Topology, Cohomology and Sheaf Theory

Topology, cohomology and sheaf theory

Tu June 16, 2010

1 Lecture 1 1.1 Manifolds Definition 1.1 (Locally Euclidean). A topological space is locally Euclidean if every p ∈ M has a neighborhood U and a homeomorphism φ : U → V , where V is an open subset of Rn. We call the pair (U, φ) a chart. Definition 1.2 (C∞ Compatible). Two charts are C∞ compatible if φ ◦ ψ−1 and ψ ◦ φ−1 are C∞ functions. Definition 1.3 (Manifold). A manifold is a locally Euclidean, Hausdorff, sec- ond countable topological space on which there existsa covering by C∞ compatible charts. Example 1.4 (The Circle). The circle is given by x2 + y2 = 1. Taking the upper and lower semicircles, along with the left and right semicircles, provides the desired cover.

1.2 Tangent Space

Let ri be coordinates on an open set in Rn and xi be ri ◦ φ. Definition 1.5 (C∞ function). f : M → R is C∞ at p if there exists (U, φ) with p ∈ U such that f ◦ φ−1 is C∞ at φ(p) ∈ Rn. Definition 1.6 (Partial Derivatives). If f ∈ C∞(M) and (U, x1, . . . , xn) is a ∂ ∂ −1 chart containing p, then we define ∂xi |pf = ∂ri |φ(p)f ◦ φ

Definition 1.7 (Tangent Space). TpM, the tangent space at p, is the vector ∂ space with basis ∂xi at p ∈ (U, φ). Theorem 1.8. Let xi and yj be different sets of coordinates at a point p. Then ∂ P i ∂ i ∂xi ∂yj = aj ∂xi , where aj = ∂yj . The proof is a direct computation. ` Definition 1.9 (Vector Field). A vector field is a function X : M → p∈M TpM P i ∂ such that Xp ∈ TpM. We can write it at p as X = a ∂xi and we call X a C∞ vector field if the ai are all C∞.

1 1.3 Differential Forms ∗ Definition 1.10 (Cotangent Space). The cotangent space is Tp M = hom(TpM, R). i ∂ It has basis (dx )p the duals of ∂xi ’s at p. Definition 1.11 (Differential 1-Form). A 1-form is a function ω : M → ` ∗ ∗ P i i p∈M Tp M such that ωp ∈ Tp M. We can write them near p as ω = a (dx )p and call them C∞ if the ai are.

Deﬁnition 1.12 (Wedge Product). Let V be a real vector space, and let α, β ∈ V ∗, then (α ∧ β)(v, w) = α(v)β(w) − α(w)β(v). It then follows that β ∧ α = −α ∧ β and α ∧ α = 0.

Deﬁnition 1.13 (Wedge Product). Let Ak(V ) be the set of alternating functions on V with k inputs. If α ∈ Ak and β ∈ A` then α ∧ β ∈ Ak+` and β ∧ α = (−1)k`α ∧ β

We define A0(V ) = R. ` ∗ Definition 1.14 (k-form). A k-form on M is a map ω : M → p∈M Ak(Tp M) ∗ ∞ such that ωp ∈ Ak(Tp M). It is called C if the coefficient functions are. We denote Ai(M) to be the C∞ i-forms on M. We have that A0(M) = C∞(M), and Ak(M) = 0 for k greater than the dimension of M.

1.4 Exterior Derivative Definition 1.15 (Derivative). If f ∈ C∞(M), define a 1-form df ∈ A1(M) i P ∂f i k on a chart (U, x ) by df = ∂xi dx . More generally, let ω ∈ A (M) by ω = P I P I aI dx , then we define dω = d(aI ) ∧ dx . Proposition 1.16. The exterior derivative d : Ak(M) → Ak+1(M) satisfies

1. Antiderivation: d(ω ∧ τ) = (dω) ∧ τ + (−1)deg τ ω ∧ dτ 2. d2 = 0.

1.5 DeRham Cohomology We deﬁne the kernel of d to be the closed k-forms Zk(M) and the image to be the exact k-forms, Bk(M).

Deﬁnition 1.17 (DeRham Cohomology). Hk(M) = Zk(M)/Bk(M). Example 1.18 (H0(M)). H0(M) is just ker d : C∞(M) → A1(M), because there is no A−1(M). So we end up with the set of locally constant functions on M, and, because M is a manifold, H0(M) is the set of all functions constant on the connected components.

2 Theorem 1.19. H0(M) = Rr where r is the number of connected component of M.

Example 1.20 (Hk(R)). An element of A1(R) is just f(x)dx where f ∈ C∞(R), and element of dA0(R) is dg = g0(x)dx. So is every C∞ function equal to g0(x) for some g ∈ C∞(R)? Yes, by the fundamental theorem of calculus, so Hk(R) is R for k = 0 and is 0 else.

2 Lecture 2 - Computation of deRham Coho- mology 2.1 Pullback of Forms If f : M → N is a C∞ map, then there is a pullback map f ∗ : Ak(N) → Ak(M). For k = 0, then f ∗(h) = h◦k. In generall,y locally ω ∈ Ak(N) can be written P I P i1 ik ∗ P ∗ ∗ i1 ω = aI dx = aI dx ∧ ... ∧ dx and we deﬁne f ω = (f aI )d(f x ) ∧ ... ∧ d(f ∗xik ). Proposition 2.1.

∗ 1. 1M = 1Ak(M) 2. (f ◦ g)∗ = g∗ ◦ f ∗ 3. f ∗d = df ∗ By 3, we know that f ∗ of a closed form is closed. If dω = 0, then d(f ∗ω) = f ∗dω = 0. Now, f ∗(dτ) = d(f ∗τ), and so exact forms pullback to exact forms. Thus, f ∗ induces a map (also denoted by f ∗ on cohomology Hk(N) → Hk(M). If f : M → N is a diﬀeomorphism, then there exists g : N → M such that ∗ ∗ ∗ ∗ f ◦ g = 1N and g ◦ f = 1M , then g ◦ f = 1H∗(N) and f ◦ g = 1H∗(M). Theorem 2.2. If f : M → N is a diﬀeomorphism, then f ∗ : H∗(N) → H∗(M) is an isomorphism.

2.2 Homological Algebra

A cochain complex C is 0 → C0 →d C1 → ... such that d2 = 0. So Hk(C) = ker dk/imdk−1 is deﬁned. j A sequence of vector spaces A →i B → C is exact at B if im(i) = ker j. A linear map f : (A, d) → (B, d) of cochain complexes is a cochain map if f ◦ d = d ◦ f (this is the data of a map in each degree). A cochain map induces f ∗ : Hk(A) → Hk(B). Theorem 2.3 (Zig-Zag Lemma). A short exact sequence of cochain complexes j 0 → A →i B → C → 0 induces a long exact sequence on cohomology ... → ∗ Hk(A) → Hk(B) → Hk(C) →d Hk+1(A) → ....

3 2.3 Mayer-Vietoris Sequence Suppose that M is covered by open sets U, V so that M = U ∪ V . Then we j have Ak(M) →i Ak(U) ⊕ Ak(V ) → Ak(U ∩ V ) by the restriction maps followed by the diﬀerence of restrictions. Theorem 2.4 (Mayer-Vietoris Sequence). The sequence above is exact.

2.4 H∗(S1)

We cover S1 by the union of two copies of R. Then H∗(U) = H∗(V ) = H∗(R) and H∗(U ∩ V ) is two copies of R, and so H∗(U ∩ V ) = H∗(R) ⊕ H∗(R). Then the Mayer-Vietoris sequence becomes 0 → R → R2 → R2 → H1(S1) → 0 → 0 → ... So by exactness, H1(S1) = im(d∗) =∼ R2/ ker d∗ = R2/im(j∗) =∼ R, because the image of j∗ is the diagonal.

2.5 Smooth Homotopy ∞ ∞ Two C maps f0, f1 : M → N are smoothly homotopic if there existsa C map F : M × [0, 1] → N such that F (x, 0) = f0(x) and f(x, 1) = f1(x) (where we say that F is C∞ if it can be extended to a C∞ function in a neighborhood of M × [0, 1] in M × R). We write f0 ∼ f1. Deﬁnition 2.5 (Homotopy Inverse). f : M → N has a homotopy inverse if there exists g : N → M such that f ◦ g ∼ 1N and g ◦ f ∼ 1M Then we say that M and N have the same homotopy type and f is a homotopy equivalence. Homotopy Axiom:Homotopic maps f, g : M → N induce the same map on cohomology f ∗ = g∗ : H∗(N) → H∗(M).

Example 2.6. Rn has the homotopy type of a point {0}. We have i the inclusion of the origin and π the unique map to the point. Then π ◦ i is the identity map on the point, and i ◦ π : Rn → Rn sends every point to 0. We claim this is homotopic to the identity. Deﬁne F : Rn × [0, 1] → Rn by F (x, t) = (1 − t)x to be the homotopy. Corollary 2.7. If f : M → N is a homotopy equivalence, then f ∗ : H∗(N) → H∗(M) is an isomorphism.

Corollary 2.8 (Poincare Lemma). H∗(Rn) = H∗(pt) is R in degree 0 and 0 else.

3 Lecture 3 - Presheaves and Cech Cohomology

Deﬁnition 3.1 (Presheaf). A presheaf on a topological space X is a function that assigns to each open U ⊂ X an abelian group F (U) and to every inclusion V V U iU : U → V a group homomorphism F (iU ) = ρV : F (U) → F (V ) such that U U V U ρU = 1F(U), and ρW = ρW ◦ ρV .

4 Example 3.2. Ak(U) is the C∞ k-forms on U. This is a presheaf on a manifold M. Example 3.3. If G is an abelian group, for every open U ⊂ X, deﬁne G(U) to be the locally constant functions f : U → G. Then G is a presheaf.

3.1 Cech Cohomology of an Open Cover

Let {Uα} be an open cover of a topological space indexed by a totally ordered set. We’ll denote intersections by putting the subscripts together. When α = 0, 1 gives the cover, we have Mayer-Vietoris, which says 0 → k Q k k A (M) → A (Ui) → A (U01) → 0. Now, let F be a presheaf on a topological space X. We then have a sequence Y Y 0 → F (X) → F (Uα) → F (Uαβ) → ... α α<β

Define Cp(U, F ) to be the term involving p open sets. Then we define p p+1 Pp+1 i δ : C (U, F ) → C (U, F ) by (δω)α0,...,αp+1 = i=0 (−1) ωα0,...,αî,...,αp+1 . It turns out that δ2 = 0, and so we define the cohomology of this complex to be the Cech cohomology Hˇ k(U, F ).

3.2 Direct Limits Deﬁnition 3.4 (Directed Set). A directed set is a set I with a binary relation < that is reﬂexive, transitive and such that any two elements have a common upper bound.

Example 3.5. Fix p ∈ X. Let I be the set of neighborhoods of p in X and say that U < V iff U ⊂ V . Fix a topological space X. Then I be the set of all open covers of X. An open cover V refines U if every V ∈ V is contained in some U ∈ U. Refinement gives a directed set structure to the set of covers. A refinement V of U can be given by a refinement map on the index sets stating which U each V is contained in. A directed system of groups is a collection {Gi}i∈I of groups indexed by a directed set U such that for all a < b we have a homomorphism fab : Ga → Gb satisfying that faa = 1 and fac = fbc ◦ fab.

∞ Example 3.6. Let I be the neighborhoods of p ∈ X. Then set GU = C (U) and say that (U, f) and (V, g) are equivalent iﬀ there exists W ⊂ U ∩ V such that f|W = g|W . We call these the germs of functions at p. ` In i∈I Gi, let ga ∈ Ga and gb ∈ Gb. Then we say ga ∼ gb if there exists c > a, b such that f (g ) = f (g ) and deﬁne lim G = ` G / ∼. ac a bc b −→i∈I i i

5 3.3 Cech Cohomology of a Topological Space For each open cover, we have Hˇ k(U, F ), and we have restrictions making it into a directed system of abelian groups. So we deﬁne Hˇ k(X, F ) to be the limit of this system.

3.4 C∞ partitions of unity Definition 3.7 (Partition of Unity). A C∞ partition of unity on a manifold ∞ P M is a collection of C functions ρα with ρα = 1 and ρα : M → [0, 1]. We define the support of a function to be the set where it is nonzero. A collection of subsets {Aα} in X is locally finite if every p ∈ X has a neighborhood that meets only finitely many of the Aα. Theorem 3.8. Given any open cover of a manifold, there exists a C∞ partition of unity with each element’s support contained in one of the open sets of the cover.

4 Lecture 4 - Sheaves and the Cech-deRham Iso- morphism

Version 6 of lecture notes on Summer School website.

Exercise 4.1. Compute H∗ of Sn, Rn \{0}, CP1, CP2 for problem session today.

Problem Session: Does H∗(U, F ) depend on the order on the index set?

4.1 Sheaves Deﬁnition 4.2 (Sheaf). A sheaf on a topological space X is a presheaf such that for any open set U and any open cover Ui of U, we have

1. Uniqueness: If s ∈ F (U) such that s|Ui = 0 for all i, then s = 0.

2. Gluing: If si ∈ F (Ui) such that si|Ui∩Uj = sj|Ui∩Uj for all i, j, then there

exists s ∈ F (U) such that s|Ui = si. Example 4.3. 1. Ak is a sheaf for any manifold. 2. Zk, the closed C∞ k-forms on a manifold is a sheaf. 3. Ap,q is a sheaf for any complex manifold.

4. R, the locally constant functions to R, is a sheaf. However, the presheaf of constant functions is not a sheaf.

6 4.2 Cohomology in Degree 0 Let F be a sheaf on a topological space X. We want to know Hˇ 0(U, F ). Let U = {Ui} be an open cover of X. Then as F is a sheaf, we have the Cech sequence. So then Hˇ 0(U, F ) = ker δ as C−1 = 0 which is just im(r) where r is the Q restriction map F (X) → F (Ui). But this is precisely F (X). Now, let V be a reﬁnement of U. Both Cech groups are isomorphic to F (X), and so they are isomorphic.

Theorem 4.4. If F is a sheaf on a topological space X, then Hˇ 0(X, F ) = F (X).

4.3 Further Computations For all q and all k > 0, Hˇ k(M, Aq) = 0. But how do we prove this? The standard method for proving that cohomology is zero is ﬁnding a map K : Cp → Cp−1 such that 1 − 0 = δK + Kδ. Applying (δK + Kδ) to a cocycle just gives δK. Therefore, this induces the zero map on cohomology, and so 1∗ = 0, which can only happen when Hq = 0. We call K a chain homotopy. There is a theorem guaranteeing that such a K exists when the cohomology is zero. ∞ Let U = {Uα} be an open cover of M, and {ρα} be a C partition of unity subordinate to U. For k ≥ 1, then K : Cp(U, F ) → Cp−1(U, F ) by P (Kω)α0,...,αp−1 = α ραωα,α0,...,αp−1 . We must check that δK+Kδ = 1 for k ≥ 1, and then we have the vanishing of the Cech cohomology, when we take F = Aq. We in fact have Hk(M, Ap,q) = 0 for k ≥ 1 on a complex manifold. But Hk(M, Ωq) is not necessarily zero!

4.4 Sheaf Morphism A morphism of sheaves is a collection of maps, one for each open set, compatible with the restriction maps We deﬁne the stalk of a presheaf Fp at p ∈ X to be lim F (U). So a −→p∈U presheaf homomorphism φp : Fp → Gp by (U, s) 7→ (U, φ(s)).

4.5 Exact Sequences of Sheaves Theorem 4.5. A short exact sequence of sheaves 0 → E → F → G → 0 induces a long exact sequence ... → Hk(X, E ) → Hk(X, F ) → Hk(X, G ) → Hk+1(X, E ) → ....

Theorem 4.6 (Poincar´eLemma). The sequence of sheaves 0 → R → A0 → ... is exact.

0 Proof. Look at 0 → R → A → im(d0) → 0, and similarly, we can make 0 → Zk−1 → Ak−1 → Zk → 0 for all k.

7 So we have 0 → Hk−1(M,Z1) → Hk(M, R) → Hk(M, A0) = 0, and we can ﬁnish the proof by basic homological manipulations.