7.1 Sheaves on manifolds

Sheaves on manifolds provide a powerful framework for studying geometric and topological properties. They generalize the concept of functions, attaching local data to each point while ensuring compatibility across overlapping regions.

This approach allows for a flexible analysis of manifolds, connecting local and global properties. Sheaves enable the study of various structures, from smooth functions to differential forms, and play a crucial role in cohomology theories and applications across mathematics.

Definition of sheaves on manifolds

• Sheaves on manifolds generalize the concept of functions on a manifold, allowing for a more flexible and powerful framework to study geometric and topological properties
• Sheaves on manifolds consist of a collection of local data (stalks) attached to each point of the manifold, along with restriction maps that ensure the local data is compatible on overlapping regions

Presheaves as precursors to sheaves

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• Presheaves are a preliminary construction that assign data (abelian groups, rings, or modules) to each open set of a manifold
• Presheaves have restriction maps between the data on different open sets, but these maps do not necessarily ensure that the local data can be glued together consistently
• Presheaves serve as a starting point for constructing sheaves by imposing additional conditions on the restriction maps

Sheaf conditions for manifolds

• For a presheaf to be a sheaf on a manifold, it must satisfy two additional conditions: the gluing axiom and the locality axiom
• The gluing axiom ensures that if local sections agree on overlaps, they can be glued together to form a global section over the union of the open sets
• The locality axiom states that if two sections are equal when restricted to every open set in a cover, then they are equal on the union of the open sets

Sheafification process for manifolds

• Sheafification is the process of turning a presheaf into a sheaf by enforcing the gluing and locality axioms
• The sheafification process involves adding new sections to the presheaf to ensure the sheaf conditions are satisfied
• The resulting sheaf is the "closest" sheaf to the original presheaf, in the sense that there is a unique morphism from the presheaf to the sheafified sheaf

Constructing sheaves on manifolds

• Sheaves on manifolds can be constructed in various ways, depending on the type of data one wants to associate with each point or open set of the manifold
• The choice of construction often depends on the specific application or problem at hand

Constant sheaves on manifolds

• Constant sheaves assign the same abelian group, ring, or module to each open set of the manifold
• The restriction maps in a constant sheaf are identity morphisms, ensuring that the local data is the same on overlapping regions
• Constant sheaves are the simplest examples of sheaves on manifolds and are often used in homological algebra and cohomology theories

Locally constant sheaves

• Locally constant sheaves assign the same data to each connected component of an open set in the manifold
• The restriction maps in a locally constant sheaf are isomorphisms between the data on connected components
• Locally constant sheaves are useful in studying covering spaces and monodromy in complex analysis and algebraic geometry

Sheaves of smooth functions

• The sheaf of smooth functions on a manifold assigns to each open set the ring of smooth real-valued functions defined on that open set
• The restriction maps are given by the restriction of functions to smaller open sets
• The sheaf of smooth functions is a fundamental example in differential geometry and plays a crucial role in the study of differential equations and vector bundles

Sheaves of differential forms

• The sheaf of differential k-forms on a manifold assigns to each open set the vector space of smooth differential k-forms defined on that open set
• The restriction maps are given by the restriction of differential forms to smaller open sets
• Sheaves of differential forms are essential in the study of de Rham cohomology, integration theory, and the topology of manifolds

Sheaf cohomology on manifolds

• Sheaf cohomology is a powerful tool for studying the global properties of sheaves on manifolds and extracting topological and geometric information
• Sheaf cohomology groups measure the obstruction to solving certain local-to-global problems, such as finding global sections or extending local sections

Čech cohomology for sheaves

• Čech cohomology is a cohomology theory for sheaves that relies on open covers of the manifold
• Given an open cover, Čech cochains are defined using the sections of the sheaf on finite intersections of open sets in the cover
• The Čech differential is defined using the restriction maps of the sheaf, and the resulting cohomology groups are independent of the choice of open cover (for fine enough covers)

De Rham cohomology vs sheaf cohomology

• De Rham cohomology is a cohomology theory for smooth manifolds that uses differential forms and the exterior derivative
• For the constant sheaf $\mathbb{R}$ and the sheaf of smooth functions, the sheaf cohomology groups are isomorphic to the de Rham cohomology groups
• This isomorphism is a consequence of the Poincaré lemma and the fine resolution of the constant sheaf by the sheaf of differential forms

Poincaré lemma for sheaves

• The Poincaré lemma states that on a contractible open subset of a manifold, every closed differential form is exact
• In the context of sheaves, the Poincaré lemma implies that the sheaf of differential forms is a fine resolution of the constant sheaf $\mathbb{R}$
• This resolution allows for the computation of sheaf cohomology using differential forms and the exterior derivative

Mayer-Vietoris sequence for sheaves

• The Mayer-Vietoris sequence is a long exact sequence that relates the sheaf cohomology groups of a manifold to the sheaf cohomology groups of two open subsets and their intersection
• The Mayer-Vietoris sequence is a powerful computational tool for determining the sheaf cohomology groups of a manifold by breaking it down into simpler pieces
• The sequence is derived from the short exact sequence of sheaves that arises from the inclusions of the open subsets into the manifold

Operations on sheaves over manifolds

• Several operations can be performed on sheaves over manifolds, allowing for the construction of new sheaves from existing ones
• These operations are functorial in nature and provide a rich structure to the category of sheaves on manifolds

Pullback sheaves on manifolds

• Given a continuous map $f: M \to N$ between manifolds and a sheaf $\mathcal{F}$ on $N$, the pullback sheaf $f^*\mathcal{F}$ on $M$ is defined by assigning to each open set $U \subset M$ the sections of $\mathcal{F}$ over the open set $f(U) \subset N$
• The restriction maps of the pullback sheaf are induced by the restriction maps of the original sheaf $\mathcal{F}$
• Pullback sheaves are contravariant functorial, meaning they reverse the direction of maps between manifolds

Pushforward sheaves on manifolds

• Given a continuous map $f: M \to N$ between manifolds and a sheaf $\mathcal{F}$ on $M$, the pushforward sheaf $f_*\mathcal{F}$ on $N$ is defined by assigning to each open set $V \subset N$ the sections of $\mathcal{F}$ over the open set $f^{-1}(V) \subset M$
• The restriction maps of the pushforward sheaf are induced by the restriction maps of the original sheaf $\mathcal{F}$
• Pushforward sheaves are covariant functorial, meaning they preserve the direction of maps between manifolds

Tensor products of sheaves

• Given two sheaves $\mathcal{F}$ and $\mathcal{G}$ of modules over a manifold $M$, the tensor product sheaf $\mathcal{F} \otimes \mathcal{G}$ is defined by assigning to each open set $U \subset M$ the tensor product of the modules $\mathcal{F}(U)$ and $\mathcal{G}(U)$
• The restriction maps of the tensor product sheaf are induced by the restriction maps of the original sheaves $\mathcal{F}$ and $\mathcal{G}$
• Tensor products of sheaves allow for the construction of new sheaves that combine the data from two given sheaves

Hom sheaves on manifolds

• Given two sheaves $\mathcal{F}$ and $\mathcal{G}$ of modules over a manifold $M$, the Hom sheaf $\mathcal{H}om(\mathcal{F}, \mathcal{G})$ is defined by assigning to each open set $U \subset M$ the module of sheaf morphisms from the restriction of $\mathcal{F}$ to $U$ to the restriction of $\mathcal{G}$ to $U$
• The restriction maps of the Hom sheaf are induced by the restriction maps of the original sheaves $\mathcal{F}$ and $\mathcal{G}$
• Hom sheaves provide a way to study the morphisms between sheaves and are useful in the construction of exact sequences and resolutions

Applications of sheaves to manifolds

• Sheaves on manifolds have numerous applications across various branches of mathematics, including geometry, topology, and mathematical physics
• The use of sheaves allows for a unifying framework to study local-to-global problems and provides powerful tools for computation and analysis

Sheaves in complex geometry

• In complex geometry, sheaves of holomorphic functions and sheaves of meromorphic functions play a crucial role in the study of complex manifolds and analytic spaces
• Sheaf cohomology in complex geometry is closely related to the theory of Hodge structures and the study of the Dolbeault complex
• Sheaves are used to formulate and solve problems in complex analysis, such as the Cousin problems and the Levi problem

Sheaves in algebraic geometry

• In algebraic geometry, sheaves of regular functions and sheaves of modules over the structure sheaf are fundamental objects in the study of algebraic varieties and schemes
• Sheaf cohomology in algebraic geometry is used to define important invariants, such as the cohomology groups of coherent sheaves and the Picard group
• Sheaves provide a framework for studying the local and global properties of algebraic varieties, such as the existence of global sections and the extension of local sections

Sheaves in differential topology

• In differential topology, sheaves of smooth functions and sheaves of differential forms are used to study the properties of smooth manifolds
• Sheaf cohomology in differential topology is closely related to de Rham cohomology and is used to compute topological invariants, such as the Betti numbers and the Euler characteristic
• Sheaves are also used in the study of vector bundles, principal bundles, and characteristic classes in differential topology

Sheaves in mathematical physics

• In mathematical physics, sheaves are used to formulate and study various physical theories, such as gauge theory, quantum field theory, and string theory
• Sheaves of observables and sheaves of fields are used to describe the local and global properties of physical systems
• Sheaf cohomology is used to study the topological and geometric aspects of physical theories, such as the classification of instantons and the anomalies in quantum field theory