Simplicial Homology and De Rham’s Theorem
Jesse A. Thorner April 20, 2009
Advisor: Dr. William Allard
Submitted in partial fulfillment of the requirements for Graduation with Distinction
Department of Mathematics Duke University Durham, North Carolina
Abstract After giving the necessary background in simplicial homology and cohomology, we will state Stokes’s theorem and show that integration of differential forms on a smooth, triangulable manifold M provides us with a homomorphism from the De Rham cohomology of M to the simplicial cohomology of M. De Rham’s theorem, which claims that this homomorphism is in fact an isomorphism, will then be stated and proved.
1 Contents
1 Acknowledgments 2
2 Introduction 3
3 Simplices 4 3.1 Geometric realization of an abstract simplicial complex ...... 4 3.2 The Simplicial Approximation Theorem...... 6
4 Homology of a Simplicial Complex 7
5 Applications of Homology Groups 9 5.1 The Euler Characteristic ...... 9 5.2 The Fundamental Group ...... 9
6 Cohomology of a Simplicial Complex 13
p 7 Duality between Hp(K) and H (K) 14
8 Differential Forms on Manifolds 16
9 Stokes’s Theorem: Integration and Cohomology 17
10 De Rham’s Theorem 18 10.1 Extension of Forms ...... 18 10.2 Surjectivity of Ip ...... 19 10.3 Injectivity of Ip ...... 22
11 Conclusion 24
1 1 Acknowledgments
I would like to gratefully acknowledge the supervision and guidance of Prof. William Allard during this work. I never would have thought that I could have learned and accomplished as much as is encompassed in this project had it not been for his confidence in me. I also thank Prof. Lenhard Ng both for agreeing to evaluate my work and for his encouragement during my classroom ex- perience with him. Additionally, I thank Prof. Kraines for his constant support for all of the undergraduates in the department.
I extend my gratitude to Dr. Joshua Davis and Dr. Amir Jafari for their encouragement dur- ing my first years at Duke, their inspiration, and the wonderful chats during office hours that first sparked my passion for pure mathematics and introduced me to the beauty of geometry. Their influence and interaction with me has been invaluable.
Next, I would like to thank my friends for listening to me and giving me advice throughout the course of this project. I especially thank Barry Wright III and Damien Wilburn for their constant support and feedback.
Finally, I thank my family; this undertaking would not have been possible without their un- conditional love and their guidance.
2 2 Introduction
In an elementary course in vector calculus, one is exposed to the gradient of a scalar field and the curl of vector field in two-dimensional Euclidean space in terms of the operator ∂ ∂ ∇ = ( , ) ∂x ∂y where x, y are the cartesian coordinates of R2. For a vector field F (x, y) = (P (x, y),Q(x, y)) : U ⊂ R2 → R2 we have ∂Q ∂P curl F = ∇ × F = − , ∂x ∂y and for a scalar field f : U ⊂ R2 → R, we have ∂f ∂f grad f = ∇f = ( , ). ∂x ∂y
From these definitions, we can see clearly that curl (grad f) = ∇ × ∇f = 0. Hence, if F = ∇g for some g ∈ C∞(U, R), then curl F = 0. But given a smooth vector field F , it is not necessarily true that curl F = 0 implies that there exists a smooth scalar field g ∈ C∞(U, R) such that F = ∇g. This is the case because of the topological obstructions. For example, consider the vector field F defined on U = R2 \{(0, 0)} given by −y x F (x, y) = ( , ). x2 + y2 x2 + y2 An easy calculation reveals that curl F = 0, and one could naively expect that Green’s theorem R would imply that the integral γ F · dr = 0 for any closed curve γ in U. But if we consider the parametrization γ given by (x, y) 7→ (cos t, sin t) where t ∈ [0, 2π], one easily calculates that R ∞ γ F · dr = 2π. This implies that there does not exist g ∈ C (U, R) such that F = ∇g. In the lan- guage of differential forms, the smooth 1-form ω = F · dr is closed (i.e. dω = 0) but there does not exist a smooth 0-form τ such that ω = dτ. This is because U is not simply connected; it is missing the origin. The topology of U obstructs F (x, y) from being the gradient of a smooth scalar function.
Define Z to be the vector space of all smooth vector fields v defined on U such that curl v = 0, and define B to be the vector space of the gradients of all smooth scalar fields defined on U. It is clear from the foregoing that B ⊂ Z, but for topological reasons B 6= Z. We can think of the quotient space Z/B as a measurement of the topological obstructions in U. The space Z/B is not trivial in this example, but if it were, then every smooth vector field define on U with a curl equal to 0 would be the gradient of a smooth potential function defined on U.
We can generalize this problem to n-dimensional manifolds and differential forms defined on them: Given a smooth k-form ω on a smooth manifold M such that dω = 0, when will ω = dτ, where τ is a smooth (k − 1)-form? In order understand this problem, we first develop tools to study simplices (which can be thought of as the building blocks of manifolds) and their topology, particularly simplicial homology. Once we do this, we relate simplicial homology to the De Rham cohomology of manifolds, which gives us a framework in which we can tackle this problem formally.
3 3 Simplices
We begin our discussion of topology of manifolds with developing some machinery to discuss simplices. A simplex can be discussed in complete abstraction, but its geometric realization is (intuitively) the arbitrary-dimensional analogue of a triangle in R2.1 Definition 3.1. A simplicial complex K is an ordered pair (V,S) consisting of a finite nonempty set V (whose elements are called vertices) and a set S of finite nonempty subsets of V (whose elements are called simplices) such that
(1) Any set consisting of exactly one vertex is a simplex.
(2) Any nonempty subset of a simplex is a simplex. A p-dimensional simplex, also called a p-simplex, is a simplex s containing p+1 vertices. If t is a (proper) subset of s, then t is called a (proper) face of s. The union of the proper faces of s is called the boundary of s, written as Bd s; the interior of s is s\Bd s. If t has dimension q < p, then t is called a q-face of s. The p-skeleton of K (written as K(p)) is set of all p-simplices of K for a given p. Condition 1 implies that there exists a bijection from K(0) to V , and Condition 2 implies that any simplex is determined by its 0-faces. If K is a simplicial complex, its dimension, denoted by dim K, is equal to sup{dim s : s ∈ K}. Since we will be considering only the case where V has finitely many vertices, dim K = max{dim s : s ∈ K}.
3.1 Geometric realization of an abstract simplicial complex
Given a simplicial complex K and its vertex set V = {v1, . . . , vm} consider the set of all functions b : V → I = [0, 1] such that for j ∈ {1, . . . , m},
(a) For any such b, {vj ∈ V : b(vj) 6= 0} is a simplex of K
m X (b) For any such b, b(vj) = 1. j=1
The real number b(vj) is called the j-th barycentric coordinate of b. We write the set of V all such functions b as |K|. Because b is a function from V to I, |K| ⊂ R . Given b1, b2 ∈ |K|, |K| has the metric v u m uX 2 d(b1, b2) = t [b1(vj) − b2(vj)] . j=1 Using barycentric coordinates, we define for any s ∈ K the closed simplex |s| by
|s| = {b ∈ |K| : b(vj) 6= 0 ⇒ vj ∈ s}.
p+1 P If s is a p-simplex, |s| is in one-to-one correspondence with the set {x ∈ R |0 ≤ xi ≤ 1, xi = 1}. Furthermore, the metric topology on (|K|, d) induces on |s| a topology that makes (|s|, d) a topo- p+1 logical space homeomorphic to the above compact convex subset of R . If s1, s2 ∈ K, then clearly s1 ∩ s2 is either empty (in which case |s1| ∩ |s2| = ∅) or a face of both s1 and s2 (in which case |s1 ∩ s2| = |s1| ∩ |s2|). Therefore, in either case (|s1| ∩ |s2|, d) is a closed set in both (|s1|, d) and (|s2|, d). Therefore, a subset A ⊂ |K| is closed in |K| if and only if A ∩ |s| is closed for every s ∈ K. Since we are only considering the case where K is a finite simplicial complex, it follows that |K| is compact.
For s ∈ K, the open simplex (s) ⊂ |K| is defined by
(s) = {b ∈ |K| : b(vj) 6= 0 ⇒ vj ∈ s for all vj ∈ V }.
1A more complete treatment of simplices can be found in [2], [4], [5], and [6].
4 Note that this is consistent with the definition of Int s.
Given a vertex v ∈ V , the star of v, written as St v, is defined by
St v = {b ∈ |K| : b(v) 6= 0}.
The reader can verify that it follows from this that St v = S{Int s : v ∈ s ∈ K}.
Because b → b(v) is a continuous map from (|K|, d) to I, St v is open in (|K|, d), and hence also in |K|. This definition can also be stated as such: St v is the union of all open simplices that contain v as a vertex.
Let K be a simplicial complex and let v0, . . . , vp be the points of a closed p-simplex |s|. Given real Pp numbers λ0, . . . , λp such that 0 ≤ λi ≤ 1 for i = 0, . . . , p and such that i=0 λi = 1, the function Pp x = i=0 λivi is again a point of |s|. Therefore, each closed simplex has a linear structure such that convex combinations of its points are again points of the closed simplex.
If X is a topological space which is a subset of some real vector space, then a continuous map f : |K| → X is linear in K if for every b ∈ |K|, P b(v )f(v ) is a point of X and vj ∈V j j f(b) = P b(v )f(v ). We now want to consider linear imbeddings of |K| in Euclidean space. vj ∈V j j n Definition 3.2. Given a set {v0, . . . , vn} of points in R , this set is said to be affinely indepen- dent if for any real scalars λk, where k ∈ {0, . . . , n}, the equations
n n X X λi = 0 and λivi = 0 k=0 k=0 imply that λk = 0 for k = 0, . . . , n. An affine transformation is a mapping α from a vector space V to a vector space W that has coefficients in the same division ring as V such that α(v) = T (v)+w, where v ∈ V , for some linear transformation T : V 7→ W and some vector w ∈ W . It is easy to verify that T is injective if and only if α is injective, and T is surjective if and only if α is surjective. An affine isomorphism is a bijective affine transformation.
Proposition 3.3. Given a set of unique points A = {a0, . . . , an}, the following are equivalent:
(1) A is an affinely independent set. (2) The set {ai − a : a 6= ai} is linearly independent for all ai ∈ A. (3) The set {ai − a : a 6= ai} is linearly independent for any ai ∈ A. Proposition 3.4. Given a transformation f, the following are equivalent:
(1) α is an affine transformation. n (2) The mapping x 7→ α(x) − α(x0) is linear for all x0 ∈ R . n (3) The mapping x 7→ α(x) − α(x0) is linear for any x0 ∈ R . n Lemma 3.5. Given a simplex s with the vertex set {v0, . . . , vp}, a linear map f : |s| → R is an imbedding if and only if it maps the vertex set of s to an affinely independent set in Rn, where n ≥ p. (These proofs are left to the reader.) A geometric realization of a simplicial complex K in Rn is a linear imbedding of |K| in Rn. Our look at simplicial homology will focus on geometric realizations of abstract simplicial complexes. From now on, for some imbedding f, an abstract simplicial complex K will be identified with f(|K|), and a simplex s with a vertex set {v0, . . . , vp} will be identified with p p X σ = f(|s|) = {x ∈ U ⊃ R |x = λivi}, i=0
5 (where λi is exactly the barycentric coordinate of x with respect to the vertex vi), and |K| will be [ identified with f(|s|). s∈K
3.2 The Simplicial Approximation Theorem.
Definition 3.6. If σ =< v0 . . . vp > is a simplicial complex, then the barycenter of σ is defined to be the point p X 1 σˆ = v . p + 1 i i=0 It is the point of Int σ all of whose barycentric coordinates with respect to the vertices of σ are equal. If general,σ ˆ is the centroid of σ.
If K is a simplicial complex, we can define a sequence of subdivisions of the skeletons of K as (0) (p) follows: Let L0 = K . In general, if Lp is a subdivision of K , let Lp+1 be the subdivision (p+1) of the K obtained by starring Lp from the barycenters of the (p + 1)-simplices of K. The union of the complexes Lp is called the is called the first barycentric subdivision of K, denoted sd(K). We can define the second barycentric subdivision of K as sd(sd(K)) = sd2(K); in general, the n-th barycentric subdivision of K is defined to be sdn(K).
We now will show that if h : |K| → |L| is a continuous map, then there is a subdivision K0 of K such that h has a simplicial approximation f : K0 → L. The proof when K is finite follows easily from the use of barycentric subdivisions.
Theorem 3.7 (The Simplicial Approximation Theorem). Let K and L be simplicial complexes; let K be finite. Given a continuous map h : |K| → |L|, there is an N such that h has a simplicial approximation f : sdN (K) → L.
Proof: Cover |K| by the open sets h−1(St w) for all vertices w ∈ L. Given this open covering A of the compact metric space K, there is a number λ such that any set of diameter less than λ lies in one of the elements of A. If there is no such λ, one could choose a sequence Sn of sets with diameter less than 1/n but does not lie in any element of A. Choose xn ∈ Sn; by compactness, some subsequence xni converges, say to x. Now x ∈ A for some A ∈ A. Because A is open, it contains Cni for sufficiently large i, contrary to our construction.
Choose N so that each simplex in sdN (K) has diameter less than λ/2. Then each star of a vertex in sdN (K) has diameter less than λ, so it lies in one of the sets h−1(St w). Then h : |K| → |L| satisfies the star condition relative to sdN (K) and L; that is, for each vertex v ∈ sdN (K) there is a vertex w ∈ L such that h(St v) ⊂ St w. Therefore, the desired simplicial approximation exists.
6 4 Homology of a Simplicial Complex
Since we will be integrating over simplices, discussion of orientation is critical. If σ is a p-simplex with a vertex set {v0, . . . , vn}, define two orderings of its vertex set to be equivalent if they differ by an even permutation. If p > 0, then this equivalence relation has two equivalence classes, each of which is called an orientation of σ. An oriented simplex is a simplex σ paired with an orientation. If v0, . . . , vp are affinely independent, let v0 . . . vp denote the simplex they span, and let < v0 . . . vp > denote the oriented simplex consisting of the simplex v0, . . . , vp and the equivalence class of the particular ordering of (v0, . . . , vp).
Definition 4.1. Let K be a simplicial complex, and let Op the oriented p-simplices of K.A p- chain on K is a function c : Op → Z such that
(1) c(−σ) = −c(σ), where σ and −σ are opposite orientations of the same simplex. (2) c(σ) = 0 for all but finitely many oriented simplices σ. If we write the law of composition in Z additively, then we add p-chains by adding their values, forming the group of (oriented) p-chains of K (denoted Cp(K, Z)). If p < 0 or p > dim K, then Cp(K, Z) is the trivial group. If σ ∈ O, the elementary chain c corresponding to σ is the function defined as follows:
(1) c(σ) = 1
(2) c(−σ) = −1
(3) c(τ) = 0 when τ ∈ O \ {σ}.
By abuse of notation, σ can be a simplex, an oriented simplex, or the elementary p-chain c cor- responding to the oriented simplex σ. We will try to maintain the notation that σ is an oriented simplex.
Cp(K, Z) is a free abelian group with a basis obtained by orienting each p-simplex and using the corresponding chains as a basis, for once all p-simplices of K are (arbitrarily) oriented, each P p-chain can be written uniquely as a finite linear combination c = niσi of each elementary chain σi. c then assigns the value ni to each σi, −ni to each −σi, and 0 to each oriented p-simplex not appearing in the summation. In fact, given an arbitrary abelian group G, the group Cp(K,G) of p-chains of K with coefficients in G can be defined as the set of all formal linear combinations X gσσ, gσ ∈ G, σ ∈ U ⊂ K subject to the relation gσ(−σ) = −gσ(σ). (We are writing the group operation in G additively.) However, for our purposes, we are only interested in the cases where G is Z or R.
Definition 4.2. Let σ =< v0 . . . vp > be an oriented p-simplex. The boundary of σ, denoted ∂pσ, is the (p − 1)-chain defined by p X i ∂pσ = (−1) < v0,..., vˆi, . . . , vp >, i=0 where xˆ means that x is deleted.
Since ∂p is obviously linear, the boundary of a p-chain is a (p − 1)-chain. For an arbitrary abelian group G and a set of oriented simplices U ⊂ K, we have that X X ∂p( gσσ) = gσ(∂pσ), σ∈U σ∈U making the boundary operator a group homomorphism from ∂p : Cp(K,G) → Cp−1(K,G).
7 Lemma 4.3. ∂p−1 ◦ ∂p = 0.
Proof: Since ∂p−1 ◦ ∂p is linear, it suffices to check this on generators < v0 . . . vp > as follows:
p X i ∂p−1(∂p < v0 . . . vp >) = ∂p−1( (−1) < v0 ... vˆi . . . vp >) i=0 p X i = (−1) ∂p−1 < v0 ... vˆi . . . vp > i=0 p p X i+j X i+j−1 = (−1) < . . . vˆj ... vˆi ··· > + (−1) < . . . vˆi ... vˆj ··· > ji p X i+j i+j−1 = ((−1) + (−1) ) < . . . vˆi ... vˆj ··· > i Now that we have a series of free abelian groups Cp(K,G) and a homomorphism ∂p between them such that ∂p−1 ◦ ∂p = 0, we can look at the chain complex ∂2 ∂1 ∂0 ∂−1 ... −→ C1(K,G) −→ C0(K,G) −→ C−1(K,G) −−→ ... As earlier, if p < 0 or p > dim K, then we let Cp(K) denote the trivial group, making the nontrivial part of the chain complex the exact sequence ∂p+1 ∂p ∂p−1 ∂1 ∂0 0 −−−→ Cp(K,G) −→ Cp−1(K,G) −−−→ ... −→ C0(K,G) −→ 0. Definition 4.4. The set ker(∂p), also denoted as Zp(K,G), is the set of p-cycles of K. The set im(∂p+1), also denoted as Bp(K,G), is the set of p-boundaries. Because ∂p ◦ ∂p+1 = 0, each boundary of a (p + 1)-chain is a p-cycle, so Bp(K,G) ⊂ Zp(K,G). We define Hp(K,G) = Zp(K,G)/Bp(K,G) and call it the p-th homology group of K. Two p-chains c1 and c2 are considered homologous if c1 − c2 = ∂p+1c for some (p + 1)-chain c. If c1 = ∂p+1c, then c1 is homologous to zero. Theorem 4.5. If K and K0 are two simplicial complexes and there exists a homeomorphism 0 0 between |K| and |K |, then Hp(K,G) is isomorphic to Hp(K ,G), making homology groups a topological invariant.2 2Proof of this landmark theorem of algebraic topology can be found in [2]. 8 5 Applications of Homology Groups Homology groups are among the most useful topological invariants of a space because of their computational tractability, making it convenient to define other topological properties of a space in terms of the space’s homology groups. 5.1 The Euler Characteristic For example, given a simplicial complex, consider the Euler characteristic χ of |K|. Originating with Euler and Descartes, the polyhedral formula χ = V − E + F = 2 relates the number of vertices V , the number of edges E, and the number of faces F of a convex polyhedron. This can be generalized using a surface of genus g to the Poincare formula χ ≡ V − E + F = χ(g), where χ(g) = 2 − 2g is the Euler characteristic for oriented compact surfaces (g = 0 yields the polyhedral formula). Since |K| is clearly an oriented compact surface, it makes sense to compute its Euler characteristic. In terms of simplices, V is the number of 0-simplices in K, E is the number of 1-simplices in K, and F is the number of 2-simplices in K. Generalizing this for surfaces of dimension greater than 2 gives us the formula dim K X χ(|K|) = (−1)p(number of p-simplices in K). p=0 dim K X p Theorem 5.1. χ(|K|) = (−1) dim Hp(K, Z). p=0 Proof: If we look further, noting that dim B−1(K, R) = dim Bdim K (K, R) = 0, we can use the fact that the number of p-simplices of Cp(K, R) is equal to dim Cp(K, R) = dim ker ∂p + dim Im ∂p = dim Zp(K, R) + dim Bp−1(K, R) to show that dim K X χ(|K|) = (−1)p(number of p-simplices in K) p=0 dim K X p = (−1) (dim Zp(K, R) − dim Bp−1(K, R)) p=0 dim K dim K X p X p = (−1) dim Zp(K, R) + (−1) dim Bp−1(K, R) p=0 p=0 dim K dim K X p X p+1 = (−1) dim Zp(K, R) + (−1) dim Bp(K, R) p=0 p=0 dim K X p = (−1) (dim Zp(K, R) − dim Bp(K, R)) p=0 dim K X p = (−1) dim Hp(K, R). p=0 Since homology is a topological invariant, the Euler characteristic is a topological invariant. 5.2 The Fundamental Group Homology has a special relationship with the fundamental group of an arcwise-connected topolog- ical space. 9 Definition 5.2. Let X be a topological space. A path in X with origin x0 and endpoint x1 is a continuous map α : [0, 1] → X such that α(0) = x0 and α(1) = x1. We let λx denote some path from x0 to x; we shall take λx0 to be the constant path. Definition 5.3. For x0, x1, x2 ∈ X, let α be a path from x0 to x1, and let β be a path from x1 to x2. The product of two paths α and β, denoted α ∗ β, is the path from x0 to x2 defined by ( 1 α(2t), 0 ≤ t ≤ 2 αβ(t) = 1 . β(2t − 1), 2 ≤ t ≤ 1 −1 −1 The inverse of α is the path α from x1 to x0 defined by α (t) = α(1 − t). Two paths α and β from x0 to x1 are homotopic (written as ≈) if there exists a continuous map F : [0, 1] × [0, 1] → X such that F (0, t1) = x0 and F (1, t1) = x1 for all t1 ∈ [0, 1]; and F (t2, 0) = α(t2) and F (t2, 1) = β(t2) for all t2 ∈ [0, 1]. The relation ≈ is clearly an equivalence relation; it is reflexive, symmetric, and transitive. If we denote the class of paths that are homo- topic to a path α under the relation ≈ as hαi, then we can define the product of two homotopy classes hαi and hβi to be hα ∗ βi wherever the path α ∗ β is defined. We can also define the inverse of a homotopy class as hαi−1 = hα−1i. If X is a topological space and x0 ∈ X, the set of ≈ equivalence classes of paths where the starting point and endpoint are both x0 forms a group under the operations of multiplication and inversion as previously defined. This group, denoted by π1(X, x0), is called the fundamental group, or first homotopy group, of the pair (X, x0). We now introduce some group theory that will help us relate the fundamental group to the first homology group. Definition 5.4. Let g, h be elements of a group G. The commutator of g and h is [g, h] = ghg−1h−1. Lemma 5.5. [g, h] = 1 if and only if g and h commute. Proof: Multiply [g, h] and 1 by hg on the right. [g, h]hg simplifies to gh, and we have gh = hg. Definition 5.6. Let G be a group. The commutator subgroup of G is the smallest subgroup of G containing all of the commutators of G. This subgroup is denoted by [G, G]. Using this, one can define an abelian group as a group with a trivial commutator subgroup. Lemma 5.7. The commutator subgroup of a group G is a normal subgroup of G. Proof: Let k, g, h ∈ G. We then compute k[g, h]k−1 = k(ghg−1g−1)k−1 = (kgk−1)(khk−1)(kg−1k−1)(kh−1k−1) = (kgk−1)(khk−1)(kgk−1)−1(khk−1)−1 = [kgk−1, khk−1] ∈ [G, G], making [G, G] closed under conjugation of elements in G. Therefore, [G, G] is normal in G. Specifically, the left and right cosets of [G, G] agree. Lemma 5.8. If G is any group, then G/[G, G] is an abelian group. Proof: For any g, h ∈ G, we compute [g[G, G], h[G, G]] = [[G, G]g, h[G, G]] = [G, G]gh[G, G][G, G]−1g−1h−1[G, G]−1 = [G, G]ghg−1h−1[G, G] = [G, G][g, h][G, G] = [G, G]. 10 Therefore, any two elements in G/[G, G] commute, making G/[G, G] abelian. We call G/[G, G] the abelianization of G, denoted G˜. Theorem 5.9. By regarding loops as 1-cycles, we obtain a homomorphism h : π1(|K|, x0) → H1(K, Z). If |K| is path-connected, then h is surjective and ker h = [π1(|K|, x0), π1(|K|, x0)]. 3 Therefore, h induces an isomorphism from π˜1(|K|, x0) onto H1(K, Z) . Proof: We can regard loops as 1-cycles by applying the simplicial approximation theorem. We denote f ≈ g for a relation of homotopy, fixing endpoints, between paths f and g. Regarding f and g as chains, f ∼ g will mean that f is homologous to g, meaning that f − g will be the boundary of a 2-chain. Here are some facts about this relation. (i) If f is a constant path, then f ∼ 0. Namely, f is a cycle since it is a loop, and since H1(point) = 0, f must then be a boundary. (ii) If f ≈ g, then f ∼ g. To see this, consider a homotopy F : [0, 1] × [0, 1] → |K| from f to g. This yields a pair of 2-simplices in |K| by subdividing the square [0, 1] × [0, 1] with orien- tation < v0v1v2v3 > into two triangles σ1 =< v0v1v3 > and σ2 =< v0v2v3 >. Let f be the loop moving along σ1 and let g be the loop moving along σ2. Then ∂(σ1 − σ2) = f − g. (iii) f ∗ g ∼ f + g. On < v0v1v2 >, put f on the edge < v0v1 > and g on the edge < v1v2 >. Then define a 2-simplex σ to be constant on the lines perpendicular to the edge < v0v2 >, resulting in f ∗ g being on the edge < v0v2 >. Then ∂σ = g − f ∗ g + f as desired. (iv) f −1 ∼ −f. This follows from (i)-(iii), which give f + f −1 ∼ f ∗ f −1 ∼ 0. By (ii), we have a well defined function φ : π1(|K|, x0) → H1(K, Z) taking homotopy classes [f] to homology classes [[f]]. This is a homomorphism; to see this, let f, g be loops and note that φ([f][g]) = φ([f ∗ g]) = [[fg]] = [[f]] + [[g]] by (iii). Our goal is to prove that φ is an isomorphism. To do this, we first define the function that will be the inverse of φ. Let ˆ −1 ˆ f be a path, and put f = λf(0) ∗ f ∗ λf(1), which is a loop at x0. Define ψ(f) = [f] ∈ π˜1(|K|, x0). This extends to a homomorphism ψ : C1(K, Z) → π˜1(|K|, x0). We need two more lemmas in order to complete the proof of the theorem. Lemma 5.10. φ takes the group B1(K, Z) of 1-boundaries into 1 ∈ π˜1(|K|, x0). Proof: Let σ =< v0v1v2 > and let f be the path along < v0v1 >, g the path along < v1v2 >, and h the path along < v2v0 >. Then ψ(∂σ) = ψ(< v0v1 > + < v1v2 > + < v2v0 >) = ψ(f + g + h) = ψ(f)ψ(g)ψ(h) = [fˆ][ˆg][hˆ] = [fˆgˆhˆ] = [λ ∗ f ∗ λ−1 ∗ λ ∗ g ∗ λ−1 ∗ λ ∗ h ∗ λ−1] v0 v1 v1 v2 v2 v0 = [λ ∗ f ∗ g ∗ h ∗ λ−1] v0 v0 = [constant] = 1. 3This follows the proof found in [1] 11 If f is a loop, then ψ ◦φ([f]) = ψ([[f]]) = [λ ∗f ∗λ−1] = [f] since λ was chosen to be a constant x0 x0 x0 path. Therefore, ψ ◦ φ is the identity function. We have yet to show that φ ◦ ψ is the identity function. The assignment x 7→ λx takes 0-simplices into 1-simplices and thus extends to a homomorphism P P P λ : C0(K, Z) → C1(K, Z) by λ nxx = λ( x nxx) = x nxλx. Lemma 5.11. If σ is a 1-simplex in |K|, then the class φ◦ψ(σ) is represented by the cycle σ+λ∂σ. Also, if c is a 1-chain, then φ ◦ ψ(c) = [[c − λ∂c]]. Proof: We compute φ ◦ ψ(σ) = φ[λ ∗ σ ∗ λ−1] = [[λ ∗ σ ∗ λ−1]] = [[λ + σ + λ−1]] = [[λ + σ − λ ]] v0 v1 v1 v0 v1 v0 v1 v0 by (iii) and (iv). The rest follows immediately. If c is a 1-cycle, then by the preceding lemma, φ ◦ ψ[[c]] = [[c]], finishing the proof of the the- orem. Theorem 5.12. Let |K| be any (n − 1)-connected based space. Then φ : πn(|K|, x0) → Hn(K, Z) is the Abelianization homomorphism if n = 1 and is an isomorphism if n > 1.4 4A proof of this theorem can be found in [3]. 12 6 Cohomology of a Simplicial Complex Definition: Let K be a simplicial complex. For 0 ≤ p ≤ dim K, let p ∗ C (K,G) = Hom(Cp(K,G),G) = (Cp(K,G)) , where V ∗ is the notation for the dual of V . Let δp : Cp(K,G) → Cp+1(K,G) be the adjoint of the p map ∂p+1 : Cp+1(K,G) → Cp(K,G). Thus δ is defined by p p p hδ c , dp+1i = hc , ∂p+1dp+1i, p p p where c is a p-cochain and dp is a p-chain. (Here, hc , cpi is used to denote the value of c evaluated p+1 p at cp.) As such, we have the relation δ ◦ δ = 0, and we can look at the chain complex 0 1 3 p−1 p 0 −→δ C1(K,G) −→δ C2(K,G) −→δ ... −−−→δ Cp(K,G) −→δ 0 As we did with homology, we define ker δp = Zp(K,G) (whose elements are called cocycles), im δp = Bp+1(K,G) (whose elements are called coboundaries), and we note that δp+1 ◦ δp = 0 p p because ∂p ◦ ∂p+1 = 0. As such, B (K,G) ⊂ Z (K,G), and we can define the p-th cohomology group of K as Hp(K,G) = Zp(K,G)/Bp(K,G). p We now derive an explicit formula exhibiting the effect of δ on elements of Cp(K,G). For oriented p p-simplices σi, σj ∈ K, let ϕσ ∈ C (K,G) be defined by i ϕσi (σj) = δj, i where δj is the Kronecker delta, and ϕσ(−σ) = −ϕσ(σ) = −1. Thus, if {σ1, . . . , σm} is a basis p p for Cp(K,G), then {ϕσ1 , . . . , ϕσm } is the dual basis for C (K,G). Since δ is linear, we need only p compute the effects of δ on the generators of a chain. So for oriented simplices σ =< v0 . . . vp > and τ =< w0 . . . wp+1 >, we have that p (δ ϕσ)(τ) = ϕσ(∂pτ) p+1 X i = ϕσ( (−1) < w0 ... wˆi . . . wp+1 >) i=0 p+1 X i = (−1) ϕσ(< w0 ... wˆi . . . wp+1 >) i=0 X = ϕ From this point forward, we will not include a subscript or superscript when referring to the boundary or coboundary operators for more tidy notation; the appropriate subscript or super- script will be made clear by the context in which each operator is used. Additionally, it will be assumed that chains and cochains will take coefficients in G = R, so we will write Cp(K,G) as Cp(K) and so on. 13 p 7 Duality between Hp(K) and H (K) Definition 7.1. If A and B are real vector spaces over a field F , then HomF (A, B) is the set of linear transformations from A into B, which is also a vector space over F . Lemma 7.2. Let C be a chain complex, and let F be a field. Let φ : Hom(Cp,F ) → HomF (Cp ⊗ F,F ) be defined by the equation hφ(f), cp ⊗ αi = hf, cpi · α, where f ∈ Hom(Cp,F ), cp ∈ Cp, and α ∈ F . Then φ is a vector space isomorphism that commutes with δ. Proof: We first check that φ is linear: hφ(f), α(cp ⊗ β)i = hφ(f), cp ⊗ αβi = hf, cpi · (αβ) = α · (hf, cpi · β) = α · hφ(f), cp ⊗ βi. To prove injectivity, suppose that φ(f) is the zero linear transformation. Then hφ(f), cp ⊗ 1i = 0 = hf, cpi · 1 = hf, cpi for all cp ∈ Cp. This implies that f is the zero homomorphism. ˜ To prove surjectivity, let φ : Cp ⊗ F → F be a linear transformation. Let us define f : Cp → F by ˜ the equation f(cp) = φ(cp ⊗ 1). It follows that f is a homomorphism of abelian groups because f(0) = 0 and ˜ f(cp + dp) = φ((cp + dp) ⊗ 1) ˜ = φ(cp ⊗ 1 + dp ⊗ 1) ˜ ˜ = φ(cp ⊗ 1) + φ(dp ⊗ 1) = f(cp) + f(dp). Furthermore, φ(f) = φ˜ since ˜ ˜ ˜ hφ(f), cp ⊗ αi = hf, cpi · α = φ(cp ⊗ 1) · α = φ((cp ⊗ 1) · α) = φ(cp ⊗ α), where the last two equalities hold because φ˜ is a linear transformation. To show that φ commutes with δ, we observe that hδφ(f), cp+1 ⊗ αi = hφ(f), (∂ ⊗ iF )(cp+1 ⊗ α)i = hf, ∂cp+1i · α = hδf, cp+1i · α = hφ(δf), cp+1 ⊗ αi. Theorem 7.3. Let C be a chain complex, and let F be a field. Then there is a natural vector space isomorphism p H (C) → HomF (Hp(C),F ). 14 Proof: We first note that if 0 → A → B → C → 0 is a short exact sequence of linear transformations between vector spaces over F , then for any vector space V over F , the dual sequence 0 → HomF (C,V ) → HomF (B,V ) → HomF (A, V ) → 0 is also exact (proof of this is left to the reader). Let E denote the chain complex C ⊗ F . Then Ep = Cp ⊗ F is a vector space over F , as are the boundaries (Bp) and cycles (Zp) in the chain complex E. Consider the short exact sequence of vector spaces 0 → Zp → Ep → Bp−1 → 0. This gives rise to a short exact dual sequence 0 → HomF (Bp−1,F ) → HomF (Ep,F ) → HomF (Zp,F ) → 0. The cochain complex in the middle is isomorphic to the cochain complex Hom(Cp,F ) by the ∗ preceding lemma. If jp : Bp → Zp is the inclusion map and its dual is written as j , then we obtain the exact sequence ∗ p ∗ 0 → coker jp−1 → H (C,F ) → ker jp → 0. We now consider the sequence jp 0 → Bp −→ Zp → Hp(E) → 0. Because it is a sequence of vector spaces and linear transformations, the dual sequence is exact: ∗ jp 0 → HomF (Hp(E),F ) → HomF (Zp,F ) −→ HomF (Bp,F ) → 0. ∗ ∗ ∼ Therefore coker jp = 0 and ker jp = HomF (Hp(C),F ), proving the theorem. This theorem shows that if K is a simplicial complex, then Hp(K) can be identified in a nat- ural way with the dual vector space of Hp(K). Since we are only considering finite-dimensional p simplicial complexes, this means that Hp(K) and H (K) are isomorphic, but the isomorphism depends on the choice of basis one uses. 15 8 Differential Forms on Manifolds We will now discuss some preliminaries about smooth manifolds. For x in a smooth manifold X, let T (X, x) denote the tangent space at x ∈ X. The cotangent space at x ∈ X is defined as T ∗(X, x). Define [ [ T (X) = T (X, x) and T ∗(X) = T ∗(X, x). x∈X x∈X T (X) is called the tangent bundle of X, and T ∗(X) is called the cotangent bundle of X. A projec- tion map π : T (X) → X is defined as follows. If v ∈ T (X), then v ∈ T (X, x) for some unique x ∈ X. Set π(v) = x. Similarly, there is a projection map from T ∗(X) to X that shall also be denoted as π. If V is an n-dimensional real vector space, then G(V ∗), the exterior algebra of V ∗, is equal to Λ0(V ∗) ⊕ Λ1(V ∗) ⊕ · · · ⊕ Λn(V ∗), where Λk(V ∗) is the space of all skew-symmetric k-linear forms on V . From these definitions, we have [ [ G(X) = G(T ∗(X, x)) and Λk(X) = Λk(T ∗(X, x)) x∈X x∈X k A k-form on X is a mapping µ : X → Λ (X) such that π ◦ µ = iX .A k-form on X is smooth if whenever V1,...,Vk are smooth vector fields, then ∞ µ(V1,...,Vk) ∈ C (X, R) where µ(V1,...,Vk)(x) = µ(x)(V1(x),...,Vk(x)). A differential form on X is a mapping ω : X → G such that π◦ω = iX ; it is smooth if its component in Λk(X) is smooth for each k. The set of smooth k-forms on X is denoted by C∞(X, Λk(X)). The set of all smooth differential forms is denoted by C∞(X, G(X)). Note that C∞(X, Λk(X)) is a free abelian group under point-wise addition and scalar multiplication. Theorem 8.1. Let X be a smooth manifold. There exists a unique linear map d : C∞(X, G(X)) → C∞(X, G(X)), called the exterior differential, such that the following properties hold: (1) d : C∞(X, Λk(X)) → C∞(X, Λk+1(X)); (2) d(f) = df (the ordinary differential) for f ∈ C∞(X, Λ0(X)); (3) if µ ∈ C∞(X, Λk(X)) and τ ∈ C∞(X, G(X)), then d(µ ∧ τ) = (dµ) ∧ τ + (−1)kµ ∧ τ; (4) d2 = 0. We can now look at the chain complex C∞(X, Λ0(X)) −→d C∞(X, Λ1(X)) −→d C∞(X, Λ2(X)) −→d ... which closely resembles the coboundary operator δ. Definition 8.2. Let X be a smooth manifold. A smooth differential form ω on X is closed if dω = 0. A form ω is exact if ω = dτ for some smooth form τ. Let Zk(X, d) denote the vector space of closed k-forms on X. Let Bk(X, d) denote the space of exact k-forms. Noting that d2 = 0, we can see that exact forms are closed forms as well. If ω = dτ, then dω = d(dτ) = d2τ = 0, implying that Bk(X, d) ⊂ Zk(X, d). Let Hk(X, d) = Zk(X, d)/Bk(X, d). Hk(X, d) is called the k-th De Rham cohomology group of X. Roughly speaking, De Rham cohomology measures the extent to which the fundamental the- orem of calculus fails in higher dimensions and on general manifolds. We now proceed to classify Hp(Rn, d) for all p ≥ 0. First, let p = 0. It is clear that Z0(Rn, d) = {constant functions on Rn} =∼ R. Since there are no p-forms when p < 0, it follows that B0(Rn, d) = 0. We now have that H0(Rn, d) =∼ R. (More generally, if a smooth manifold X is n-connected, then similar arguments show that H0(X, d) =∼ Rn.) The following landmark theorem completes our classification. 16 Theorem 8.3 (Poincare’s Lemma). 5 If an open set U ⊂ Rn is homotopic to a point x ∈ U (i.e., U is contractible), then Hp(U, d) = 0 for all p > 0. 5A proof of this theorem can be found in [8]. 17 9 Stokes’s Theorem: Integration and Cohomology Definition 9.1. A smoothly triangulated manifold is a triple (X, K, h) where X is a C∞ manifold, K is a simplicial complex, and h : |K| → X is a homeomorphism such that for each σ ∈ K, there is an open subset U ⊂ X with a coordinate system χ : U → Rn such that h(supp(σ)) (which makes sense when we look at σ as a simplicial complex in its own right) is a subset of U and χ ◦ h|supp(σ) is equal to A|supp(σ), where A is an affine transformation. Here, A is an affine transformation from the vector space associated with σ, denoted by V(σ) = {b ∈ |K| : v∈ / σ ⇒ b(v) = 0}, to R. ˜ Given a smoothly triangulated manifold (X, K, h), we can see that homomorphisms fp : p p ∞ p H (X, d) → H (K) are defined whenever there is given a sequence of linear maps fp : C (X, Λ (X)) → p p p C (K) such that δ ◦ fp = fp+1 ◦ d for all p. Then, fp(Z (X, d)) ⊂ Z (K), because dω = 0 (hence ω ∈ Cp(X, d) implies that δ(fp(ω)) = fp+1(dω) = fp+1(0) = 0. p p p−1 Also, fp(B (X, d)) ⊂ B (K), because ω = dτ for some τ ∈ C (X, d) implies that fp(ω) = fp(dτ) = δ(fp−1τ) ∈ Im δ We now can proceed to define a sequence of linear maps ∞ p p Ip : C (X, Λ (X)) → C (K), ∞ p where Ip is integration. For ω ∈ C (X, Λ (X)), Ip(ω) will be linear functional on Cp(K). Thus it suffices to specify the values of Ip(ω) on the basis elements of Cp(K), that is, on the oriented p-simplices σ, where p ≤ dim X = n. To do this, consider a coordinate system x : U → Rn where and U is an open set containing supp(σ). Note that x(supp(σ)) is a geometric p-simplex in Rn. We then choose an oriented p-simplex η in Rp and an affine map P : Rp → Rn such that −1 P (supp(η)) = x(supp(σ)) and x(σ) and P (η) agree in orientation. If we set hσ = x ◦ P , then ] ] hσ(ω), (where denotes pullback) is a smooth p-form on U. We define Ip(ω)(σ) to be the integral of this p-form over σ: Z ] Ip(ω)(σ) = hσ(ω). σ In other words, let (r1, . . . , rp) denote the coordinates in the plane of σ consistent with the ori- entation of σ; if σ =< v0 . . . vp >, let (r1, . . . , rp) be the coordinates relative to the ordered basis {v1 − v0, . . . , vp − v0}. Then ] hσ(ω) = gdr1 ∧ · · · ∧ drp for some continuous function g on U, and Z Ip(ω)(σ) = gdr1 . . . drp, σ which is the Riemann integral. Note that this integral is independent of the homeomorphism h. Theorem 9.2. δ ◦ Ip = Ip+1 ◦ d. This is exactly Stokes’s theorem. For any given smooth p-form ω and oriented (p + 1)-simplex σ, Z Z ] ] Ip+1(dω)(σ) = (hσ) (dω) = d(hσ(ω)) σ σ Z ] = hσ(ω) = Ip(ω)(∂σ) = δ ◦ Ip(ω)(σ). ∂σ p p Ip has now been shown to be a homomorphism from H (X, d) to H (K). 18 10 De Rham’s Theorem We now want to show that for smoothly triangulated manifolds (X, K, h), the De Rham cohomology of X is isomorphic to the simplicial cohomology of K, making the De Rham cohomology of X dual to the simplicial homology of K. Theorem 10.1. (De Rham’s Theorem): Let (X, K, h) be a smoothly triangulated manifold. Then p p Ip : H (X, d) → H (K) is an isomorphism for each p where 0 ≤ p ≤ dim X. In order for us to prove De Rham’s theorem, we need to prove three lemmas pertaining to 6 extension of forms on simplices, surjectivity of Ip, and injectivity of Ip. This particular approach is motivated by geometric construction rather than algebraic manipulation. 10.1 Extension of Forms In order to prove De Rham’s Theorem theorem, we need to prove a preliminary lemma allowing us to extend forms that are defined near the boundary of a simplex through the neighborhood of a simplex. Let σ be a p-simplex in Rn, n ≥ p. Lemma 10.2. (ar): Let ω be a closed smooth r-form near ∂σ with r ≥ 0, s ≥ 1. Suppose that Z ω = 0 if p = r + 1. ∂σ Then there is a closed smooth form ω0 near σ which equals ω near ∂σ. (br): Let ω be a closed smooth r-form near σ with r ≥ 1, p ≥ 1, and let ξ be a smooth (r − 1)-form near ∂σ such that dξ = ω near ∂σ. Suppose that Z Z ξ = ω if r = p. ∂σ σ Then there is a smooth form ξ0 near σ such that ξ0 = ξ near ∂σ and dξ0 = ω near σ. Proof: We will prove, for all s, that (a0) ⇒ (b1) ⇒ (a1) ⇒ (b2) ⇒ (a2) ⇒ ... (a0): First, note that ω is closed. If dω = 0, then ω is constant on any connected subset of σ. 1 Now, suppose that p = 1 and that σ =< v0v1 >. Then Z 0 = ω = ω(v1) − ω(v0) ∂σ 0 so ω(v0) = ω(v1). Since σ is connected, ω is constant and ω ≡ ω(v0). We can then let ω = ω(v0). Now let p > 1. Since dω = 0, ω is constant near ∂σ, and we can choose ω0 be be that constant. (ar−1) ⇒ (br): ω is a closed r-form (r ≥ 1) defined on an open set containing σ. By the Poincare lemma, ω is exact near σ; that is, there exists a smooth (r − 1)-form ξ1 defined near σ such that dξ1 = ω near σ. In general, ξ1 will not be equal to ξ near ∂σ. Consider the difference η = ξ1 − ξ near ∂σ. It is closed since, near ∂σ, d(ξ1 − ξ) = ω − ω = 0. Furthermore, if p = (r − 1) + 1 = r, then Z Z Z Z Z Z Z η = ξ1 − ξ = dξ1 − ξ = ω − ξ = 0. ∂σ ∂σ ∂σ σ ∂σ σ ∂σ 6This proof is based upon the proof of de Rham’s theorem found in [5] and [7]. 19 0 We can now apply (ar−1) to the form η. There exists a smooth closed (r − 1)-form η defined near σ such that η0 = η near ∂σ. Let ξ0 = ξ − η0. Then ξ0 is a smooth (r − 1)-form defined near σ such 0 0 0 0 that ξ = ξ1 − η = ξ near ∂σ, and dξ = dξ1 − dη = ω − 0 = ω near σ. 0 (br) ⇒ (ar) for r > 0: Let σ =< v0 . . . vp >; set σ =< v1 . . . vp >. Let Q be the union of all proper faces of σ with v0 as a vertex. Any open set containing Q contains a star-shaped neigh- borhood U of Q. By (br), there is a smooth form ξ0 such that dξ0 = ω in U. In particular, this holds in ∂σ0. If p − 1 = s, then by letting A = ∂σ − σ0, we have ∂A = ∂(∂σ − σ0) = ∂2σ − ∂σ0 = −∂σ0. Therefore, Z Z Z Z Z Z ω − ξ0 = ω − ξ0 = ω + ξ0 σ0 ∂σ0 σ0 −∂A σ0 ∂A Z Z Z Z Z Z = ω + dξ0 = ω + ω = ω = ω = 0. σ0 A σ0 A σ0+A ∂σ 0 0 We can now use (br), using σ , to assume the existence of a smooth form ξ1 near σ such that 0 0 0 0 ξ1 = ξ0 near ∂σ and dξ1 = ω near σ . There exists a neighborhood U of ∂σ in which ξ0 and 0 0 ξ1 are defined and equal to each other. Let ξ be their common value here, and set ξ = ξ0 near 0 0 0 0 0 0 Q \ U and ξ = ξ1 near σ \ U . Then ξ is a smooth form such that dξ = ω near ∂σ. Since there is a smooth form ξ near σ which equals ξ0 near ∂σ. By setting ω0 = dξ near σ, we obtain a form with the required properties. 10.2 Surjectivity of Ip We must now show that Ip is both surjective and injective. We first prove surjectivity. Lemma 10.3. : There exists a sequence of linear maps p ∞ p Φp : C (K) → C (X, Λ (X)) (0 ≤ p ≤ dim X) with the following properties: (1) d ◦ Φp = Φp+1 ◦ δ. (2) Ip ◦ Φp = identity 0 0 0 (3) If c denotes the 0-cochain such that c (v) = 1 for each vertex v ∈ K, then Φ0(c ) = 1. (4) If σ is an oriented p-simplex of K, then the 1-form Φp(ϕσ) is identically zero in a neigh- borhood of X\ St σ. Proof: We will use the following notation throughout this proof: P = {0, . . . , p}, J = {j0, . . . , jp}, M = {1, . . . , m}, V = {k ∈ M :< vkvj0 . . . vjp >∈ K}, and Mi = M\{ji}. We shall identify |K| and X through the homeomorphism h (i.e. |K| = X and h = identity). We begin by constructing a partition of unity, subordinate to the covering {St v : v ∈ K(0)} of X. Let v1, . . . , vm denote the vertices of K. For each j ∈ {1, . . . , m}, let bj denote the jth barycentric coordinate function on |K| = X and let Fj = {x ∈ X : bj(x) ≥ 1/(n + 1)} and Gj = {x ∈ X : bj(x) ≤ 1/(n + 2)}, where n = dim X. Then Fj and Gj are closed sets in X with the following properties: 20 (a) Fj ⊂ St vj. (b) X\ St vj ⊂ Gj. c c (c) Fj ∩ Gj = ∅, and Fj ⊂ Gj, where Gj = X \ Gj. (d) Since Fj is a closed set on the compact space X, Fj is compact. Therefore, a smooth function c gj can be found which is greater than 0 on Fj and equal to 0 outside the open set Gj. (e) The closed sets Fj cover X. (Given x ∈ X, then x ∈ Int σ for some simplex σ =< v0 . . . vp >, P 1 where p ≤ n. Now bj(x) = 0 for j∈ / J and i∈P bji (x) = 1. Since p + 1 ≤ n + 1, bj(x) ≥ n+1 for some j ∈ J . Thus, x ∈ Fj for this j.) In particular, for each x ∈ X, gj(x) 6= 0 for some j. C Furthermore, the open sets Gj for an open covering of X. P gj(x) (f) Since, from property 5, j∈M gj > 0, we can define functions φj(x) = P that k∈M gk(x) are defined and smooth on X. Furthermore, {φj} is a smooth partition of unity on X subordinate C P C C to {Gj }; that is, j∈M φj = 1, and φj vanishes outside Gj . Since Gj ⊂ St vj, the partition of unity {φj} is also subordinate to the open covering {St vj}. We define Φp in terms of {φj}. Since Φp is to be linear, it suffices to specify the values of Φp p on the generators ϕσ of C (K). For σ =< vj0 . . . vjp >, we define Φp(ϕσ) to be the p-form X i Φp(ϕσ) = p! (−1) φji dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp . i∈P We now verify Properties 1-4. Property 1: Clearly, d ◦ Φp(ϕσ) = (p + 1)!dφj0 ∧ · · · ∧ dφjp . On the other hand, Φp+1 ◦ δ(ϕσ) X = Φ ( ϕ ) p+1 Claim: If the vertices vk, vj0 , . . . , vjp are distinct and yet are not the vertices of a (p + 1)-simplex of K, then φkdφj0 ∧ · · · ∧ dφjp ≡ 0. If x∈ / St vk, then φk(x) = 0. If x ∈ St vk, then bk(x) 6= 0. But now bji (x) = 0 for some i ∈ {0, . . . , p}, for otherwise bk(x) 6= 0, bj0 (x) 6= 0, ... , bjp (x) 6= 0, making < vkvj0 . . . vjp > a (p + 1)-simplex. This is a contradiction; using this particular value of i, let 1 U = {y ∈ X : b (y) = }. ji n + 2 Then U is an open set in X containing x, and φji , which is identically 0 on U because U ⊂ Gji . Hence dφji ≡ 0 on U, and, in particular, dφji (x) = 0, verifying the claim. Applying this result to the expression for Φp+1 ◦ δ(ϕσ) yields X X φkdφj0 ∧ · · · ∧ dφjp = φkdφj0 ∧ · · · ∧ dφjp = A, k∈V k∈M\J 21 since those terms on the right-hand side which do not appear on the left are identically zero. Fur- thermore, X X i − (−1) φji dφk ∧ dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp k∈V i∈P X i X = − (−1) φji dφk ∧ dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp i∈P k∈V X i X = − (−1) φji dφk ∧ dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp i∈P k∈M\J X i X = − (−1) φji dφk ∧ dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp i∈P k∈Mi X i X = − (−1) φji ( dφk) ∧ dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp i∈P k∈Mi X i = − (−1) φji (−dφji ) ∧ dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp i∈P X = φji dφj0 ∧ · · · ∧ dφjp i∈P X = φkdφj0 ∧ · · · ∧ dφjp = B, k∈J P using the fact that d( k∈M φk) = d(1) = 0. We now have that Φp+1 ◦ δ(ϕσ) = (p + 1)!(A + B) X X = (p + 1)!( φkdφj0 ∧ · · · ∧ dφjp + φkdφj0 ∧ · · · ∧ dφjp ) k∈M\J k∈J X = (p + 1)!( φk)dφj0 ∧ · · · ∧ dφjp k∈M = (p + 1)!dφj0 ∧ · · · ∧ dφjp = d ◦ Φp(ϕσ), just as we require. Property 3: Since Φ0(ϕ 0 X X Φ0(c ) = ϕ Property 4: Suppose σ =< vj0 . . . vjp >. Then X i Φp(ϕσ) = p! (−1) φji dφj0 ∧ · · · ∧ dφdji ∧ · · · ∧ dφjp . i∈P 1 Note that if x ∈ X is such that bjk (x) < n+2 for some k ∈ P, then x ∈ Gjk , so that φjk and dφjk , and hence Φp(ϕσ), are zero at x. Thus Φp(ϕσ) is identically zero on 1 {x ∈ X|b (x) < for some k ∈ P}, jk n + 2 which is an open set containing X\ St σ. Property 2: This proof will be by induction. For p = 0, I0 ◦ Φ0(ϕ [I0 ◦ Φ0(ϕ 22 But note that φj(vk) = 0 for k 6= j since vk ∈/ St vj and φj = 0 outside St vj. Furthermore, X 1 = φj(vk) = φk(vk) (for each k). j∈M Hence j [I0 ◦ Φ0(ϕ Since this holds for all j and k, I0 ◦ Φ0 = identity, as required. Now assume Property 2 for dimension p − 1. For σ and τ oriented p-simplices of K, Z [Ip ◦ Φp(ϕσ)](τ) = Φp(ϕσ). τ We must show that this equals 1 if σ = τ and 0 if σ 6= τ. That this is zero for σ 6= τ is a consequence of Property 4 since Φp(ϕσ) is identically zero in a neighborhood of X \ St σ ⊂ τ. So R 0 we need only check that σ Φp(ϕσ) = 1. For this, let σ =< vj0 . . . vjp > and σ =< vj1 . . . vjp >. Then Z Z Z Φp(δϕσ0 ) = d[Φp−1(ϕσ0 )] = Φp−1(ϕσ0 ). σ σ ∂σ But ∂σ = σ0 plus an alternating sum of other oriented (p − 1)-simplices, so Z Z Φp−1(ϕτ ) = Φp−1(ϕτ ) = 1 ∂σ τ by induction. Hence Z Z Z Φp(δϕτ ) = Φp(ϕσ + terms of type ϕτ for τ 6= σ) = Φp(ϕσ) = [Ip ◦ Φp(ϕσ)](σ), σ σ σ completing the proof. 10.3 Injectivity of Ip We now proceed to prove the injectivity of Ip, completing the proof of De Rham’s theorem. p−1 Lemma 10.4. Let ω be a closed p-form on X. Suppose Ip(ω) = δ(c) for some c ∈ C (K). Then there exists a (p − 1)-form τ on X such that Ip−1(τ) = c and dτ = ω. Proof: We shall construct a sequence τ0, τ1, . . . , τn (n = dim X) of (p − 1)-forms such that (k) (1) τk is defined in a neighborhood of the k-skeleton K of K, (k) (2) dτk = ω near K , (k) (3) τk = τk−1 near K , and (4) Ip(τp−1) = c. Note that this will prove the lemma because for each oriented (p − 1)-simplex σ of |K| and each k ≥ p − 1, Z Z Ip−1(τk)(σ) = τk = τp−1 = Ip−1(τp−1)(σ) = c(σ). σ σ 23 (0) To construct τ0, cover K by a collection of mutually disjoint balls. Since ω is closed, ω must be exact in each of these balls because of the Poincare lemma. Hence there exists a smooth (p − 1)- 0 0 0 form τ0, defined on the union of these balls, such that dτ0 = ω there. If p 6= 1, take τ0 = τ0. If p = 1, we want I0(τ0) = c. But for a vertex vj of |K|, Z 0 0 0 I0(τ0)(< vj >) = τ0 = τ0(vj) 0 0 0 Let aj = c(vj) − τ0(vj), and define τ0 on the ball about vj by τ0 = τ0 + aj. Then dτ0 = dτ0 = ω (0) near K , and I0(τ0) = c as required. Now assume that τk−1 has been constructed with Properties 1-4. To construct τk, note that if we can find, for each k-simplex σ, a smooth (p − 1)-form τk(σ) defined in a neighborhood of σ k−1 such that d(τk(σ)) = ω near σ and τk(σ) = τk−1 near σ (a (k − 1)-simplex), then glueing will 0 yield a smooth (p − 1)-form τk satisfying Properties 1-3. To construct τk(σ), we shall apply (b1) from Lemma 8.2. Note that ω is a smooth, closed p- k−1 form defined near σ and that τk−1 is a smooth (p − 1)-form defined near σ such that dτk−1 = ω near σk−1. Furthermore, if k = p, then Z Z p ω = Ip(ω)(σ) = δ c(σ) = Ik−1(τk−1)(∂σ) = τk−1. σ ∂σ We can now apply Lemma 8.2, particularly (bp). There exists a smooth (p − 1)-form τk(σ) near k−1 0 σ such that τk(σ) = τk−1 near σ and d(τk(σ)) = ω near σ. This constructs τk satisfying 0 0 Properties 1-3. If k 6= p − 1, set τk = τk. If k = p − 1, we have τp−1 satisfying Properties 1-3, and 0 we want τp−1 such that Ip−1(τp−1) = c. Let c1 = c−Ip−1(τp−1), and define τp−1 in a neighborhood of K(p−1) by 0 τp−1 = τp−1 + Φp−1(c1), where Φ was defined in Lemma 9.3. For each r and each oriented r-simplex σ, note that Φr(ϕσ) is identically zero on a neighbor- hood of X\ St σ. In particular, it is zero near K(r−1) for each r-cochain c. Applying this with r = p, then with r = p − 1, we find 0 0 0 dτp−1 = dτp−1 + d ◦ Φp−1(c1) = dτp−1 + Φp ◦ δ(c1) = dτp−1 = ω near K(p−1) and 0 0 τp−1 = τp−1 + Φp−1(c1) = τp−1 = τp−2 (p−2) near K . Thus τp−1 satisfies Properties 1-3 with k = p − 1. Property 4 is also satisfied: Ip−1(τp−1) = Ip−1(τp−1) + Ip−1 ◦ Φp−1(c1) = (c − c1) + c1 = c. 24 11 Conclusion Lemmas 9.3 and 9.4 are now proven. Therefore, Ip is an isomorphism between the De Rham cohomology Hp(X, d) and the simplicial cohomology Hp(K). In other words, De Rham’s theo- rem shows that the simplicial homology groups (with coefficients in R) of a smoothly triangulated manifold (X, K, h) are dual to the De Rham cohomology groups of X. In particular, these groups are independent of the triangulation (K, h) of X. Let us now look at the problem posed in the introduction again. We want to know when a closed 1-form in the punctured plane is exact. Let v be a curl free vector field on U = R2 \ (0, 0). R Let α = v, where γ1(t) = (cos t, sin t) and t ∈ [0, 2π] is a generating loop for H1(U, ). Since γ1 R R ω = 2π, where γ1 −y x ω = dx + dy, x2 + y2 x2 + y2 it follows that Z α (v − ω) = 0. γ1 2π Since γ1 is a generator for H1(U, R), and the line integral over a sum of loops is the sum of the line integrals over those loops, it follows that Z α (v − ω) = 0 nγ1 2π 2 for all n ∈ Z. Since every loop γ in R \ (0, 0) is equivalent to γn = nγ1 for some integer n, and α R α v − 2π ω is curl free, it follows that γ (v − 2π ω) = 0 for every loop γ in U. Thus, by De Rham’s α theorem, we have that v − 2π ω = ∇φ for some smooth function φ on U, and the difference between a curl free vector field on U and the gradient of a smooth function on U is ω multiplied by a real constant. Therefore, the vector space Z/B (as defined in the introduction), which is equal to H1(U), is a one-dimensional real vector space and is therefore isomorphic to R. Since Hp(U, d) is ∼ dual to Hp(K) where K triangulates U, it follows that H1(K) = R. This means that there is only one equivalence class of 1-cycles in U that contains no 1-boundaries. This is class of 1-cycles that encompass the origin, which one could intuitively expect. We can now see that in some cases, it is much easier to analyze homology and cohomology of a triangulation of a manifold rather than looking at differential forms on the manifold itself. In general, for some p ∈ Rn, R, k = 0 k n 0, 1 ≤ k ≤ n − 2 H (R \{p}) = . R, k = n − 1 0, k = n 25 References [1] Glen E. Bredon, Topology and Geometry, Springer, 1986 [2] Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001 [3] J. P. May, A Concise Course in Algebraic Topology, University of Chicago Press, 1999 [4] James R. Munkres, Elements of Algebraic Topology, Westview Pess, 1993 [5] M. Singer and J. A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer, 1976 [6] Edwin H. Spanier, Algebraic Topology, Springer, 1994 [7] Hassler Whitney, Geometric Integration Theory, Princeton University Press, 1957 [8] William K. Allard, Math 204 Notes: The Poincare Lemma, 26