Differential Forms II: Manifolds and Cohomology develops the language and tools of differential geometry on smooth manifolds, then uses them to build a bridge from calculus to topology. The course begins by defining smooth manifolds and the tangent and cotangent bundles that replace the familiar linear structure of Euclidean space. From there it introduces differential forms as the natural objects for differentiation and integration on manifolds, leading to orientations, integration, and the general Stokes’ theorem as the central analytic principle tying local computations to global results.

The later chapters shift from analysis to topology through de Rham cohomology, which measures global obstructions to solving differential equations on manifolds. The Poincaré lemma provides the local exactness theory that makes cohomology meaningful, while the Mayer-Vietoris sequence and explicit computations show how to assemble local data into global invariants. The course culminates in de Rham’s theorem, which identifies de Rham cohomology with singular cohomology with real coefficients, and then explores applications and further directions that show how differential forms connect geometry, topology, and analysis in a unified framework.

# Introduction

This course continues the exterior calculus of Differential Forms I by moving from open subsets of $\mathbb R^n$ to smooth manifolds. The central question is how much of calculus depends on coordinates, and how much survives on spaces that only look Euclidean locally. We shall build the language needed to integrate forms on manifolds, prove the manifold version of Stokes theorem, and then use closed and exact forms to extract topological information.

The course has two intertwined themes. The analytic theme is that differentiation, pullback, integration, and Stokes theorem can be formulated without a preferred coordinate system. The topological theme is that the failure of a closed form to be exact records global structure, leading to de Rham cohomology and ultimately to de Rham theorem.

## The Problem of Coordinate-Free Calculus

In Differential Forms I, forms lived on open sets $U \subset \mathbb R^n$, where vectors, covectors, and coordinates are available from the start. Many geometric spaces are not single open subsets of Euclidean space: spheres, tori, projective spaces, and matrix groups require several coordinate systems. The first problem of the course is to formulate calculus so that changing coordinates changes the description but not the object being studied.

[explanation: Local Descriptions and Global Objects] A smooth manifold is a space that can be studied through coordinate charts, but the charts are not part of the final geometric answer. A function, vector field, differential form, or integral must therefore have local coordinate formulae that agree on overlaps. This is the same discipline already present in exterior calculus on $\mathbb R^n$: the coordinate expression of a form may change, while the alternating multilinear object it represents remains the same.

The guiding test for every definition in the first part of the course is compatibility under change of coordinates. If a construction is meaningful only after choosing a chart and does not transform correctly on chart overlaps, it is not an intrinsic construction on the manifold. [/explanation]

The simplest place where this issue appears is already one-dimensional. A circle can be drawn in the plane, but its intrinsic calculus should not depend on choosing a preferred point at which to cut it open.

[example: The Circle Requires More Than One Coordinate] Consider $S^1=\{(x,y)\in \mathbb R^2:x^2+y^2=1\}$. A single global real coordinate would be a homeomorphism from $S^1$ onto an open subset of $\mathbb R$, but $S^1$ is compact and no nonempty open subset of $\mathbb R$ is compact, so such a coordinate cannot exist. Let $N=(0,1)$ and $S=(0,-1)$. Stereographic coordinates give charts \begin{align*} \varphi_N(x,y)=\frac{x}{1-y}\quad\text{on }S^1\setminus\{N\}, \qquad \varphi_S(x,y)=\frac{x}{1+y}\quad\text{on }S^1\setminus\{S\}. \end{align*} Solving for the inverse of the first chart gives \begin{align*} \varphi_N^{-1}(t)=\left(\frac{2t}{t^2+1},\frac{t^2-1}{t^2+1}\right), \end{align*} and on the overlap $S^1\setminus\{N,S\}$ the transition map is \begin{align*} (\varphi_S\circ \varphi_N^{-1})(t)=\frac{1}{t}. \end{align*} The inverse transition has the same formula, so both transition maps are smooth on $\mathbb R\setminus\{0\}$. Calculus on $S^1$ is therefore built from local coordinates, and the smooth transition function $t\mapsto 1/t$ is what lets derivatives and differential forms agree on the overlap. [/example]

This example foreshadows the general pattern: a manifold is locally Euclidean, but global questions come from how the local pieces fit together.

## Smooth Manifolds as the Setting

What hypotheses on a space make it a suitable domain for calculus? Local Euclidean behaviour is necessary, but topology also matters: we need enough open sets to separate points, and enough countability to support partitions of unity and integration theory. The first chapter after this introduction therefore begins with topological and smooth manifolds.

[definition: Topological Manifold] A topological manifold of dimension $n$ is a topological space $M$ such that: 1. $M$ is Hausdorff; 2. $M$ is second-countable; 3. every point $p \in M$ has an open neighbourhood $U \subset M$ and a homeomorphism $\varphi: U \to \varphi(U) \subset \mathbb R^n$, where $\varphi(U)$ is open. [/definition]

A pair $(U,\varphi)$ as in the definition is called a coordinate chart. The Hausdorff and second-countability assumptions prevent pathological spaces from entering the theory and ensure that the usual tools of analysis behave as expected.

[definition: Smooth Atlas] A smooth atlas on a topological manifold $M$ is a collection of coordinate charts $\{(U_i,\varphi_i)\}_{i \in I}$ such that: 1. $M = \bigcup_{i \in I} U_i$; 2. for every $i,j \in I$, the transition map \begin{align*} \varphi_j \circ \varphi_i^{-1}: \varphi_i(U_i \cap U_j) \to \varphi_j(U_i \cap U_j) \end{align*} is smooth whenever $U_i \cap U_j \ne \varnothing$. [/definition]

The transition maps are the mechanism by which Euclidean calculus becomes manifold calculus. Once they are smooth, derivatives and differential forms transform in a controlled way from chart to chart.

[example: The Standard Smooth Structure on the Sphere] Let $N=(0,\ldots,0,1)$ and $S=(0,\ldots,0,-1)$, and write a point of $\mathbb R^{n+1}$ as $(u,z)$ with $u\in\mathbb R^n$ and $z\in\mathbb R$. Define \begin{align*} \varphi_N(u,z)=\frac{u}{1-z} \end{align*} on $S^n\setminus\{N\}$ and \begin{align*} \varphi_S(u,z)=\frac{u}{1+z} \end{align*} on $S^n\setminus\{S\}$. If $a=\varphi_N(u,z)$, then $u=a(1-z)$ and $|u|^2+z^2=1$, so \begin{align*} |a|^2(1-z)^2+z^2&=1,\\ |a|^2(1-z)^2-(1-z)(1+z)&=0,\\ (1-z)\bigl(|a|^2(1-z)-(1+z)\bigr)&=0. \end{align*} Since $(u,z)\ne N$, we have $1-z\ne0$, hence \begin{align*} |a|^2(1-z)&=1+z,\\ |a|^2-1&=(|a|^2+1)z,\\ z&=\frac{|a|^2-1}{|a|^2+1}. \end{align*} Therefore \begin{align*} u &=a\left(1-\frac{|a|^2-1}{|a|^2+1}\right)\\ &=a\left(\frac{2}{|a|^2+1}\right)\\ &=\frac{2a}{|a|^2+1}, \end{align*} so \begin{align*} \varphi_N^{-1}(a)=\left(\frac{2a}{|a|^2+1},\frac{|a|^2-1}{|a|^2+1}\right). \end{align*} This point lies on $S^n$ because \begin{align*} \left|\frac{2a}{|a|^2+1}\right|^2+\left(\frac{|a|^2-1}{|a|^2+1}\right)^2 &=\frac{4|a|^2}{(|a|^2+1)^2}+\frac{(|a|^2-1)^2}{(|a|^2+1)^2}\\ &=\frac{4|a|^2+|a|^4-2|a|^2+1}{(|a|^2+1)^2}\\ &=\frac{|a|^4+2|a|^2+1}{(|a|^2+1)^2}\\ &=1. \end{align*} The same calculation with $1+z$ in place of $1-z$ gives \begin{align*} \varphi_S^{-1}(b)=\left(\frac{2b}{|b|^2+1},\frac{1-|b|^2}{|b|^2+1}\right). \end{align*} On the overlap $S^n\setminus\{N,S\}$, the coordinate $a=\varphi_N(u,z)$ is nonzero. Indeed, if $a=0$, then $u=0$, and $|u|^2+z^2=1$ gives $z=\pm1$, so the point is either $N$ or $S$. For $a\in\mathbb R^n\setminus\{0\}$, \begin{align*} (\varphi_S\circ\varphi_N^{-1})(a) &=\frac{\frac{2a}{|a|^2+1}}{1+\frac{|a|^2-1}{|a|^2+1}}\\ &=\frac{\frac{2a}{|a|^2+1}}{\frac{2|a|^2}{|a|^2+1}}\\ &=\frac{a}{|a|^2}. \end{align*} Similarly, for $b\in\mathbb R^n\setminus\{0\}$, \begin{align*} (\varphi_N\circ\varphi_S^{-1})(b) &=\frac{\frac{2b}{|b|^2+1}}{1-\frac{1-|b|^2}{|b|^2+1}}\\ &=\frac{\frac{2b}{|b|^2+1}}{\frac{2|b|^2}{|b|^2+1}}\\ &=\frac{b}{|b|^2}. \end{align*} Each component of $a\mapsto a/|a|^2$ is the rational function $a_i/(a_1^2+\cdots+a_n^2)$, whose denominator is nonzero on $\mathbb R^n\setminus\{0\}$. Thus both transition maps are smooth on the overlap. The two stereographic charts cover $S^n$, and their smooth transition maps make them a smooth atlas on the sphere. [/example]

## Tangent and Cotangent Bundles