1 Tuesday, January 12

1.1 Background

From Phil’s email:

Personally, I found the following online references particularly useful:

1.2 Introduction

Recall that a topological manifold (or \(C^0\) manifold) \(X\) is a Hausdorff topological space locally homeomorphic to \({\mathbb{R}}^n\) with a countable topological base, so we have charts \(\phi_u: U\to {\mathbb{R}}^n\) which are homeomorphisms from open sets covering \(X\).

\(S^1\) is covered by two charts homeomorphic to intervals:

Maps that are merely continuous are poorly behaved, so we may want to impose extra structure. This can be done by imposing restrictions on the transition functions, defined as \begin{align*} t_{uv} \mathrel{\vcenter{:}}=\varphi_V \to \varphi_U ^{-1} : \varphi_U(U \cap V) \to \varphi_V(U \cap V) .\end{align*}

Is this a strictly increasing hierarchy? It’s not clear e.g. that every \(C^k\) manifold is PL.

Consider \({\mathbb{R}}^n\) as a topological manifold: are any two smooth structures on \({\mathbb{R}}^n\) diffeomorphic?

Fix a copy of \({\mathbb{R}}\) and form a single chart \({\mathbb{R}}\xrightarrow{\operatorname{id}} {\mathbb{R}}\). There is only a single transition function, the identity, which is smooth. But consider \begin{align*} X &\to {\mathbb{R}}\\ t &\mapsto t^3 .\end{align*} This is also a smooth structure on \(X\), since the transition function is the identity. This yields a different smooth structure, since these two charts don’t like in the same maximal atlas. Otherwise there would be a transition function of the form \(t_{VU}: t\mapsto t^{1/3}\), which is not smooth at zero. However, the map \begin{align*} X &\to X \\ t &\mapsto t^3 .\end{align*} defines a diffeomorphism between the two smooth structures.

\({\mathbb{R}}\) admits a unique smooth structure.

Let \(\tilde {\mathbb{R}}\) be some exotic \({\mathbb{R}}\), i.e. a smooth manifold homeomorphic to \({\mathbb{R}}\). Cover this by coordinate charts to the standard \({\mathbb{R}}\):

There exists a cover which is locally finite and supports a partition of unity: a collection of smooth functions \(f_i: U_i \to {\mathbb{R}}\) with \(f_i \geq 0\) and \({\operatorname{supp}}f \subseteq U_i\) such that \(\sum f_i = 1\) (i.e., bump functions). It is also a purely topological fact that \(\tilde {\mathbb{R}}\) is orientable.

So we have bump functions:

Take a smooth vector field \(V_i\) on \(U_i\) everywhere aligning with the orientation. Then \(\sum f_i V_i\) is a smooth nowhere vector field on \(X\) that is nowhere zero in the direction of the orientation. Taking the associated flow \begin{align*} {\mathbb{R}}&\to \tilde {\mathbb{R}}\\ t &\mapsto \varphi(t) .\end{align*} such that \(\varphi'(t) = V(\varphi(t))\). Then \(\varphi\) is a smooth map that defines a diffeomorphism. This follows from the fact that the vector field is everywhere positive.

To understand smooth structures on \(X\), we should try to solve differential equations on \(X\).

Note that here we used the existence of a global frame, i.e. a trivialization of the tangent bundle, so this doesn’t quite work for e.g. \(S^2\).

What is the difference between all of the above structures? Are there obstructions to admitting any particular one?

  1. (Munkres) Every \(C^1\) structure gives a unique \(C^k\) and \(C^ \infty\) structure.1

  2. (Grauert) Every \(C^ \infty\) structure gives a unique real-analytic structure.

  3. Every PL manifold admits a smooth structure in \(\dim X \leq 7\), and it’s unique in \(\dim X\leq 6\), and above these dimensions there exists PL manifolds with no smooth structure.

  4. (Kirby–Siebenmann) Let \(X\) be a topological manifold of \(\dim X\geq 5\), then there exists a cohomology class \(\operatorname{ks}(X) \in H^4(X; {\mathbb{Z}}/2{\mathbb{Z}})\) which is 0 if and only if \(X\) admits a PL structure. Moreover, if \(\operatorname{ks}(X) = 0\), then (up to concordance) the set of PL structures is given by \(H^3(X; {\mathbb{Z}}/2{\mathbb{Z}})\).

  5. (Moise) Every topological manifold in \(\dim X\leq 3\) admits a unique smooth structure.

  6. (Smale et al.): In \(\dim X\geq 5\), the number of smooth structures on a topological manifold \(X\) is finite. In particular, \({\mathbb{R}}^n\) for \(n \neq 4\) has a unique smooth structure. So dimension 4 is interesting!

  7. (Taubes) \({\mathbb{R}}^4\) admits uncountably many non-diffeomorphic smooth structures.

  8. A compact oriented smooth surface \(\Sigma\), the space of complex-analytic structures is a complex orbifold2 of dimension \(3g-2\) where \(g\) is the genus of \(\Sigma\), up to biholomorphism (i.e. moduli).

Kervaire-Milnor: \(S^7\) admits 28 smooth structures, which form a group.

2 Friday, January 15

Let \begin{align*} V &\mathrel{\vcenter{:}}=\left\{{a^2 + b^2 + c^2 + d^3 + e^{6k-1} = 0}\right\} \subseteq {\mathbb{C}}^5 \\ S_\varepsilon&\mathrel{\vcenter{:}}=\left\{{ {\left\lvert {a} \right\rvert}^2 + {\left\lvert {b} \right\rvert}^2 + {\left\lvert {c} \right\rvert}^2 + {\left\lvert {d} \right\rvert}^2 + {\left\lvert {e} \right\rvert}^2}\right\} .\end{align*} Then \(V_k \cap S_\varepsilon\cong S^7\) is a homeomorphism, and taking \(k=1,2,\cdots, 28\) yields the 28 smooth structures on \(S^7\). Note that \(V_k\) is the cone over \(V_k \cap S_\varepsilon\).

? Admits a smooth structure, and \(\mkern 1.5mu\overline{\mkern-1.5muV\mkern-1.5mu}\mkern 1.5mu_k \subseteq {\mathbb{CP}}^5\) admits no smooth structure.

Is every triangulable manifold PL, i.e. homeomorphic to a simplicial complex?

No! Given a simplicial complex, there is a notion of the combinatorial link \(L_V\) of a vertex \(V\):

It turns out that there exist simplicial manifolds such that the link is not homeomorphic to a sphere, whereas every PL manifold admits a “PL triangulation” where the links are spheres.

What’s special in dimension 4? Recall the Kirby-Siebenmann invariant \(\operatorname{ks}(x) \in H^4(X; {\mathbb{Z}}_2)\) for \(X\) a topological manifold where \(\operatorname{ks}(X) = 0 \iff X\) admits a PL structure, with the caveat that \(\dim X \geq 5\). We can use this to cook up an invariant of 4-manifolds.

Let \(X\) be a topological 4-manifold, then \begin{align*} \operatorname{ks}(X) \mathrel{\vcenter{:}}=\operatorname{ks}(X \times{\mathbb{R}}) .\end{align*}

Recall that in \(\dim X\geq 7\), every PL manifold admits a smooth structure, and we can note that \begin{align*} H^4(X; {\mathbb{Z}}_2) = H^4(X \times{\mathbb{R}}; {\mathbb{Z}}_2) = {\mathbb{Z}}_2, .\end{align*} since every oriented 4-manifold admits a fundamental class. Thus \begin{align*} \operatorname{ks}(X) = \begin{cases} 0 & X \times{\mathbb{R}}\text{ admits a PL and smooth structure} \\ 1 & X \times{\mathbb{R}}\text{ admits no PL or smooth structures }. \end{cases} \end{align*}

\(\operatorname{ks}(X) \neq 0\) implies that \(X\) has no smooth structure, since \(X \times{\mathbb{R}}\) doesn’t. Note that it was not known if this invariant was nonzero for a while!

Note that \(H^2(X; {\mathbb{Z}})\) admits a symmetric bilinear form \(Q_X\) defined by \begin{align*} {\left\langle { \alpha},~{ \beta} \right\rangle} \mapsto \int_X \alpha\wedge \beta = \alpha \smile\beta([X]) \in {\mathbb{Z}} .\end{align*} where \([X]\) is the fundamental class.

3 Main Theorems for the Course

Proving the following theorems is the main goal of this course.

If \(X, Y\) are compact oriented topological 4-manifolds, then \(X\cong Y\) are homeomorphic if and only if \(\operatorname{ks}(X) = \operatorname{ks}(Y)\) and \(Q_X \cong Q_Y\) are isometric, i.e. there exists an isometry \begin{align*} \varphi: H^2(X; {\mathbb{Z}}) \to H^2(Y; {\mathbb{Z}}) .\end{align*} that preserves the two bilinear forms in the sense that \({\left\langle {\varphi \alpha},~{ \varphi \beta} \right\rangle} = {\left\langle { \alpha},~{ \beta} \right\rangle}\).

Conversely, every unimodular bilinear form appears as \(H^2(X; {\mathbb{Z}})\) for some \(X\), i.e. the pairing induces a map \begin{align*} H^2(X; {\mathbb{Z}}) &\to H^2(X; {\mathbb{Z}}) {}^{ \check{} }\\ \alpha \mapsto {\left\langle { \alpha },~{ {-}} \right\rangle} .\end{align*} which is an isomorphism. This is essentially a classification of simply-connected 4-manifolds.

Note that preservation of a bilinear form is a stand-in for “being an element of the orthogonal group,” where we only have a lattice instead of a full vector space.

There is a map \(H^2(X; {\mathbb{Z}}) \xrightarrow{PD} H_2(X; {\mathbb{Z}})\) from Poincaré , where we can think of elements in the latter as closed surfaces \([\Sigma]\), and \begin{align*} {\left\langle { \Sigma_1 },~{ \Sigma_2 } \right\rangle} = \text{signed number of intersections points of } \Sigma_1 \pitchfork\Sigma_2 .\end{align*} Note that Freedman’s theorem is only about homeomorphism, and is not true smoothly. This gives a way to show that two 4-manifolds are homeomorphic, but this is hard to prove! So we’ll black-box this, and focus on ways to show that two smooth 4-manifolds are not diffeomorphic, since we want homeomorphic but non-diffeomorphic manifolds.

The signature of a topological 4- manifold is the signature of \(Q_X\), where we note that \(Q_X\) is a symmetric nondegenerate bilinear form on \(H^2(X; {\mathbb{R}})\) and for some \(a, b\) \begin{align*} (H^2(X; {\mathbb{R}}), Q_x) \xrightarrow{\text{isometric}} {\mathbb{R}}^{a, b} .\end{align*} where \(a\) is the number of \(+1\)s appearing in the matrix and \(b\) is the number of \(-1\)s. This is \({\mathbb{R}}^{ab}\) where \(e_i^2 = 1, i=1\cdots a\) and \(e_i^2 = -1, i=a+1, \cdots b\), and is thus equipped with a specific bilinear form corresponding to the Gram matrix of this basis. \begin{align*} \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & \ddots & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{bmatrix} = I_{a\times a} \oplus -I_{b \times b} .\end{align*} Then the signature is \(a-b\), the dimension of the positive-definite space minus the dimension of the negative-definite space.

Suppose \({\left\langle { \alpha},~{\alpha} \right\rangle} \in 2{\mathbb{Z}}\) and \(\alpha\in H^2(X; {\mathbb{Z}})\) and \(X\) a simply connected smooth 4-manifold. Then 16 divides \(\operatorname{sig}(X)\).

Note that Freedman’s theorem implies that there exists topological 4-manifolds with no smooth structure.

Let \(X\) be a smooth simply-connected 4-manifold. If \(a=0\) or \(b=0\), then \(Q_X\) is diagonalizable and there exists an orthonormal basis of \(H^2(X; {\mathbb{Z}})\).

This comes from Gram-Schmidt, and restricts what types of intersection forms can occur.

3.1 Warm Up: \({\mathbb{R}}^2\) Has a Unique Smooth Structure

Last time we showed \({\mathbb{R}}^1\) had a unique smooth structure, so now we’ll do this for \({\mathbb{R}}^2\). The strategy of solving a differential equation, we’ll now sketch the proof.

A Riemannian metric \(g\in \operatorname{Sym}^2 T^*X\) for \(X\) a smooth manifold is a metric on every \(T_p X\) given by \begin{align*} g_p: T_pX \times T_p X &\to {\mathbb{R}}\\ g(v, v) \geq 0, g(v,v) = 0 \iff v=0 .\end{align*}

An almost complex structure is a \(J\in \mathop{\mathrm{End}}(TX)\) such that \(J^2 = -\operatorname{id}\).

Let \(e\in T_p X\) and \(e\neq 0\), then if \(X\) is a surface then \(\left\{{e, Je}\right\}\) is a basis of \(T_p X\).

This is a basis because if \(Je\) and \(e\) are parallel, then ??? In particular, \(J_p\) is determined by a point in \({\mathbb{R}}^2\setminus\left\{{\text{the }x{\hbox{-}}\text{axis}}\right\}\)

3.1.1 Sketch of Proof

Let \(\tilde {\mathbb{R}}^2\) be an exotic \({\mathbb{R}}^2\).

3.1.1.1 Step 1

Choose a metric on \(\tilde {\mathbb{R}}^2\) \(g \mathrel{\vcenter{:}}=\sum f_I g_i\) with \(g_i\) metrics on coordinate charts \(U_i\) and \(f_i\) a partition of unity.

3.1.1.2 Step 2

Find an almost complex structure on \(\tilde {\mathbb{R}}^2\). Choosing an orientation of \(\tilde {\mathbb{R}}^2\), \(g\) defines a unique almost complex structure \(J_p e \mathrel{\vcenter{:}}= f\in T_p \tilde {\mathbb{R}}^2\) such that

This is because after choosing \(e\), there are two orthogonal vectors, but only one choice yields an oriented basis.

3.1.1.3 Step 3

We then apply a theorem:

Any almost complex structure on a surface comes from a complex structure, in the sense that there exist charts \(\varphi_i: U_i \to {\mathbb{C}}\) such that \(J\) is multiplication by \(i\).

So \(d \varphi(J \cdot e) = i \cdot d \varphi_i (e)\), and \((\tilde {\mathbb{R}}^2, J)\) is a complex manifold. Since it’s simply connected, the Riemann Mapping Theorem shows that it’s biholomorphic to \({\mathbb{D}}\) or \({\mathbb{C}}\), both of which are diffeomorphic to \({\mathbb{R}}^2\).

See the Newlander-Nirenberg theorem, a result in complex geometry.

4 Lecture 3 (Wednesday, January 20)

Today: some background material on sheaves, bundles, connections.

4.1 Sheaves

Recall that if \(X\) is a topological space, a presheaf of abelian groups \(\mathcal{F}\) is an assignment \(U\to \mathcal{F}(U)\) of an abelian group to every open set \(U \subseteq X\) together with a restriction map \(\rho_{UV}: \mathcal{F}(U) \to \mathcal{F}(V)\) for any inclusion \(V \subseteq U\) of open sets. This data has to satisfying certain conditions:

  1. \(\mathcal{F}(\emptyset) = 0\), the trivial abelian group.

  2. \(\rho_{UU}: \mathcal{F}(U) \to \mathcal{F}(U) = \operatorname{id}_{\mathcal{F}(U) }\)

  3. Compatibility if restriction is taken in steps: \(U \subseteq V \subseteq W \implies \rho_{VW} \circ \rho_{UV} = \rho_{UW}\).

We say \(\mathcal{F}\) is a sheaf if additionally:

  1. Given \(s_i \in \mathcal{F}(U_i)\) such that \(\rho_{U_i \cap U_j} (s_i) = \rho_{U_i \cap U_j} (s_j)\) implies that there exists a unique \(s\in \mathcal{F}(\bigcup_i U_i)\) such that \(\rho_{U_i}(s) = s_i\).

Let \(X\) be a topological manifold, then \(\mathcal{F}\mathrel{\vcenter{:}}= C^0({-}, {\mathbb{R}})\) the set of continuous functionals form a sheaf. We have a diagram

Link to diagram

Property (d) holds because given sections \(s_i \in C^0(U_i; {\mathbb{R}})\) agreeing on overlaps, so \({ \left.{{s_i}} \right|_{{U_i \cap U_j}} } = { \left.{{s_j}} \right|_{{U_i \cap U_j}} }\), there exists a unique \(s\in C^0(\bigcup_i U_i; {\mathbb{R}})\) such that \({ \left.{{s}} \right|_{{U_i}} } = s_i\) for all \(i\) – continuous functions glue.

Recall that we discussed various structures on manifolds: PL, continuous, smooth, complex-analytic, etc. We can characterize these by their sheaves of functions, which we’ll denote \({\mathcal{O}}\). For example, \({\mathcal{O}}\mathrel{\vcenter{:}}= C^0({-}; {\mathbb{R}})\) for topological manifolds, and \({\mathcal{O}}\mathrel{\vcenter{:}}= C^ \infty ({-}; {\mathbb{R}})\) is the sheaf for smooth manifolds. Note that this also works for PL functions, since pullbacks of PL functions are again PL. For complex manifolds, we set \({\mathcal{O}}\) to be the sheaf of holomorphic functions.

Let \(A\in {\mathsf{Ab}}\) be an abelian group, then \(\underline{A}\) is the sheaf defined by setting \(\underline{A}(U)\) to be the locally constant functions \(U\to A\). E.g. let \(X \in {\mathsf{Mfd}}_{{\mathsf{Top}}}\) be a topological manifold, then \(\underline{{\mathbb{R}}}(U) = {\mathbb{R}}\) if \(U\) is connected since locally constant \(\implies\) globally constant in this case.

Note that the presheaf of constant functions doesn’t satisfy (d)! Take \({\mathbb{R}}\) and a function with two different values on disjoint intervals:

Note that \({ \left.{{s_1}} \right|_{{U_1 \cap U_2}} } = { \left.{{s_2}} \right|_{{U_1 \cap U_2}} }\) since the intersection is empty, but there is no constant function that restricts to the two different values.

4.2 Bundles

Let \(\pi: \mathcal{E}\to X\) be a vector bundle, so we have local trivializations \(\pi ^{-1} (U) \xrightarrow{h_u} Y^d \times U\) where we take either \(Y={\mathbb{R}}, {\mathbb{C}}\), such that \(h_v \circ h_u ^{-1}\) preserves the fibers of \(\pi\) and acts linearly on each fiber of \(Y\times(U \cap V)\). Define \begin{align*} t_{UV}: U \cap V \to \operatorname{GL}_d(Y) \end{align*} where we require that \(t_{UV}\) is continuous, smooth, complex-analytic, etc depending on the context.

There are two \({\mathbb{R}}^1\) bundles over \(S^1\):

Note that the Mobius bundle is not trivial, but can be locally trivialized.

We abuse notation: \(\mathcal{E}\) is also a sheaf, and we write \(\mathcal{E}(U)\) to be the set of sections \(s: U\to \mathcal{E}\) where \(s\) is continuous, smooth, holomorphic, etc where \(\pi \circ s = \operatorname{id}_U\). I.e. a bundle is a sheaf in the sense that its sections form a sheaf.

The trivial line bundle gives the sheaf \({\mathcal{O}}\) : maps \(U \xrightarrow{s} U\times Y\) for \(Y={\mathbb{R}}, {\mathbb{C}}\) such that \(\pi \circ s = \operatorname{id}\) are the same as maps \(U\to Y\).

An \({\mathcal{O}}{\hbox{-}}\)module is a sheaf \(\mathcal{F}\) such that \(\mathcal{F}(U)\) has an action of \(\mathcal{O}(U)\) compatible with restriction.

If \(\mathcal{E}\) is a vector bundle, then \(\mathcal{E}(U)\) has a natural action of \({\mathcal{O}}(U)\) given by \(f\curvearrowright s \mathrel{\vcenter{:}}= fs\), i.e. just multiplying functions.

The locally constant sheaf \(\underline{{\mathbb{R}}}\) is not an \({\mathcal{O}}{\hbox{-}}\)module: there isn’t natural action since the sections of \({\mathcal{O}}\) are generally non-constant functions, and multiplying a constant function by a non-constant function doesn’t generally give back a constant function.

We’d like a notion of maps between sheaves:

A morphism of sheaves \(\mathcal{F} \to \mathcal{G}\) is a group morphism \(\varphi(U): \mathcal{F}(U) \to \mathcal{G}(U)\) for all opens \(U \subseteq X\) such that the diagram involving restrictions commutes:

Let \(X = {\mathbb{R}}\) and define the skyscraper sheaf at \(p \in {\mathbb{R}}\) as \begin{align*} {\mathbb{R}}_p(U) \mathrel{\vcenter{:}}= \begin{cases} {\mathbb{R}}& p\in U \\ 0 & p\not\in U. \end{cases} .\end{align*}

The \({\mathcal{O}}(U){\hbox{-}}\)module structure is given by \begin{align*} {\mathcal{O}}(U) \times{\mathcal{O}}(U) &\to {\mathbb{R}}_p(U) \\ (f, s) &\mapsto f(p) s .\end{align*} This is not a vector bundle since \({\mathbb{R}}_p(U)\) is not an infinite dimensional vector space, whereas the space of sections of a vector bundle is generally infinite dimensional (?). Alternatively, there are arbitrarily small punctured open neighborhoods of \(p\) for which the sheaf makes trivial assignments.

Let \(X = {\mathbb{R}}\in {\mathsf{Mfd}}_{\operatorname{Sm}}\) viewed as a smooth manifold, then multiplication by \(x\) induces a morphism of structure sheaves: \begin{align*} (x \cdot): {\mathcal{O}}&\to {\mathcal{O}}\\ s & \mapsto x\cdot s \end{align*} for any \(x\in {\mathcal{O}}(U)\), noting that \(x\cdot s\in {\mathcal{O}}(U)\) again.

Check that \(\ker \varphi\) is naturally a sheaf and \(\ker(\varphi)(U) = \ker (\varphi(U)): \mathcal{F}(U) \to \mathcal{G}(U)\)

Here the kernel is trivial, i.e. on any open \(U\) we have \((x\cdot):{\mathcal{O}}(U) \hookrightarrow{\mathcal{O}}(U)\) is injective. Taking the cokernel \(\operatorname{coker}(x\cdot)\) as a presheaf, this assigns to \(U\) the quotient presheaf \({\mathcal{O}}(U) / x{\mathcal{O}}(U)\), which turns out to be equal to \({\mathbb{R}}_0\). So \({\mathcal{O}}\to {\mathbb{R}}_0\) by restricting to the value at \(0\), and there is an exact sequence \begin{align*} 0 \to {\mathcal{O}}\xrightarrow{(x\cdot)} {\mathcal{O}}\to {\mathbb{R}}_0 \to 0 .\end{align*}

This is one reason sheaves are better than vector bundles: the category is closed under taking quotients, whereas quotients of vector bundles may not be vector bundles.

5 Lecture 4 (Friday, January 22)

5.1 The Exponential Exact Sequence

Let \(X = {\mathbb{C}}\) and consider \({\mathcal{O}}\) the sheaf of holomorphic functions and \({\mathcal{O}}^{\times}\) the sheaf of nonvanishing holomorphic functions. The former is a vector bundle and the latter is a sheaf of abelian groups. There is a map \(\exp: {\mathcal{O}}\to {\mathcal{O}}^{\times}\), the exponential map, which is the data \(\exp(U): {\mathcal{O}}(U) \to {\mathcal{O}}^{\times}(U)\) on every open \(U\) given by \(f\mapsto e^f\). There is a kernel sheaf \(2\pi i \underline{{\mathbb{Z}}}\), and we get an exact sequence \begin{align*} 0 \to 2\pi i \underline{{\mathbb{Z}}} \to {\mathcal{O}}\xrightarrow{\exp} {\mathcal{O}}^{\times}\to \operatorname{coker}(\exp) \to 0 .\end{align*}

What is the cokernel sheaf here?

Let \(U\) be a contractible open set, then we can identify \({\mathcal{O}}^{\times}(U) / \exp({\mathcal{O}}^{\times}(U)) = 1\).

Any \(f\in {\mathcal{O}}^{\times}(U)\) has a logarithm, say by taking a branch cut, since \(\pi_1(U) =0 \implies \log f\) has an analytic continuation. Consider the annulus \(U\) and the function \(z\in {\mathcal{O}}^{\times}(U)\), then \(z\not\in \exp({\mathcal{O}}(U))\) – if \(z=e^f\) then \(f=\log(z)\), but \(\log(z)\) has monodromy on \(U\):

Thus on any sufficiently small open set, \(\operatorname{coker}(\exp) = 1\). This is only a presheaf: there exists an open cover of the annulus for which \({ \left.{{z}} \right|_{{U_i}} }\), and so the naive cokernel doesn’t define a sheaf. This is because we have a locally trivial section which glues to \(z\), which is nontrivial.

Redefine the cokernel so that it is a sheaf. Hint: look at sheafification, which has the defining property \(\mathop{\mathrm{Hom}}_{{\mathsf{Presh}}}(\mathcal{G}, \mathcal{F}^{\mathsf{Presh}}) =\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}}( \mathcal{G}, \mathcal{F}^{{\mathsf{Sh}}})\) for any sheaf \(\mathcal{G}\).

The global sections sheaf of \(\mathcal{F}\) on \(X\) is given by \(H^0( X; \mathcal{F}) = \mathcal{F}(X)\).

Given vector bundles \(V, W\), we have constructions \(V \oplus W, V \otimes W, V {}^{ \check{} }, \mathop{\mathrm{Hom}}(V, W) = V {}^{ \check{} }\otimes W, \operatorname{Sym}^n V, \bigwedge^p V\), and so on. Some of these work directly for sheaves:

\(\mathop{\mathrm{Hom}}(V, W)\) will denote the global homomorphisms \(\mathscr{H}\kern-2pt\operatorname{om}(V, W)(X)\), which is a sheaf.

Let \(X^n \in {\mathsf{Mfd}}_{{\operatorname{sm}}}\) and let \(\Omega^p\) be the sheaf of smooth \(p{\hbox{-}}\)forms, i.e \(\bigwedge^p T {}^{ \check{} }\), i.e. \(\Omega^p(U)\) are the smooth \(p\) forms on \(U\), which are locally of the form \(\sum f_{i_1, \cdots, i_p} (x_1, \cdots, x_n) dx_{i_1} \wedge dx_{i_2} \wedge \cdots dx_{i_p}\) where the \(f_{i_1, \cdots, i_p}\) are smooth functions.

Take \(X= S^1\), writing this as \({\mathbb{R}}/{\mathbb{Z}}\), we have \(\Omega^1(X) \ni dx\). There are two coordinate charts which differ by a translation on their overlaps, and \(dx(x + c) =dx\) for \(c\) a constant:

Check that on a torus, \(dx_i\) is a well-defined 1-form.

Note that there is a map \(d: \Omega^p \to \Omega^{p+1}\) where \(\omega\mapsto d \omega\).

\(d\) is not a map of \({\mathcal{O}}{\hbox{-}}\)modules: \(d(f\cdot \omega) = f\cdot \omega + {\color{red} df \wedge \omega}\), where the latter is a correction term. In particular, it is not a map of vector bundles, but is a map of sheaves of abelian groups since \(d ( \omega_1 + \omega_2) = d( \omega_1 ) + d ( \omega_2)\), making \(d\) a sheaf morphism.

Let \(X \in {\mathsf{Mfd}}_{\mathbb{C}}\), we’ll use the fact that \(TX\) is complex-linear and thus a \({\mathbb{C}}{\hbox{-}}\)vector bundle.

Note that \(\Omega^p\) for complex manifolds is \(\bigwedge^p T {}^{ \check{} }\), and so if we want to view \(X \in {\mathsf{Mfd}}_{\mathbb{R}}\) we’ll write \(X_{{\mathbb{R}}}\). \(TX_{\mathbb{R}}\) is then a real vector bundle of rank \(2n\).

\(\Omega^p\) will denote holomorphic \(p{\hbox{-}}\)forms, i.e. local expressions \(\sum f_I(z_1, \cdots, z_n) \bigwedge dz_I\). For example, \(e^zdz\in \Omega^1({\mathbb{C}})\) but \(z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu dz\) is not, where \(dz = dx + idy\). We’ll use a different notation when we allow the \(f_I\) to just be smooth: \(A^{p, 0}\), the sheaf of \((p, 0){\hbox{-}}\)forms. Then \(z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu dz\in A^{1, 0}\).

Note that \(T {}^{ \check{} }X_{\mathbb{R}}\otimes _{\mathbb{C}}= A^{1, 0} \oplus A^{0, 1}\) since there is a unique decomposition \(\omega = fdz + gd\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\) where \(f,g\) are smooth. Then \(\Omega^d X_{\mathbb{R}}\otimes_{\mathbb{R}}{\mathbb{C}}= \bigoplus _{p+q=d} A^{p, q}\). Note that \(\Omega_{\setminus}^p \neq A^{p, q}\) and these are really quite different: the former are more like holomorphic bundles, and the latter smooth. Moreover \(\dim \Omega^p(X) < \infty\), whereas \(\Omega_{\setminus}^1\) is infinite-dimensional.

6 Principal \(G{\hbox{-}}\)Bundles and Connections (Monday, January 25)

Let \(G\) be a (possibly disconnected) Lie group. Then a principal \(G{\hbox{-}}\)bundle \(\pi:P\to X\) is a space admitting local trivializations \(h_u: \pi ^{-1} (U) \to G \times U\) such that the transition functions are given by left multiplication by a continuous function \(t_{UV}: U \cap V \to G\).

Setup: we’ll consider \(TX\) for \(X\in {\mathsf{Mfd}}_{\operatorname{Sm}}\), and let \(g\) be a metric on the tangent bundle given by \begin{align*} g_p: T_pX^{\otimes 2} \to {\mathbb{R}} ,\end{align*} a symmetric bilinear form with \(g_p(u, v) \geq 0\) with equality if and only if \(v=0\).

Define \(\mathop{\mathrm{Frame}}_p(X) \mathrel{\vcenter{:}}=\left\{{\text{bases of } T_p X}\right\}\), and \(\mathop{\mathrm{Frame}}(X) \mathrel{\vcenter{:}}=\bigcup_{p\in X} \mathop{\mathrm{Frame}}_p(X)\).

More generally, \(\mathop{\mathrm{Frame}}(\mathcal{E})\) can be defined for any vector bundle \(\mathcal{E}\), so \(\mathop{\mathrm{Frame}}(X) \mathrel{\vcenter{:}}=\mathop{\mathrm{Frame}}(TX)\). Note that \(\mathop{\mathrm{Frame}}(X)\) is a principal \(\operatorname{GL}_n({\mathbb{R}}){\hbox{-}}\)bundle where \(n\mathrel{\vcenter{:}}=\operatorname{rank}(\mathcal{E})\). This follows from the fact that the transition functions are fiberwise in \(\operatorname{GL}_n({\mathbb{R}})\), so the transition functions are given by left-multiplication by matrices.

A principal \(G{\hbox{-}}\)bundle admits a \(G{\hbox{-}}\)action where \(G\) acts by right multiplication: \begin{align*} P \times G \to P \\ ( (g, x), h) \mapsto (gh, x) .\end{align*} This is necessary for compatibility on overlaps. Key point: the actions of left and right multiplication commute.

The orthogonal frame bundle of a vector bundle \(\mathcal{E}\) equipped with a metric \(g\) is defined as \(\mathop{\mathrm{OFrame}}_p(\mathcal{E}) \mathrel{\vcenter{:}}=\left\{{\text{orthonormal bases of } \mathcal{E}_p}\right\}\), also written \(O_r({\mathbb{R}})\) where \(r \mathrel{\vcenter{:}}=\operatorname{rank}( \mathcal{E})\).

The fibers \(P_x \to \left\{{x}\right\}\) of a principal \(G{\hbox{-}}\)bundle are naturally torsors over \(G\), i.e. a set with a free transitive \(G{\hbox{-}}\)action.

Let \(\mathcal{E}\to X\) be a complex vector bundle. Then a Hermitian metric is a hermitian form on every fiber, i.e.  \begin{align*} h_p: \mathcal{E}_p \times\overline{\mathcal{E}_p } \to {\mathbb{C}} .\end{align*} where \(h_p(v, \mkern 1.5mu\overline{\mkern-1.5muv\mkern-1.5mu}\mkern 1.5mu ) \geq 0\) with equality if and only if \(v=0\). Here we define \(\overline{\mathcal{E}_p}\) as the fiber of the complex vector bundle \(\overline{\mathcal{E}}\) whose transition functions are given by the complex conjugates of those from \(\mathcal{E}\).

Note that \(\mathcal{E}, \overline{\mathcal{E}}\) are genuinely different as complex bundles. There is a conjugate-linear map given by conjugation, i.e. \(L(cv) = \mkern 1.5mu\overline{\mkern-1.5muc\mkern-1.5mu}\mkern 1.5mu L(v)\), where the canonical example is \begin{align*} {\mathbb{C}}^n &\to {\mathbb{C}}^n \\ (z_1, \cdots, z_n) &\mapsto (\mkern 1.5mu\overline{\mkern-1.5muz_1\mkern-1.5mu}\mkern 1.5mu, \cdots, \mkern 1.5mu\overline{\mkern-1.5muz_n\mkern-1.5mu}\mkern 1.5mu) .\end{align*}

We define the unitary frame bundle \(\mathop{\mathrm{UFrame}}(\mathcal{E}) \mathrel{\vcenter{:}}=\bigcup_p \mathop{\mathrm{UFrame}}(\mathcal{E})_p\), where at each point this is given by the set of orthogonal frames of \(\mathcal{E}_p\) given by \((e_1, \cdots, e_n)\) where \(h(e_i , \mkern 1.5mu\overline{\mkern-1.5mue_j\mkern-1.5mu}\mkern 1.5mu ) = \delta_{ij}\).

This is a principal \(G{\hbox{-}}\)bundle for \(G = U_r({\mathbb{C}})\), the invertible matrices \(A_{/{\mathbb{C}}}\) satisfy \(A \overline{A}^t = \operatorname{id}\).

For \(G={\mathbb{Z}}/2{\mathbb{Z}}\) and \(X= S^1\), the Möbius band is a principal \(G{\hbox{-}}\)bundle:

For \(G={\mathbb{Z}}/2{\mathbb{Z}}\), for any (possibly non-oriented) manifold \(X\) there is an orientation principal bundle \(P\) which is locally a set of orientations on \(U\), i.e.  \begin{align*} P\mathrel{\vcenter{:}}=\left\{{(x, O) {~\mathrel{\Big|}~}x\in X,\, O \text{ is an orientation of }T_p X}\right\} .\end{align*} Note that \(P\) is an oriented manifold, \(P\to X\) is a local isomorphism, and has a canonical orientation. (?) This can also be written as \(P = \mathop{\mathrm{Frame}}(X) / \operatorname{GL}_n^+({\mathbb{R}})\), since an orientation can be specified by a choice of \(n\) linearly independent vectors where we identify any two sets that differ by a matrix of positive determinant.

Let \(P\to X\) be a principal \(G{\hbox{-}}\)bundle and let \(G\to \operatorname{GL}(V)\) be a continuous representation. The associated bundle is defined as \begin{align*} P\times_G V = \left\{{(p, v){~\mathrel{\Big|}~}p\in P,\, v\in V}\right\} / \sim && \text{where } (p, v) \sim (pg, g ^{-1} v) ,\end{align*} which is well-defined since there is a right action on the first component and a left action on the second.

Note that \(\mathop{\mathrm{Frame}}(\mathcal{E})\) is a \(\operatorname{GL}_r({\mathbb{R}}){\hbox{-}}\)bundle and the map \(\operatorname{GL}_r({\mathbb{R}}) \xrightarrow{\operatorname{id}} \operatorname{GL}({\mathbb{R}}^r)\) is a representation. At every fiber, we have \(G \times_G V = (p, v)/\sim\) where there is a unique representative of this equivalence class given by \((e, pv)\). So \(P\times_G V_p \to \left\{{p}\right\} \cong V_x\).

Show that \(\mathop{\mathrm{Frame}}(\mathcal{E}) \times_{\operatorname{GL}_r({\mathbb{R}})} {\mathbb{R}}^r \cong \mathcal{E}\). This follows from the fact that the transition functions of \(P \times_G V\) are given by left multiplication of \(t_{UV}: U \cap V \to G\), and so by the equivalence relation, \(\operatorname{im}t_{UV} \in \operatorname{GL}(V)\).

Suppose that \(M^3\) is an oriented Riemannian 3-manifold. Them \(TM\to \mathop{\mathrm{Frame}}(M)\) which is a principal \({\operatorname{SO}}(3){\hbox{-}}\)bundle. The universal cover is the double cover \({\operatorname{SU}}(2) \to {\operatorname{SO}}(3)\), so can the transition functions be lifted? This shows up for spin structures, and we can get a \({\mathbb{C}}^2\) bundle out of this.

7 Wednesday, January 27

7.1 Bundles and Connections

Let \(\mathcal{E}\to X\) be a vector bundle, then a connection on \(\mathcal{E}\) is a map of sheaves of abelian groups \begin{align*} \nabla: \mathcal{E}\to \mathcal{E}\otimes\Omega^1_X \end{align*} satisfying the Leibniz rule: \begin{align*} \nabla (fs) = f \nabla s + s\otimes ds \end{align*} for all opens \(U\) with \(f\in {\mathcal{O}}(U)\) and \(s\in \mathcal{E}(U)\). Note that this works in the category of complex manifolds, in which case \(\nabla\) is referred to as a holomorphic connection.

A connection \(\nabla\) induces a map \begin{align*} \tilde{\nabla}: \mathcal{E}\otimes\Omega^p &\to \mathcal{E}\otimes\Omega^{p+1} \\ s \otimes \omega &\mapsto \nabla s \wedge w + s\otimes d \omega .\end{align*} where \(\wedge: \Omega^p \otimes\Omega^1 \to \Omega^{p+1}\). The standard example is \begin{align*} d: {\mathcal{O}}&\to \Omega^1 \\ f &\mapsto df .\end{align*} where the induced map is the usual de Rham differential.

Prove that the curvature of \(\nabla\), i.e. the map \begin{align*} F_{\nabla} \mathrel{\vcenter{:}}=\nabla \circ \nabla: \mathcal{E}\to \mathcal{E}\otimes\Omega^2 \end{align*} is \({\mathcal{O}}{\hbox{-}}\)linear, so \(F_{\nabla}(fs) = f\nabla \circ \nabla(s)\). Use the fact that \(\nabla s \in \mathcal{E}\otimes\Omega^1\) and \(\omega \in \Omega^p\) and so \(\nabla s \otimes \omega \in \mathcal{E} \Omega^1 \otimes \Omega^p\) and thus reassociating the tensor product yields \(\nabla s \wedge \omega \in \mathcal{E}\otimes\Omega^{p+1}\).

Why is this called a connection?

This gives us a way to transport \(v\in \mathcal{E}_p\) over a path \(\gamma\) in the base, and \(\nabla\) provides a differential equation (a flow equation) to solve that lifts this path. Solving this is referred to as parallel transport. This works by pairing \(\gamma'(t) \in T_{ \gamma(t) } X\) with \(\Omega^1\), yielding \(\nabla s = ( \gamma'(t)) = s( \gamma(t))\) which are sections of \(\gamma\).

Note that taking a different path yields an endpoint in the same fiber but potentially at a different point, and \(F_\nabla = 0\) if and only if the parallel transport from \(p\) to \(q\) depends only on the homotopy class of \(\gamma\).

Note: this works for any bundle, so can become confusing in Riemannian geometry when all of the bundles taken are tangent bundles!

The Levi-Cevita connection \(\nabla^{LC}\) on \(TX\), which depends on a metric \(g\). Taking \(X=S^2\) and \(g\) is the round metric, there is nonzero curvature:

In general, every such transport will be rotation by some vector, and the angle is given by the area of the enclosed region.

A connection is flat if \(F_\nabla = 0\). A section \(s \in \mathcal{E}(U)\) is flat if it is given by \begin{align*} L(U) \mathrel{\vcenter{:}}=\left\{{ s\in \mathcal{E}(U) {~\mathrel{\Big|}~}\nabla s = 0}\right\} .\end{align*}

Show that if \(\nabla\) is flat then \(L\) is a local system: a sheaf that assigns to any sufficiently small open set a vector space of fixed dimension. An example is the constant sheaf \(\underline{{\mathbb{C}}^d}\). Furthermore \({\operatorname{rank}}(L) = {\operatorname{rank}}(\mathcal{E})\).

Given a local system, we can construct a vector bundle whose transition functions are the same as those of the local system, e.g. for vector bundles this is a fixed matrix, and in general these will be constant transition functions. Equivalently, we can take \(L\otimes_{\mathbb{R}}{\mathcal{O}}\), and \(L\otimes 1\) form flat sections of a connection.

7.2 Sheaf Cohomology

Let \(\mathcal{F}\) be a sheaf of abelian groups on a topological space \(X\), and let \(\mathfrak{U} \mathrel{\vcenter{:}}=\left\{{U_i}\right\} \rightrightarrows X\) be an open cover of \(X\). Let \(U_{i_1, \cdots, i_p} \mathrel{\vcenter{:}}= U_{i_1} \cap U_{i_2} \cap\cdots \cap U_{i_p}\). Then the Čech Complex is defined as \begin{align*} C_{\mathfrak{U}}^p(X, \mathcal{F}) \mathrel{\vcenter{:}}=\prod_{i_1 < \cdots < i_p} \mathcal{F}(U_{i_1, \cdots, i_p}) \end{align*} with a differential \begin{align*} {{\partial}}^p: C_{\mathfrak{U}}^p(X, \mathcal{F}) &\to C_{\mathfrak{U}}^{p+1}(X \mathcal{F}) \\ \sigma &\mapsto ({{\partial}}\sigma)_{i_0, \cdots, i_p} \mathrel{\vcenter{:}}=\prod_j (-1)^j { \left.{{\sigma_{i_0, \cdots, \widehat{i_j}, \cdots, i_p}}} \right|_{{U_{i_0, \cdots, i_p}}} } \end{align*} where we’ve defined this just on one given term in the product, i.e. a \(p{\hbox{-}}\)fold intersection.

Check that \({{\partial}}^2 = 0\).

The Čech cohomology \(H^p_{\mathfrak{U}}(X, \mathcal{F})\) with respect to the cover \(\mathfrak{U}\) is defined as \(\ker {{\partial}}^p/\operatorname{im}{{\partial}}^{p-1}\). It is a difficult theorem, but we write \(H^p(X, \mathcal{F})\) for the Čech cohomology for any sufficiently refined open cover when \(X\) is assumed paracompact.

Consider \(S^1\) and the constant sheaf \(\underline{{\mathbb{Z}}}\):

ere we have \begin{align*} C^0(S^1, \underline{{\mathbb{Z}}}) = \underline{{\mathbb{Z}}}(U_1) \oplus \underline{{\mathbb{Z}}}(U_2) = \underline{{\mathbb{Z}}} \oplus \underline{{\mathbb{Z}}} ,\end{align*} and \begin{align*} C^1(S^1, {\mathbb{Z}}) = \bigoplus_{\substack{ \text{double} \\ \text{intersections}} } \underline{{\mathbb{Z}}}(U_{ij}) \underline{{\mathbb{Z}}}(U_{12}) = \underline{{\mathbb{Z}}}(U_1 \cap U_{2}) = \underline{{\mathbb{Z}}} \oplus \underline{{\mathbb{Z}}} .\end{align*} We then get \begin{align*} C^0(S^1, \underline{{\mathbb{Z}}}) &\xrightarrow{{{\partial}}} C^1(S^1, \underline{{\mathbb{Z}}}) \\ {\mathbb{Z}}\oplus {\mathbb{Z}}&\to {\mathbb{Z}}\oplus {\mathbb{Z}}\\ (a, b) &\mapsto (a-b, a-b) ,\end{align*}

Which yields \(H^*(S^1, \underline{{\mathbb{Z}}}) = [{\mathbb{Z}}, {\mathbb{Z}}, 0, \cdots]\).

8 Sheaf Cohomology (Friday, January 29)

Last time: we defined the Čech complex \(C_{\mathfrak{U} }^p(X, \mathcal{F} ) \mathrel{\vcenter{:}}=\prod_{i_1, \cdots, i_p} \mathcal{F} (U_{i_1} \cap\cdots \cap U_{i_p})\) for \(\mathfrak{U}\mathrel{\vcenter{:}}=\left\{{U_i}\right\}\) is an open cover of \(X\) and \(F\) is a sheaf of abelian groups.

If \(\mathfrak{U}\) is a sufficiently fine cover then \(H^p_{\mathfrak{U}}(X, \mathcal{F})\) is independent of \(\mathfrak{U}\), and we call this \(H^p(X; \mathcal{F})\).

Recall that we computed \(H^p(S^1, \underline{{\mathbb{Z}}} = [{\mathbb{Z}}, {\mathbb{Z}}, 0, \cdots]\).

Let \(X\) be a paracompact and locally contractible topological space. Then \(H^p(X, \underline{{\mathbb{Z}}}) \cong H^p_{{\operatorname{Sing}}}(X, \underline{{\mathbb{Z}}})\). This will also hold more generally with \(\underline{{\mathbb{Z}}}\) replaced by \(\underline{A}\) for any \(A\in {\mathsf{Ab}}\).

We say \(\mathcal{F}\) is acyclic on \(X\) if \(H^{> 0 }(X; \mathcal{F}) = 0\).

How to visualize when \(H^1(X; \mathcal{F}) = 0\):

On the intersections, we have \(\operatorname{im}{{\partial}}^0 = \left\{{ (s_{i} - s_{j})_{ij} {~\mathrel{\Big|}~}s_i \in \mathcal{F}(U_i)}\right\}\), which are cocycles. We have \(C^1(X; \mathcal{F})\) are collections of sections of \(\mathcal{F}\) on every double overlap. We can check that \(\ker {{\partial}}^1 = \left\{{ (s_{ij}) {~\mathrel{\Big|}~}s_{ij} - s_{ik} + s_{jk} = 0}\right\}\), which is the cocycle condition. From the exercise from last class, \({{\partial}}^2 = 0\).

Let \(X\) be a paracompact Hausdorff space and let \begin{align*} 0 \to \mathcal{F}_1 \xrightarrow{\varphi} \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \end{align*} be a SES of sheaves of abelian groups, i.e. \(\mathcal{F}_3 = \operatorname{coker}(\varphi)\) and \(\varphi\) is injective. Then there is a LES in cohomology:

For \(X\) a manifold, we can define a map and its cokernel sheaf:

\begin{align*} 0 \to \underline{{\mathbb{Z}}} \xrightarrow{\cdot 2} \underline{{\mathbb{Z}}} \to \underline{{\mathbb{Z}}/2{\mathbb{Z}}} \to 0 .\end{align*} Using that cohomology of constant sheaves reduces to singular cohomology, we obtain a LES in homology:

Suppose \(0 \to \mathcal{F}\to I_0 \xrightarrow{d_0} I_1 \xrightarrow{d_1} I_2 \xrightarrow{d_2} \cdots\) is an exact sequence of sheaves, so on any sufficiently small set kernels equal images., and suppose \(I_n\) is acyclic for all \(n\geq 0\). This is referred to as an acyclic resolution. Then the homology can be computed at \(H^p(X; \mathcal{F}) = \ker (I_p(X) \to I_{p+1}(X)) / \operatorname{im}(I_{p-1}(X) \to I_p(X) )\).

Note that locally having kernels equal images is different than satisfying this globally!

This is a formal consequence of the existence of the LES. We can split the LES into a collection of SESs of sheaves:

\begin{align*} 0 \to \mathcal{F}\to I_0 \xrightarrow{d_0} \operatorname{im}(d_0) \to 0 && \operatorname{im}(d_0) = \ker(d_1) \\ 0 \to \ker(d_1) \hookrightarrow I_1 \to I_1/\ker (d_1) = \operatorname{im}(d_1) && \operatorname{im}(d_1) = \ker(d_2) \\ .\end{align*} Note that these are all exact sheaves, and thus only true on small sets. So take the associated LESs. For the SES involving \(I_0\), we obtain: