🔢Algebraic Topology Unit 11 – Sheaves and Sheaf Cohomology

Key Concepts and Definitions

• Presheaf $\mathcal{F}$ assigns to each open set $U$ of a topological space $X$ an abelian group $\mathcal{F}(U)$
• Sheaf extends the concept of a presheaf by requiring local information to uniquely determine global information
• Restriction maps $\rho_{V,U}: \mathcal{F}(U) \to \mathcal{F}(V)$ for each inclusion $V \subseteq U$ of open sets
  • Restriction maps satisfy the composition property $\rho_{W,V} \circ \rho_{V,U} = \rho_{W,U}$ for $W \subseteq V \subseteq U$
• Stalks $\mathcal{F}_x$ at a point $x \in X$ consist of germs, equivalence classes of sections over neighborhoods of $x$
• Exact sequence is a sequence of morphisms between objects (abelian groups or sheaves) where the image of each morphism equals the kernel of the next
• Short exact sequence $0 \to A \to B \to C \to 0$ captures essential information about the objects and their relationships
• Cohomology measures the global properties of a topological space by associating abelian groups to the space

Topological Foundations

• Topological space $(X, \tau)$ consists of a set $X$ and a collection $\tau$ of subsets of $X$ called open sets satisfying certain axioms
  • $\emptyset$ and $X$ are open
  • Arbitrary union of open sets is open
  • Finite intersection of open sets is open
• Basis $\mathcal{B}$ for a topology is a collection of open sets such that every open set can be expressed as a union of elements in $\mathcal{B}$
• Continuous function $f: X \to Y$ between topological spaces preserves open sets, i.e., the preimage of an open set is open
• Hausdorff space is a topological space where any two distinct points have disjoint neighborhoods
• Compact space is a topological space where every open cover has a finite subcover
  • Heine-Borel theorem characterizes compact subsets of Euclidean space as closed and bounded sets
• Paracompact space is a Hausdorff space where every open cover has a locally finite open refinement

Introduction to Sheaves

• Sheaf $\mathcal{F}$ on a topological space $X$ assigns an abelian group $\mathcal{F}(U)$ to each open set $U \subseteq X$
  • Elements of $\mathcal{F}(U)$ are called sections over $U$
• Sheaf axioms ensure local information uniquely determines global information
  • (Identity) For every open set $U$, the restriction map $\rho_{U,U}$ is the identity on $\mathcal{F}(U)$
  • (Locality) If $\{U_i\}$ is an open cover of $U$ and $s, t \in \mathcal{F}(U)$ such that $\rho_{U_i,U}(s) = \rho_{U_i,U}(t)$ for all $i$, then $s = t$
  • (Gluing) If $\{U_i\}$ is an open cover of $U$ and $s_i \in \mathcal{F}(U_i)$ with $\rho_{U_i \cap U_j, U_i}(s_i) = \rho_{U_i \cap U_j, U_j}(s_j)$, then there exists a unique $s \in \mathcal{F}(U)$ such that $\rho_{U_i,U}(s) = s_i$ for all $i$
• Constant sheaf $\underline{A}$ assigns the abelian group $A$ to every open set with identity restriction maps
• Locally constant sheaf assigns a fixed abelian group to each connected component of the space

Sheaf Operations and Properties

• Morphism of sheaves $\varphi: \mathcal{F} \to \mathcal{G}$ is a collection of group homomorphisms $\varphi(U): \mathcal{F}(U) \to \mathcal{G}(U)$ for each open set $U$ that commute with restriction maps
  • Sheaf morphisms form a category with sheaves as objects and morphisms as arrows
• Kernel sheaf $\ker(\varphi)$ assigns to each open set $U$ the subgroup $\ker(\varphi(U)) \subseteq \mathcal{F}(U)$
• Image sheaf $\operatorname{im}(\varphi)$ assigns to each open set $U$ the subgroup $\operatorname{im}(\varphi(U)) \subseteq \mathcal{G}(U)$
• Cokernel sheaf $\operatorname{coker}(\varphi)$ assigns to each open set $U$ the quotient group $\mathcal{G}(U) / \operatorname{im}(\varphi(U))$
• Exact sequence of sheaves $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ induces a long exact sequence of cohomology groups
• Direct sum of sheaves $\mathcal{F} \oplus \mathcal{G}$ assigns to each open set $U$ the direct sum of abelian groups $\mathcal{F}(U) \oplus \mathcal{G}(U)$
• Tensor product of sheaves $\mathcal{F} \otimes \mathcal{G}$ assigns to each open set $U$ the tensor product of abelian groups $\mathcal{F}(U) \otimes \mathcal{G}(U)$

Cohomology Basics

• Cochain complex is a sequence of abelian groups $C^i$ and homomorphisms (differentials) $d^i: C^i \to C^{i+1}$ such that $d^{i+1} \circ d^i = 0$
  • Cocycles $Z^i = \ker(d^i)$ and coboundaries $B^i = \operatorname{im}(d^{i-1})$ satisfy $B^i \subseteq Z^i$
• Cohomology groups $H^i(C^\bullet) = Z^i / B^i$ measure the "gap" between cocycles and coboundaries
• Cochain map $f: C^\bullet \to D^\bullet$ between cochain complexes is a collection of homomorphisms $f^i: C^i \to D^i$ that commute with the differentials
  • Cochain maps induce homomorphisms on cohomology $H^i(f): H^i(C^\bullet) \to H^i(D^\bullet)$
• Short exact sequence of cochain complexes $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ induces a long exact sequence in cohomology
  • Connecting homomorphism $\delta: H^i(C^\bullet) \to H^{i+1}(A^\bullet)$ measures the obstruction to lifting cocycles
• Homotopy equivalence between cochain complexes induces isomorphisms on cohomology
• Cohomology with coefficients in an abelian group $G$ is defined by applying the functor $\operatorname{Hom}(-, G)$ to a chain complex

Sheaf Cohomology Theory

• Sheaf cohomology $H^i(X, \mathcal{F})$ associates abelian groups to a sheaf $\mathcal{F}$ on a topological space $X$
  • Measures the global obstructions to solving local problems related to $\mathcal{F}$
• Čech cohomology is defined using Čech cochains, functions assigning sections to intersections of open sets in a cover
  • Čech differential $\delta: \check{C}^i(\mathcal{U}, \mathcal{F}) \to \check{C}^{i+1}(\mathcal{U}, \mathcal{F})$ is defined by alternating sums of restrictions
  • Čech cohomology groups $\check{H}^i(\mathcal{U}, \mathcal{F}) = \ker(\delta^i) / \operatorname{im}(\delta^{i-1})$
  • Refined covers lead to isomorphic Čech cohomology groups, defining the Čech cohomology $\check{H}^i(X, \mathcal{F})$
• Derived functor cohomology is defined using injective resolutions and right derived functors
  • Injective sheaf $\mathcal{I}$ satisfies the lifting property for sheaf morphisms
  • Injective resolution $0 \to \mathcal{F} \to \mathcal{I}^0 \to \mathcal{I}^1 \to \cdots$ allows the computation of right derived functors $R^i\Gamma(X, \mathcal{F})$
• Čech and derived functor cohomology agree for paracompact Hausdorff spaces and lead to the same abstract cohomology theory
• Cohomology of a constant sheaf $\underline{A}$ recovers the singular cohomology $H^i(X, A)$ with coefficients in $A$

Applications in Algebraic Topology

• Sheaf cohomology provides a unified framework for various cohomology theories in algebraic topology
• de Rham theorem relates de Rham cohomology (defined using differential forms) to singular cohomology
  • de Rham complex $\Omega^\bullet(X)$ of smooth differential forms on a manifold $X$ computes the de Rham cohomology $H^i_{dR}(X)$
  • de Rham sheaf $\mathcal{A}^i$ assigns to each open set $U$ the vector space of smooth $i$-forms on $U$
  • Poincaré lemma states that the de Rham complex is locally exact, implying that $\mathcal{A}^\bullet$ is a resolution of the constant sheaf $\underline{\mathbb{R}}$
  • de Rham theorem: $H^i_{dR}(X) \cong H^i(X, \mathbb{R})$ for smooth manifolds $X$
• Dolbeault theorem relates Dolbeault cohomology (defined using complex differential forms) to sheaf cohomology on complex manifolds
  • Dolbeault complex $A^{p,q}(X)$ of $(p,q)$-forms on a complex manifold $X$ computes the Dolbeault cohomology $H^{p,q}_{\bar{\partial}}(X)$
  • Dolbeault sheaf $\mathcal{O}^{p,q}$ assigns to each open set $U$ the vector space of $(p,q)$-forms on $U$
  • Dolbeault theorem: $H^{p,q}_{\bar{\partial}}(X) \cong H^q(X, \Omega^p)$, where $\Omega^p$ is the sheaf of holomorphic $p$-forms
• Hodge theorem decomposes the cohomology of a compact Kähler manifold into a direct sum of Dolbeault cohomology groups
• Sheaf cohomology also plays a role in the study of vector bundles, characteristic classes, and intersection theory

Problem-Solving Techniques

• Compute sheaf cohomology using Čech cochains and covers
  • Choose a suitable open cover of the space and calculate the Čech complex
  • Determine the kernels and images of the Čech differentials to find the cohomology groups
• Use injective resolutions to compute derived functor cohomology
  • Find an injective resolution of the sheaf and apply the global section functor
  • Calculate the cohomology of the resulting complex
• Exploit the long exact sequence in cohomology induced by a short exact sequence of sheaves
  • Break down a complicated sheaf into simpler pieces using short exact sequences
  • Use the induced long exact sequence to relate the cohomology groups of the sheaves
• Apply the Mayer-Vietoris sequence to compute cohomology using a cover of the space
  • Decompose the space into simpler pieces using a suitable cover
  • Use the Mayer-Vietoris sequence to relate the cohomology of the space to the cohomology of the pieces and their intersections
• Utilize spectral sequences to compute cohomology in stages
  • Set up a spectral sequence (Leray, Serre, etc.) that converges to the desired cohomology groups
  • Compute the terms of the spectral sequence and analyze the differentials to determine the limit terms
• Identify vanishing theorems and cohomological dimension to simplify computations
  • Use theorems like Cartan's theorem B, Kodaira vanishing, or Grauert's theorem to deduce the vanishing of certain cohomology groups
  • Determine the cohomological dimension of the space to limit the range of non-zero cohomology groups