6.3 Čech homology and cohomology

Čech homology and cohomology are powerful tools for studying topological spaces. They use open covers and nerves to capture local information, making them ideal for analyzing complex structures. These theories complement singular homology, offering unique insights into spatial properties.

Čech theories satisfy important axioms like homotopy invariance and excision. They're great for computing topological invariants like Betti numbers and Euler characteristics. Čech cohomology is particularly useful for studying continuous functions and local properties of spaces.

Čech homology and cohomology

Definition using open covers and nerves

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• An open cover of a topological space X is a collection of open sets {Uᵢ} whose union is the entire space X
• The nerve of an open cover U = {Uᵢ} is a simplicial complex N(U) where each k-simplex corresponds to a non-empty intersection of k+1 open sets from the cover
• The Čech complex Č(U) is the chain complex associated with the nerve N(U)
  • Its homology groups are the Čech homology groups Ȟₙ(X; G) with coefficients in an abelian group G
• Čech cohomology groups Ȟⁿ(X; G) are defined as the cohomology groups of the cochain complex Č*(U; G) associated with the nerve N(U) and the coefficient group G
• The Čech (co)homology groups are independent of the choice of open cover, provided the open cover is sufficiently fine
  • Each point in X should be contained in an open set of the cover

Properties of Čech (co)homology

• Homotopy invariance: If f, g: X → Y are homotopic maps, then the induced homomorphisms f*, g*: Ȟₙ(X; G) → Ȟₙ(Y; G) are equal
• Exactness: For any pair (X, A) of topological spaces, there is a long exact sequence relating the Čech homology groups of X, A, and the quotient space X/A
• Excision: If U ⊂ A ⊂ X and the closure of U is contained in the interior of A, then the inclusion (X - U, A - U) ↪ (X, A) induces isomorphisms Ȟₙ(X - U, A - U; G) ≅ Ȟₙ(X, A; G) for all n
• Dimension axiom: For a one-point space P, Ȟ₀(P; G) ≅ G and Ȟₙ(P; G) = 0 for n ≠ 0

Čech vs Singular homology

Relationship between Čech and singular (co)homology

• For "nice" spaces (CW complexes, manifolds), Čech homology and cohomology are isomorphic to singular homology and cohomology, respectively
• The isomorphism is established using a direct limit construction over all open covers of the space, with refinement of covers as the directed set
• For general topological spaces, Čech (co)homology may differ from singular (co)homology
  • Čech theory captures more local information about the space

Similarities and differences

• Both Čech and singular (co)homology satisfy the Eilenberg-Steenrod axioms for (co)homology theories
  • These axioms include homotopy invariance, exactness, excision, and the dimension axiom
• Čech (co)homology is defined using open covers and nerves, while singular (co)homology is defined using singular simplices and their boundaries
• Čech (co)homology is better suited for studying local properties of spaces, while singular (co)homology is more straightforward for computations on simplicial complexes and CW complexes

Axioms for Čech homology

Eilenberg-Steenrod axioms

• Homotopy invariance: Homotopic maps induce equal homomorphisms on Čech homology groups
• Exactness: Long exact sequence relating the Čech homology groups of a pair (X, A) and the quotient space X/A
• Excision: Inclusion of a "nice" pair (X - U, A - U) into (X, A) induces isomorphisms on Čech homology groups
• Dimension axiom: Čech homology of a one-point space P is isomorphic to the coefficient group G in dimension 0 and trivial in other dimensions

Proofs of the axioms

• The proofs involve using the properties of open covers, nerves, and the associated chain complexes
• Homotopy invariance follows from the fact that homotopic maps induce chain homotopic maps on the Čech complex
• Exactness is derived from the short exact sequence of chain complexes associated with a pair (X, A)
• Excision is proved by showing that the open covers of X and A can be chosen to satisfy the excision condition
• The dimension axiom is a direct consequence of the definition of Čech homology for a one-point space

Applications of Čech homology

Computing topological invariants

• The kth Betti number βₖ(X) is the rank of the kth Čech homology group Ȟₖ(X; Z)
  • It counts the number of k-dimensional "holes" in the space X (genus for surfaces, number of connected components for k=0)
• The Euler characteristic χ(X) is the alternating sum of the Betti numbers: χ(X) = ∑ₖ (-1)ᵏ βₖ(X)
  • It is a topological invariant that can be used to distinguish non-homeomorphic spaces (orientable vs non-orientable surfaces)

Studying continuous functions

• Čech cohomology can be used to study the existence of certain continuous functions on a space
  • Nowhere vanishing functions are related to the zeroth cohomology group (non-vanishing vector fields on manifolds)
  • Functions with prescribed zeros and poles are related to higher cohomology groups (meromorphic functions on complex manifolds)
• The cup product in Čech cohomology provides a way to study the multiplicative structure of cohomology and its relation to the topology of the space
  • It can detect the non-triviality of certain cohomology classes (powers of the fundamental class of a manifold)

Local properties of spaces

• Čech (co)homology can be applied to study the local properties of spaces
  • Local homology groups at a point capture the (co)homological behavior of the space near that point (local structure of singularities)
• Čech cohomology can be used to define sheaf cohomology, which is a powerful tool for studying the global properties of sheaves on a space (algebraic geometry, complex analysis)