation T (M) of M Reason: Corollary to the Theorem implies ker(d k ) = im(d k 1 ) where d is the restriction of the exterior derivative in the kernel. which in turn implies ker( k ) im( k 1 ) = im(d k) im(d k 1 ). Note: #{σ T (M) : dimσ = k} = dim R Σ k = dim R Σ k. Theorem: Euler characteristic χ(m) := n k=1 ( 1)k ker(d dim k ) R im(d k 1 ) = χ(t (M)). Corollary: χ(t (M)) does not depend on triang. T (M) of M.

17 Proof of the corollary dim R Σ k = dim R (Im( k )) + dim R ker( k ) im( k 1 ) + dim RIm( k 1 ). Therefore χ(m) = n k=0 ( 1)k ker( k dim ) R ) = χ(t (M)). im( k 1

18 Extension of Forms Theorem Proof: by induction on k. Outline: Show (a 0 ) holds, then (a k 1 ) = (b k ), and finally, (b k ) = (a k ). (a 0 ) : Say ω Ω 0 (U( σ)) closed. Then ω is locally constant. If s > 1, then ω const in U( σ) so we can let let ω = ω in U(σ). If s = 1 then σ = p 0 p 1 as an 1 simplex; also it is given that σ ω = 0. But σ ω = ω(p 1) ω(p 0 ) = 0 so we can let ω = ω. (a k 1 ) = (b k ): Say ω, α are as in (b k ). Poincare lemma gives α Ω k 1 (U(σ)), dα = ω U(σ). Let α α =: β Ω k 1 (U( σ)).

19 Then β is closed in U( (σ)). If s = k then σ β = α σ α = σ ω σ dα = 0. Applying (a k 1 ) to β we get a closed form β Ω k 1 (U(σ)) such that β U( σ) = β. Then α := ( β + α ) Ω k 1 (U(σ)) is as required in (b k ). (b k ) = (a k ): Say σ = (p 0... p s ) and ω are as in (a k ), k > 0. Also, let σ := (p 1... p s ) P, where P is he union of proper faces of σ with p 0 as a vertex. Then ω is defined and closed in a ngbhd U(P); clearly U(P) St(p 0 ) and it is star-shaped.

20 Poincare lemma gives α Ω k 1 (U(P)) s.t. dα = ω U(P) ; this holds in particular in some nbhd U( σ ) U(P). For s = k + 1 define A := ( σ σ ) Σ k. Then A = σ, and hence σ ω σ α = σ ω + A dα = σ ω = 0. Applying now (b k) to simplex σ we get α Ω k 1 (U(σ )) such that α U( σ )) = α and d α = ω U(σ ). Shrink U(P) so that U(P) U(σ ) U( σ ), let U( σ) := U(P) U(σ ) and set α Ω k 1 (U( σ)) by α = α on U(P) and α = α on U(σ ). Extending α using partition

21 of unity to Ω k 1 (U(σ)) gives the closed form (required by (a k )) ω := d α since ω = d α σ = ω by construction of α and α.