10.3 Čech cohomology

Čech cohomology uses open covers to measure global topological properties of spaces through local data. It assigns abelian groups to open sets and defines cohomology groups from a cochain complex. This approach bridges local and global information, making it useful for various mathematical applications.

Čech cohomology relates to other theories like singular and de Rham cohomology. It's often more computable, especially for spaces with nice covers. The Čech-de Rham complex connects combinatorial and differential form approaches, showcasing Čech cohomology's versatility in topology and geometry.

Definitions of Čech cohomology

• Čech cohomology is a cohomology theory for topological spaces that uses open covers to construct a cochain complex
• It provides a way to measure the global topological properties of a space by studying local data from open covers
• The Čech cohomology groups are defined as the cohomology groups of the Čech cochain complex

Presheaves vs sheaves

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• A presheaf is a functor from the category of open sets of a topological space to the category of abelian groups
  • It assigns an abelian group to each open set and a restriction homomorphism to each inclusion of open sets
• A sheaf is a presheaf that satisfies the gluing axiom and the locality axiom
  • The gluing axiom ensures that sections can be patched together consistently on overlaps
  • The locality axiom guarantees that sections are determined by their values on a cover
• Sheaves capture the idea of local-to-global passage and are central to the construction of Čech cohomology

Čech nerve of a cover

• Given an open cover $\mathcal{U} = \{U_i\}_{i \in I}$ of a topological space $X$, the Čech nerve $\check{N}(\mathcal{U})$ is a simplicial set
• The $n$-simplices of $\check{N}(\mathcal{U})$ are $(n+1)$-fold intersections of open sets from the cover
  • $\check{N}(\mathcal{U})_n = \coprod_{i_0, \ldots, i_n \in I} U_{i_0} \cap \ldots \cap U_{i_n}$
• The face and degeneracy maps are induced by the inclusions and identity maps of the open sets
• The Čech nerve encodes the combinatorial data of the open cover and is used to define the Čech cochain complex

Simplicial cohomology of Čech nerve

• The Čech cochain complex $\check{C}^\bullet(\mathcal{U}; \mathcal{F})$ is defined as the simplicial cohomology of the Čech nerve with coefficients in a sheaf $\mathcal{F}$
  • $\check{C}^n(\mathcal{U}; \mathcal{F}) = \prod_{i_0, \ldots, i_n \in I} \mathcal{F}(U_{i_0} \cap \ldots \cap U_{i_n})$
• The coboundary maps are induced by the alternating sum of the restriction maps of the sheaf $\mathcal{F}$
• The Čech cohomology groups $\check{H}^\bullet(X; \mathcal{F})$ are defined as the cohomology groups of the Čech cochain complex
  • $\check{H}^n(X; \mathcal{F}) = H^n(\check{C}^\bullet(\mathcal{U}; \mathcal{F}))$
• The Čech cohomology groups are independent of the choice of cover, provided the cover is sufficiently fine

Relation to other cohomology theories

• Čech cohomology is one of several cohomology theories for topological spaces, each with its own advantages and applications
• It is important to understand the relationships between these theories and how they compare in terms of computability and functoriality

Singular vs Čech cohomology

• Singular cohomology is defined using singular cochains, which are dual to singular chains
  • Singular chains are formal linear combinations of continuous maps from standard simplices to the space
• Čech cohomology, on the other hand, is defined using open covers and the associated Čech cochain complex
• For a wide class of spaces (e.g., manifolds, CW complexes), singular and Čech cohomology are isomorphic
  • This isomorphism is established via a double complex that relates singular and Čech cochains
• However, Čech cohomology has the advantage of being more easily computable in certain situations

Čech-de Rham complex

• For smooth manifolds, there is a natural comparison between Čech cohomology and de Rham cohomology
• The Čech-de Rham complex is a double complex that combines the Čech cochain complex and the de Rham complex
  • The horizontal differential is the Čech coboundary, and the vertical differential is the de Rham differential
• The cohomology of the total complex of the Čech-de Rham complex is isomorphic to both Čech and de Rham cohomology
  • This isomorphism is a manifestation of the de Rham theorem, which relates sheaf cohomology and de Rham cohomology
• The Čech-de Rham complex provides a bridge between the combinatorial approach of Čech cohomology and the differential forms approach of de Rham cohomology

Leray's theorem for Čech cohomology

• Leray's theorem is a powerful tool for computing the Čech cohomology of a space using a cover by acyclic open sets
• An open set $U$ is acyclic with respect to a sheaf $\mathcal{F}$ if $\check{H}^i(U; \mathcal{F}) = 0$ for all $i > 0$
• Leray's theorem states that if $\mathcal{U}$ is a cover of $X$ by acyclic open sets, then the Čech cohomology of $X$ is isomorphic to the cohomology of the Čech cochain complex
  • $\check{H}^\bullet(X; \mathcal{F}) \cong H^\bullet(\check{C}^\bullet(\mathcal{U}; \mathcal{F}))$
• This theorem simplifies the computation of Čech cohomology by reducing it to a combinatorial problem on the nerve of the cover

Computation of Čech cohomology

• Computing Čech cohomology groups can be challenging, but there are several tools and techniques that can be employed depending on the properties of the space and the cover
• These computational methods often involve spectral sequences, exact sequences, or explicit descriptions of the Čech cochain complex

Čech cohomology of spheres

• The Čech cohomology groups of the $n$-sphere $S^n$ can be computed using a cover by two open sets, such as the complement of the north and south poles
• For the sheaf of constant functions $\underline{\mathbb{R}}$, the Čech cohomology groups are:
  • $\check{H}^0(S^n; \underline{\mathbb{R}}) \cong \mathbb{R}$
  • $\check{H}^n(S^n; \underline{\mathbb{R}}) \cong \mathbb{R}$
  • $\check{H}^i(S^n; \underline{\mathbb{R}}) \cong 0$ for $i \neq 0, n$
• This computation shows that the Čech cohomology of spheres agrees with the singular cohomology and captures the topological invariants of spheres

Mayer-Vietoris sequence for Čech cohomology

• The Mayer-Vietoris sequence is a long exact sequence that relates the Čech cohomology of a space to the Čech cohomology of two open subsets and their intersection
• For a space $X$ and open subsets $U, V \subset X$ such that $X = U \cup V$, the Mayer-Vietoris sequence is:
  • $\ldots \to \check{H}^i(X; \mathcal{F}) \to \check{H}^i(U; \mathcal{F}) \oplus \check{H}^i(V; \mathcal{F}) \to \check{H}^i(U \cap V; \mathcal{F}) \to \check{H}^{i+1}(X; \mathcal{F}) \to \ldots$
• This sequence is a powerful tool for computing Čech cohomology by breaking down the space into simpler pieces
• It can be used iteratively for covers with more than two open sets, leading to a spectral sequence that converges to the Čech cohomology of the space

Acyclicity of contractible covers

• A key property that simplifies the computation of Čech cohomology is the acyclicity of contractible covers
• If a space $X$ is covered by contractible open sets $\{U_i\}_{i \in I}$, then the Čech cohomology of $X$ is isomorphic to the cohomology of the nerve of the cover
  • $\check{H}^\bullet(X; \mathcal{F}) \cong H^\bullet(N(\{U_i\}_{i \in I}); \mathcal{F})$
• This result follows from the fact that contractible sets are acyclic and Leray's theorem
• Many spaces of interest, such as manifolds and CW complexes, admit covers by contractible open sets, making this a useful tool for computing their Čech cohomology

Applications of Čech cohomology

• Čech cohomology has numerous applications in topology, geometry, and other areas of mathematics
• Its ability to capture global topological information from local data makes it a versatile tool for studying a wide range of problems

Classification of line bundles

• Čech cohomology can be used to classify line bundles over a topological space $X$
• The isomorphism classes of line bundles over $X$ are in bijection with the elements of the first Čech cohomology group with coefficients in the sheaf of non-vanishing continuous functions $\mathcal{C}^\times$
  • $\{\text{Line bundles over } X\} / \cong \leftrightarrow \check{H}^1(X; \mathcal{C}^\times)$
• This classification result is a consequence of the fact that line bundles are locally trivial and can be constructed by gluing together local trivializations using transition functions
• The Čech cohomology group $\check{H}^1(X; \mathcal{C}^\times)$ encodes the obstruction to globally trivializing a line bundle

Brouwer fixed point theorem

• The Brouwer fixed point theorem states that any continuous function from the $n$-dimensional ball $B^n$ to itself has a fixed point
• Čech cohomology provides a proof of this theorem using the concept of the degree of a map
• The degree of a continuous map $f: S^n \to S^n$ is an integer that measures the number of times the image of $f$ wraps around the sphere, counted with orientation
• The key steps in the proof are:
  • Extend the map $f$ to a map $\tilde{f}: B^{n+1} \to B^{n+1}$ by radial projection
  • Show that the degree of $\tilde{f}|_{S^n}$ is equal to the Kronecker pairing of the pullback of the generator of $\check{H}^n(S^n; \mathbb{Z})$ with the fundamental class of $S^n$
  • Use the fact that the degree of the identity map is 1 to conclude that $\tilde{f}$ has a fixed point, and thus $f$ has a fixed point
• This proof demonstrates the power of Čech cohomology in capturing the essential topological features of maps between spaces

Topological invariance of Euler characteristic

• The Euler characteristic is a topological invariant that can be defined for a wide class of spaces, including polyhedra and manifolds
• For a triangulated space $X$, the Euler characteristic $\chi(X)$ is defined as the alternating sum of the number of simplices in each dimension
  • \chi(X) = \sum_{i=0}^{\dim X} (-1)^i \#\{\text{i-simplices in } X\}
• Čech cohomology provides a proof of the topological invariance of the Euler characteristic
• The key steps in the proof are:
  • Express the Euler characteristic as the alternating sum of the dimensions of the Čech cohomology groups with coefficients in a field $\mathbb{F}$
    • $\chi(X) = \sum_{i=0}^{\infty} (-1)^i \dim_{\mathbb{F}} \check{H}^i(X; \mathbb{F})$
  • Use the isomorphism between Čech and singular cohomology to show that this expression is independent of the triangulation
• This proof illustrates the relationship between Čech cohomology and other topological invariants, and how it can be used to establish their properties

Refinements of Čech cohomology

• While Čech cohomology is a powerful tool, there are several refinements and variations that can be used to address specific problems or to improve its computational efficiency
• These refinements often involve modifying the type of covers used or the way in which the Čech cochain complex is constructed

Closed covers vs open covers

• The standard definition of Čech cohomology uses open covers of the space, but it can also be defined using closed covers
• For a closed cover $\mathcal{C} = \{C_i\}_{i \in I}$ of a space $X$, the Čech cohomology groups $\check{H}^\bullet(X, \mathcal{C}; \mathcal{F})$ are defined as the cohomology of the Čech cochain complex associated to the cover
• The relationship between Čech cohomology defined using open and closed covers depends on the properties of the space and the sheaf
  • For paracompact Hausdorff spaces and soft sheaves, the two definitions agree
  • In general, the closed cover definition may yield different results or require additional assumptions
• The choice between open and closed covers often depends on the specific application and the available computational tools

Alternating Čech cohomology

• Alternating Čech cohomology is a variation of Čech cohomology that uses alternating cochains instead of the standard Čech cochains
• For an open cover $\mathcal{U} = \{U_i\}_{i \in I}$ and a sheaf $\mathcal{F}$, the alternating Čech cochain complex is defined as
  • $\check{C}_{\text{alt}}^n(\mathcal{U}; \mathcal{F}) = \{c \in \prod_{i_0, \ldots, i_n \in I} \mathcal{F}(U_{i_0} \cap \ldots \cap U_{i_n}) \mid c \text{ is alternating}\}$
• A cochain $c$ is alternating if it changes sign whenever two indices are swapped
  • $c_{i_0, \ldots, i_k, \ldots, i_l, \ldots, i_n} = -c_{i_0, \ldots, i_l, \ldots, i_k, \ldots, i_n}$
• The alternating Čech cohomology groups $\check{H}_{\text{alt}}^\bullet(X; \mathcal{F})$ are the cohomology groups of the alternating Čech cochain complex
• Alternating Čech cohomology has better functorial properties than standard Čech cohomology and is more closely related to singular cohomology

Relation to Alexander-Spanier cohomology

• Alexander-Spanier cohomology is another cohomology theory for topological spaces that is defined using open covers and local sections
• For a space $X$ and a sheaf $\mathcal{F}$, the Alexander-Spanier cochain complex is defined as
  • $C_{\text{AS}}^n(X; \mathcal{F}) = \prod_{(x_0, \ldots, x_n) \in X^{n+1}} \mathcal{F}_{x_0, \ldots, x_n}$
  • Here, $\mathcal{F}_{x_0, \ldots, x_n}$ denotes the stalk of $\mathcal{F}$ at the point $(x_0, \ldots, x_n)$
• The Alexander-Spanier cohomology groups $H_{\text{AS}}^\bullet(X; \mathcal{F})$ are the cohomology groups of this cochain complex
• For paracompact Hausdorff spaces and soft sheaves, Alexander-Spanier cohomology agrees with Čech cohomology
  • $\check{H}^\bullet(X; \mathcal{F}) \cong H_{\text{AS}}^\bullet(X; \mathcal{F})$
• Alexander