🔁Elementary Differential Topology Unit 13 – De Rham Cohomology & Mayer-Vietoris Sequence

Key Concepts and Definitions

• De Rham cohomology groups $H^k_{dR}(M)$ measure the global properties of a smooth manifold $M$ using differential forms
• Differential forms generalize the notion of functions and provide a way to integrate over submanifolds
  • 0-forms are smooth functions, 1-forms are dual to vector fields, and k-forms are antisymmetric multilinear maps
• The exterior derivative $d$ is a linear operator that maps k-forms to (k+1)-forms and satisfies $d^2 = 0$
• Closed forms have zero exterior derivative ($d\omega = 0$), while exact forms are the exterior derivative of another form ($\omega = d\alpha$)
• The k-th de Rham cohomology group is defined as the quotient of closed k-forms modulo exact k-forms: $H^k_{dR}(M) = \frac{\ker d_k}{\operatorname{im} d_{k-1}}$
• The Mayer-Vietoris sequence relates the cohomology of a manifold to the cohomology of its subspaces
• Homotopy invariance states that the de Rham cohomology depends only on the homotopy type of the manifold

Historical Context and Motivation

• De Rham cohomology was introduced by Georges de Rham in the 1930s as a way to study the global properties of manifolds
• It provides a link between differential geometry and algebraic topology
• Historically, it was motivated by the study of differential equations on manifolds and the desire to generalize the fundamental theorem of calculus
• De Rham's theorem establishes an isomorphism between de Rham cohomology and singular cohomology, connecting differential forms and topological invariants
• The Mayer-Vietoris sequence, named after Walther Mayer and Leopold Vietoris, was developed as a computational tool in algebraic topology
  • It allows for the computation of cohomology groups by breaking a space into simpler pieces
• The sequence has been widely used in various areas of mathematics, including complex analysis, algebraic geometry, and physics

Differential Forms and Exterior Derivative

• A differential k-form $\omega$ on a smooth manifold $M$ is a smooth section of the k-th exterior power of the cotangent bundle $\Lambda^k(T^*M)$
  • In local coordinates $(x^1, \ldots, x^n)$, a k-form can be written as $\omega = \sum_{i_1 < \ldots < i_k} \omega_{i_1 \ldots i_k}(x) dx^{i_1} \wedge \ldots \wedge dx^{i_k}$
• The exterior derivative $d$ is a generalization of the differential of a function
  • For a k-form $\omega$, the exterior derivative $d\omega$ is a (k+1)-form defined by $(d\omega)_{i_1 \ldots i_{k+1}} = \sum_{j=1}^{k+1} (-1)^{j-1} \frac{\partial \omega_{i_1 \ldots \hat{i_j} \ldots i_{k+1}}}{\partial x^{i_j}}$
• The exterior derivative satisfies the following properties:
  • Linearity: $d(\alpha + \beta) = d\alpha + d\beta$
  • Graded Leibniz rule: $d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta$, where $\alpha$ is a k-form
  • Nilpotency: $d^2 = 0$
• The kernel of $d$ consists of closed forms, while the image of $d$ consists of exact forms
  • The de Rham complex is the sequence of differential forms connected by the exterior derivative: $0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \ldots \xrightarrow{d} \Omega^n(M) \to 0$

De Rham Cohomology Groups

• The k-th de Rham cohomology group $H^k_{dR}(M)$ is the quotient vector space of closed k-forms modulo exact k-forms
  • $H^k_{dR}(M) = \frac{\ker d_k}{\operatorname{im} d_{k-1}} = \frac{Z^k(M)}{B^k(M)}$, where $Z^k(M)$ are closed k-forms and $B^k(M)$ are exact k-forms
• The elements of $H^k_{dR}(M)$ are equivalence classes of closed k-forms that differ by an exact form
  • Two closed forms $\omega_1$ and $\omega_2$ represent the same cohomology class if $\omega_1 - \omega_2 = d\alpha$ for some (k-1)-form $\alpha$
• The dimension of $H^k_{dR}(M)$ is called the k-th Betti number $b_k(M)$ and is a topological invariant of the manifold
• The de Rham cohomology groups satisfy the following properties:
  • Homotopy invariance: If $M$ and $N$ are homotopy equivalent, then $H^k_{dR}(M) \cong H^k_{dR}(N)$ for all $k$
  • Poincaré duality: For a compact oriented n-manifold $M$, there is an isomorphism $H^k_{dR}(M) \cong H^{n-k}_{dR}(M)$
  • Künneth formula: For two manifolds $M$ and $N$, there is an isomorphism $H^k_{dR}(M \times N) \cong \bigoplus_{i+j=k} H^i_{dR}(M) \otimes H^j_{dR}(N)$

Mayer-Vietoris Sequence: Structure and Purpose

• The Mayer-Vietoris sequence is a long exact sequence that relates the cohomology of a manifold to the cohomology of its subspaces
• Given an open cover $\{U, V\}$ of a manifold $M$, the Mayer-Vietoris sequence is:
  • $\ldots \to H^k_{dR}(U \cap V) \xrightarrow{\delta} H^k_{dR}(U) \oplus H^k_{dR}(V) \xrightarrow{\alpha} H^k_{dR}(M) \xrightarrow{\beta} H^{k+1}_{dR}(U \cap V) \to \ldots$
• The maps in the sequence are:
  • $\delta$: the coboundary map, induced by the exterior derivative
  • $\alpha$: the map induced by the inclusions $U \hookrightarrow M$ and $V \hookrightarrow M$
  • $\beta$: the map induced by the restriction of forms from $M$ to $U \cap V$
• The exactness of the sequence means that the kernel of each map is equal to the image of the previous map
• The Mayer-Vietoris sequence is a powerful tool for computing the cohomology of a manifold by breaking it into simpler pieces
  • It allows for the calculation of cohomology groups using induction on the number of open sets in a cover
• The sequence can be generalized to covers with more than two open sets using the Čech cohomology

Applications in Topology

• De Rham cohomology and the Mayer-Vietoris sequence have numerous applications in topology and geometry
• They can be used to compute topological invariants such as Betti numbers and Euler characteristics
  • The Euler characteristic $\chi(M)$ of a compact manifold $M$ can be expressed as the alternating sum of Betti numbers: $\chi(M) = \sum_{k=0}^n (-1)^k b_k(M)$
• The cup product in de Rham cohomology provides a way to study the multiplicative structure of cohomology rings
  • For two closed forms $\omega \in H^k_{dR}(M)$ and $\eta \in H^l_{dR}(M)$, their cup product is the cohomology class of $\omega \wedge \eta \in H^{k+l}_{dR}(M)$
• De Rham cohomology can be used to define characteristic classes of vector bundles, such as the Chern classes and the Euler class
  • These classes provide obstructions to the existence of certain geometric structures on manifolds
• The Mayer-Vietoris sequence is a key tool in the proof of the Poincaré duality theorem and the de Rham theorem
• Cohomology groups can be used to study the existence and uniqueness of differential equations on manifolds

Computational Techniques and Examples

• Computing de Rham cohomology groups involves finding closed forms and identifying exact forms
• For simple manifolds, such as the n-sphere $S^n$ or the n-torus $T^n$, the cohomology groups can be calculated directly
  • For the n-sphere: $H^k_{dR}(S^n) = \begin{cases} \mathbb{R}, & k = 0 \text{ or } n \\ 0, & \text{otherwise} \end{cases}$
  • For the n-torus: $H^k_{dR}(T^n) \cong \mathbb{R}^{\binom{n}{k}}$
• The Mayer-Vietoris sequence can be used to compute cohomology groups inductively
  • Example: Consider the torus $T^2$ as the union of two annuli $U$ and $V$. The Mayer-Vietoris sequence yields:
    • $0 \to H^0_{dR}(T^2) \to H^0_{dR}(U) \oplus H^0_{dR}(V) \to H^0_{dR}(U \cap V) \to H^1_{dR}(T^2) \to \ldots$
    • By calculating the cohomology groups of $U$, $V$, and $U \cap V$, one can determine the cohomology of $T^2$
• Spectral sequences, such as the Leray-Serre spectral sequence, can be used to compute cohomology groups in more complex situations
• Computational algebraic topology software, such as CHomP and Perseus, can be used to calculate cohomology groups for simplicial complexes

Connections to Other Areas of Mathematics

• De Rham cohomology and the Mayer-Vietoris sequence have connections to various areas of mathematics
• In complex analysis, the Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds
  • It is defined using the Dolbeault operators $\partial$ and $\bar{\partial}$ instead of the exterior derivative
• In algebraic geometry, the algebraic de Rham cohomology is a cohomology theory for algebraic varieties
  • It is related to the algebraic version of the Mayer-Vietoris sequence, the Mayer-Vietoris spectral sequence
• In physics, de Rham cohomology appears in the study of gauge theories and quantum field theory
  • The de Rham complex is related to the BRST complex in the quantization of gauge theories
• In differential topology, the de Rham cohomology is related to the study of characteristic classes and obstruction theory
  • The Chern-Weil homomorphism relates the Chern classes of a vector bundle to the de Rham cohomology of the base manifold
• The Mayer-Vietoris sequence is a special case of the long exact sequence in homology and cohomology associated with a short exact sequence of chain complexes
• Sheaf cohomology, a generalization of de Rham cohomology, is a powerful tool in algebraic geometry and complex analysis
  • The Mayer-Vietoris sequence for sheaf cohomology is a key computational tool in these areas