🍃Sheaf Theory Unit 7 – Sheaves on Manifolds & Vector Bundles

Key Concepts and Definitions

• Manifolds are topological spaces that locally resemble Euclidean space and provide a foundation for studying geometric objects
• Sheaves are mathematical structures that assign data (e.g., functions, vectors) to open sets of a topological space in a way that is compatible with restrictions
• Presheaves are similar to sheaves but may not satisfy the gluing axiom, which ensures local data can be uniquely patched together
• Vector bundles are fiber bundles whose fibers are vector spaces, allowing for the study of local linear structures on manifolds
  • Tangent bundles and cotangent bundles are examples of vector bundles associated with a manifold
• Sections of a sheaf or vector bundle assign a value (e.g., function, vector) to each point in the base space, generalizing the concept of functions on a manifold
• Morphisms between sheaves or vector bundles are structure-preserving maps that commute with restrictions and provide a way to compare and relate different sheaves or bundles

Manifolds: A Quick Refresher

• Manifolds are topological spaces that are locally Euclidean, meaning each point has a neighborhood homeomorphic to an open subset of Euclidean space
• Charts are homeomorphisms from open subsets of the manifold to open subsets of Euclidean space, providing local coordinate systems
• Transition functions between overlapping charts ensure compatibility and allow for the definition of smooth structures on manifolds
• Smooth manifolds are equipped with a smooth structure, enabling the study of calculus and differential geometry
  • Smooth functions on a manifold are those whose local representations in charts are smooth (infinitely differentiable)
• Tangent spaces are vector spaces attached to each point of a manifold, capturing the notion of infinitesimal directions and velocities
• Tangent bundles are vector bundles whose fibers are the tangent spaces at each point, providing a global structure for studying vector fields and differential equations on manifolds

Introduction to Sheaves

• Sheaves are mathematical objects that assign algebraic structures (e.g., rings, modules) to open sets of a topological space, allowing for the study of local-to-global properties
• Presheaves are assignments of algebraic structures to open sets that satisfy the restriction axiom: given open sets $U \subset V$, there is a restriction map $\rho_{V,U}: \mathcal{F}(V) \to \mathcal{F}(U)$ that is compatible with composition
• Sheaves are presheaves that additionally satisfy the gluing axiom: if $\{U_i\}$ is an open cover of $U$ and $s_i \in \mathcal{F}(U_i)$ are sections that agree on overlaps (i.e., $\rho_{U_i,U_i \cap U_j}(s_i) = \rho_{U_j,U_i \cap U_j}(s_j)$), then there exists a unique $s \in \mathcal{F}(U)$ such that $\rho_{U,U_i}(s) = s_i$ for all $i$
  • The gluing axiom ensures that local data can be uniquely patched together to form global data
• Sheaf morphisms are natural transformations between sheaves that commute with restriction maps, providing a way to compare and relate different sheaves
• Sheaf cohomology is a powerful tool for studying global properties of sheaves and the underlying topological space, generalizing concepts like de Rham cohomology

Vector Bundles: The Basics

• Vector bundles are fiber bundles whose fibers are vector spaces, allowing for the study of local linear structures on manifolds
• A vector bundle consists of a total space $E$, a base space $B$, and a projection map $\pi: E \to B$ such that each fiber $\pi^{-1}(b)$ is a vector space and the local trivialization condition is satisfied
  • Local trivialization means that each point in the base space has a neighborhood $U$ such that $\pi^{-1}(U)$ is isomorphic to $U \times \mathbb{R}^n$ (or $U \times V$ for a fixed vector space $V$)
• Transition functions between local trivializations are smooth maps that take values in the general linear group $GL(n,\mathbb{R})$ (or $GL(V)$), ensuring compatibility between fibers
• Sections of a vector bundle assign a vector in the fiber to each point in the base space, generalizing the concept of vector fields on manifolds
• Operations on vector bundles include Whitney sum (direct sum of fibers), tensor product, and dual bundle, which allow for the construction of new bundles from existing ones
• Tangent bundles and cotangent bundles are important examples of vector bundles associated with a smooth manifold, capturing infinitesimal directions and differential forms, respectively

Sheaves on Manifolds

• Sheaves on manifolds provide a framework for studying local-to-global properties of geometric objects and structures
• The sheaf of smooth functions on a manifold $M$, denoted by $C^\infty_M$, assigns to each open set $U \subset M$ the ring of smooth functions $C^\infty(U)$, with restriction maps given by function restriction
  • The sheaf of smooth functions is a fundamental example of a sheaf on a manifold and plays a crucial role in differential geometry
• The sheaf of sections of a vector bundle $E \to M$, denoted by $\Gamma(E)$, assigns to each open set $U \subset M$ the $C^\infty(U)$-module of smooth sections of $E$ over $U$, with restriction maps given by section restriction
  • The sheaf of sections captures the local behavior of a vector bundle and allows for the study of global properties through sheaf cohomology
• Sheaf cohomology on manifolds, particularly for the sheaf of sections of a vector bundle, provides invariants that encode topological and geometric information about the manifold and the bundle
• The de Rham complex is an important example of a sheaf complex on a manifold, whose cohomology groups are isomorphic to the singular cohomology groups with real coefficients

Sections and Global Sections

• Sections of a sheaf $\mathcal{F}$ on a topological space $X$ are elements of $\mathcal{F}(U)$ for some open set $U \subset X$, representing local data or assignments
• Global sections of a sheaf $\mathcal{F}$ are elements of $\mathcal{F}(X)$, i.e., sections defined on the entire space $X$
  • The set of global sections of a sheaf $\mathcal{F}$ is denoted by $\Gamma(X,\mathcal{F})$ or $H^0(X,\mathcal{F})$
• For the sheaf of smooth functions $C^\infty_M$ on a manifold $M$, global sections are precisely the smooth functions on the entire manifold
• For the sheaf of sections $\Gamma(E)$ of a vector bundle $E \to M$, global sections are the smooth sections of $E$ defined on the entire manifold $M$
• The existence of global sections for a given sheaf depends on the topological and geometric properties of the underlying space and the sheaf itself
  • Sheaf cohomology, particularly the first cohomology group $H^1(X,\mathcal{F})$, measures the obstruction to the existence of global sections
• Extending local sections to global sections is a fundamental problem in sheaf theory and often requires additional conditions or the use of partition of unity arguments on manifolds

Operations on Sheaves and Vector Bundles

• Sheaf morphisms are natural transformations between sheaves that commute with restriction maps, allowing for the comparison and relation of different sheaves
  • Sheaf morphisms can be used to construct exact sequences of sheaves, which provide valuable information about the sheaves involved
• Pullback (or inverse image) of a sheaf $\mathcal{F}$ on $Y$ along a continuous map $f: X \to Y$ is a sheaf $f^*\mathcal{F}$ on $X$ that captures the behavior of $\mathcal{F}$ under precomposition with $f$
  • Pullback is a contravariant operation that allows for the transfer of sheaves between spaces
• Pushforward (or direct image) of a sheaf $\mathcal{F}$ on $X$ along a continuous map $f: X \to Y$ is a sheaf $f_*\mathcal{F}$ on $Y$ that captures the behavior of $\mathcal{F}$ under postcomposition with $f$
  • Pushforward is a covariant operation that allows for the transfer of sheaves between spaces
• Tensor product of vector bundles $E \to M$ and $F \to M$ is a vector bundle $E \otimes F \to M$ whose fibers are the tensor products of the fibers of $E$ and $F$
  • The sheaf of sections of the tensor product bundle is isomorphic to the tensor product of the sheaves of sections of the individual bundles
• Whitney sum (or direct sum) of vector bundles $E \to M$ and $F \to M$ is a vector bundle $E \oplus F \to M$ whose fibers are the direct sums of the fibers of $E$ and $F$
  • The sheaf of sections of the Whitney sum bundle is isomorphic to the direct sum of the sheaves of sections of the individual bundles
• Dual bundle of a vector bundle $E \to M$ is a vector bundle $E^* \to M$ whose fibers are the dual vector spaces of the fibers of $E$
  • The sheaf of sections of the dual bundle is isomorphic to the sheaf of $C^\infty_M$-linear maps from the sheaf of sections of $E$ to the sheaf of smooth functions $C^\infty_M$

Applications and Examples

• Sheaves and vector bundles have numerous applications in geometry, topology, and mathematical physics
• The tangent bundle $TM$ of a smooth manifold $M$ is a vector bundle whose fibers are the tangent spaces at each point, allowing for the study of vector fields and differential equations on manifolds
  • The sheaf of sections of the tangent bundle is the sheaf of vector fields on the manifold
• The cotangent bundle $T^*M$ of a smooth manifold $M$ is a vector bundle whose fibers are the cotangent spaces (dual of tangent spaces) at each point, allowing for the study of differential forms and integration on manifolds
  • The sheaf of sections of the cotangent bundle is the sheaf of differential 1-forms on the manifold
• The de Rham complex is a sheaf complex on a manifold $M$ that encodes the differential structure and leads to the de Rham cohomology groups $H^k_{dR}(M)$, which are isomorphic to the singular cohomology groups with real coefficients
  • The de Rham theorem establishes an important link between differential geometry and algebraic topology
• Characteristic classes, such as Chern classes and Pontryagin classes, are cohomology classes associated with vector bundles that provide topological invariants of the bundle and the underlying manifold
  • These classes are often defined using the sheaf cohomology of certain sheaves associated with the vector bundle, such as the sheaf of differential forms with coefficients in the endomorphism bundle
• Gauge theory, which plays a crucial role in mathematical physics, studies connections and curvature on principal bundles and associated vector bundles
  • The sheaf of sections of an associated vector bundle is a fundamental object in gauge theory, and its cohomology groups often have physical interpretations
• Sheaf cohomology on complex manifolds, particularly for the sheaf of holomorphic functions and the sheaf of holomorphic sections of a holomorphic vector bundle, is a powerful tool in complex geometry and leads to important invariants such as Hodge numbers and Dolbeault cohomology groups