4.3 Čech cohomology

Čech cohomology offers a unique approach to studying topological spaces using open covers. It assigns cohomology groups to spaces, capturing global information from local data. This method is particularly useful for analyzing sheaves and vector bundles.

Čech cohomology connects to singular cohomology for certain spaces, providing an alternative computational tool. It's especially valuable in algebraic geometry and sheaf theory, where it helps classify principal bundles and compute sheaf cohomology.

Čech cohomology definition

• Čech cohomology is a cohomology theory for topological spaces that uses open covers to define cohomology groups
• It provides an alternative approach to singular cohomology and is particularly useful for studying sheaves and vector bundles

Presheaves on topological spaces

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• A presheaf $\mathcal{F}$ on a topological space $X$ assigns an abelian group $\mathcal{F}(U)$ to each open set $U \subseteq X$
• Presheaves capture local-to-global properties of topological spaces
  • Restriction maps $\rho_{V,U}: \mathcal{F}(U) \to \mathcal{F}(V)$ for open sets $V \subseteq U$ relate the data assigned to different open sets
• Examples of presheaves include the presheaf of continuous functions and the presheaf of differential forms

Čech cohomology groups

• Čech cohomology groups $\check{H}^n(X; \mathcal{F})$ are defined for a topological space $X$ and a presheaf $\mathcal{F}$
  • They measure the global cohomological information captured by the presheaf $\mathcal{F}$
• Constructed using open covers of $X$ and the nerve of the cover
• The $n$-th Čech cohomology group is the direct limit of the cohomology groups of the nerves of increasingly fine open covers

Čech cohomology vs singular cohomology

• Čech cohomology and singular cohomology are different approaches to defining cohomology groups for topological spaces
• For "nice" spaces (e.g., CW complexes), Čech cohomology and singular cohomology agree
  • This allows for the use of Čech cohomology in situations where it is more convenient or computationally tractable
• Čech cohomology is particularly well-suited for studying sheaves and their cohomology

Constructing Čech cohomology

• The construction of Čech cohomology involves several key steps, each building upon the previous one
• The goal is to define cohomology groups that capture global information about a topological space using local data from open covers

Open covers of topological spaces

• An open cover of a topological space $X$ is a collection $\mathcal{U} = \{U_i\}_{i \in I}$ of open sets such that $X = \bigcup_{i \in I} U_i$
• Open covers provide a way to break down a space into smaller, more manageable pieces
• Refining an open cover $\mathcal{U}$ means finding another open cover $\mathcal{V}$ such that for each $V \in \mathcal{V}$, there exists a $U \in \mathcal{U}$ with $V \subseteq U$

Nerve of an open cover

• The nerve $N(\mathcal{U})$ of an open cover $\mathcal{U} = \{U_i\}_{i \in I}$ is a simplicial complex
  • The vertices of $N(\mathcal{U})$ correspond to the open sets $U_i$
  • A collection of vertices $\{U_{i_0}, \ldots, U_{i_n}\}$ forms an $n$-simplex if and only if $U_{i_0} \cap \cdots \cap U_{i_n} \neq \emptyset$
• The nerve captures the combinatorial structure of the open cover

Čech complex of a cover

• The Čech complex $\check{C}^\bullet(\mathcal{U}; \mathcal{F})$ is a cochain complex associated to an open cover $\mathcal{U}$ and a presheaf $\mathcal{F}$
• It is defined using the nerve $N(\mathcal{U})$ and the presheaf data $\mathcal{F}(U_{i_0} \cap \cdots \cap U_{i_n})$
• The coboundary maps in the Čech complex are induced by the restriction maps of the presheaf and the alternating sum of face maps in the nerve

Čech cohomology as direct limit

• The Čech cohomology groups $\check{H}^n(X; \mathcal{F})$ are defined as the direct limit of the cohomology groups of the Čech complexes $\check{C}^\bullet(\mathcal{U}; \mathcal{F})$ over increasingly fine open covers $\mathcal{U}$
• This direct limit construction ensures that Čech cohomology is independent of the choice of open cover
  • Refining an open cover induces a map between the corresponding Čech complexes, which in turn induces a map on cohomology
• The direct limit captures the global cohomological information by considering all possible open covers of the space

Properties of Čech cohomology

• Čech cohomology satisfies several important properties that make it a useful tool in algebraic topology and geometry
• These properties often mirror those of singular cohomology, allowing for the use of Čech cohomology in situations where it is more convenient or computationally tractable

Homotopy invariance of Čech cohomology

• Čech cohomology is homotopy invariant: if $f, g: X \to Y$ are homotopic continuous maps, then they induce the same map on Čech cohomology
  • $f^* = g^*: \check{H}^n(Y; \mathcal{F}) \to \check{H}^n(X; f^*\mathcal{F})$
• This property allows for the study of topological spaces up to homotopy equivalence using Čech cohomology

Mayer-Vietoris sequence for Čech cohomology

• The Mayer-Vietoris sequence is a long exact sequence that relates the Čech cohomology of a space $X$ to the Čech cohomology of two open subsets $U, V \subseteq X$ that cover $X$
• It provides a way to compute the Čech cohomology of a space by breaking it down into simpler pieces
  • $\cdots \to \check{H}^n(U \cap V; \mathcal{F}) \to \check{H}^n(U; \mathcal{F}) \oplus \check{H}^n(V; \mathcal{F}) \to \check{H}^n(X; \mathcal{F}) \to \check{H}^{n+1}(U \cap V; \mathcal{F}) \to \cdots$
• The Mayer-Vietoris sequence is a powerful tool for computing Čech cohomology in practice

Čech cohomology of CW complexes

• For CW complexes, Čech cohomology agrees with singular cohomology
  • $\check{H}^n(X; \mathcal{F}) \cong H^n(X; \mathcal{F})$ for a CW complex $X$ and a presheaf $\mathcal{F}$
• This allows for the use of Čech cohomology to study CW complexes, which are a broad class of topological spaces
• Many results and techniques from singular cohomology can be applied to Čech cohomology in this setting

Čech cohomology with compact supports

• Čech cohomology with compact supports $\check{H}_c^n(X; \mathcal{F})$ is a variant of Čech cohomology that only considers compactly supported cochains
• It is useful for studying non-compact spaces and their cohomological properties
  • For example, Poincaré duality for non-compact manifolds can be formulated using Čech cohomology with compact supports
• Čech cohomology with compact supports is related to the usual Čech cohomology via a long exact sequence involving the cohomology of the "ends" of the space

Applications of Čech cohomology

• Čech cohomology has numerous applications in various areas of mathematics, including algebraic topology, sheaf theory, differential geometry, and algebraic geometry
• Its ability to capture global information using local data makes it a valuable tool for studying geometric and algebraic structures

Classifying principal bundles with Čech cohomology

• Principal $G$-bundles over a topological space $X$ can be classified by the Čech cohomology group $\check{H}^1(X; G)$, where $G$ is the sheaf of continuous functions into the group $G$
• This classification provides a way to understand the global structure of principal bundles using cohomological data
• The classification can be extended to higher-dimensional nonabelian cohomology, which captures more intricate geometric structures

Čech cohomology in sheaf theory

• Čech cohomology is closely related to sheaf cohomology, which is a central tool in sheaf theory
• For a sheaf $\mathcal{F}$ on a topological space $X$, the Čech cohomology groups $\check{H}^n(X; \mathcal{F})$ agree with the sheaf cohomology groups $H^n(X; \mathcal{F})$ under mild assumptions
• Čech cohomology provides a concrete way to compute sheaf cohomology, which is used to study the global properties of sheaves

Čech cohomology and de Rham cohomology

• For smooth manifolds, Čech cohomology with coefficients in the sheaf of smooth functions is closely related to de Rham cohomology
• The de Rham theorem states that the de Rham cohomology groups $H^n_{dR}(M)$ are isomorphic to the Čech cohomology groups $\check{H}^n(M; \mathbb{R})$ with coefficients in the constant sheaf $\mathbb{R}$
• This relationship allows for the use of Čech cohomology to study differential geometric properties of manifolds

Čech cohomology in algebraic geometry

• In algebraic geometry, Čech cohomology is used to define sheaf cohomology for algebraic varieties
• It plays a crucial role in the study of coherent sheaves and their cohomological properties
  • For example, the dimensions of the Čech cohomology groups of a coherent sheaf on a projective variety are related to its Hilbert polynomial
• Čech cohomology is also used in the construction of étale cohomology, which is a powerful tool for studying algebraic varieties over arbitrary fields

Computational techniques

• Computing Čech cohomology groups can be challenging, especially for complex topological spaces
• Several computational techniques have been developed to make these calculations more tractable and to relate Čech cohomology to other cohomology theories

Calculating Čech cohomology of spheres

• The Čech cohomology groups of the $n$-sphere $S^n$ can be computed using a standard open cover consisting of two open sets
• The nerve of this cover is a simplex, and the Čech complex reduces to a simple cochain complex
  • $\check{H}^k(S^n; \mathbb{Z}) \cong \begin{cases} \mathbb{Z}, & k = 0,n \\ 0, & \text{otherwise} \end{cases}$
• This calculation demonstrates the power of Čech cohomology in capturing global topological information using a simple open cover

Čech cohomology of projective spaces

• The Čech cohomology groups of real and complex projective spaces can be computed using a standard open cover and the Mayer-Vietoris sequence
• For the real projective space $\mathbb{RP}^n$, the Čech cohomology groups with $\mathbb{Z}/2\mathbb{Z}$ coefficients are
  • $\check{H}^k(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$ for $0 \leq k \leq n$
• For the complex projective space $\mathbb{CP}^n$, the Čech cohomology groups with $\mathbb{Z}$ coefficients are
  • $\check{H}^{2k}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}$ for $0 \leq k \leq n$, and $\check{H}^{odd}(\mathbb{CP}^n; \mathbb{Z}) \cong 0$

Leray's theorem for computing Čech cohomology

• Leray's theorem provides a way to compute the Čech cohomology of a space $X$ using a continuous map $f: X \to Y$ and the Čech cohomology of the space $Y$
• It states that there is a spectral sequence with $E_2^{p,q} = \check{H}^p(Y; \mathcal{H}^q(f; \mathcal{F}))$ that converges to $\check{H}^{p+q}(X; \mathcal{F})$, where $\mathcal{H}^q(f; \mathcal{F})$ is the $q$-th direct image sheaf of $\mathcal{F}$ under $f$
• Leray's theorem is particularly useful when the Čech cohomology of $Y$ and the direct image sheaves are easier to compute than the Čech cohomology of $X$ directly

Spectral sequences and Čech cohomology

• Spectral sequences are a powerful algebraic tool for computing cohomology groups, and they can be used in conjunction with Čech cohomology
• The Čech-to-derived spectral sequence relates the Čech cohomology of a space $X$ with coefficients in a sheaf $\mathcal{F}$ to the derived functor cohomology of $\mathcal{F}$
  • It has $E_2^{p,q} = \check{H}^p(X; \mathcal{H}^q(\mathcal{F}))$ and converges to the derived functor cohomology $H^{p+q}(X; \mathcal{F})$
• Other spectral sequences, such as the Leray spectral sequence and the Grothendieck spectral sequence, also involve Čech cohomology and provide tools for computing cohomology groups in various settings