Cellular homology is a powerful tool for understanding the topological structure of spaces. It uses the CW complex structure to compute homology groups, which measure "holes" in different dimensions. This approach is often more efficient than simplicial homology.

The computation of cellular homology involves defining chain complexes based on cell attachments. By analyzing these complexes and applying key axioms, we can determine the homology groups of a space, revealing its fundamental topological features.

Cellular Homology Groups

Cellular Chain Complex and Homology

• The cellular chain complex C(X) is defined using the cellular boundary maps $∂ₙ: Cₙ(X) → Cₙ₋₁(X)$, which connect the n-dimensional cells to the (n-1)-dimensional cells
• The n-th cellular homology group $Hₙ(X)$ is defined as the quotient $ker(∂ₙ) / im(∂ₙ₊₁)$, where $ker(∂ₙ)$ is the kernel of $∂ₙ$ and $im(∂ₙ₊₁)$ is the image of $∂ₙ₊₁$
• The homology groups $Hₙ(X)$ measure the "holes" in the complex X that are not filled by the (n+1)-cells, modulo the boundaries of the (n+1)-cells
  • For example, $H₁(X)$ measures 1-dimensional holes (loops) that are not boundaries of 2-cells

Cellular Chain Groups and Boundary Maps

• For a CW complex X, the cellular chain groups $Cₙ(X)$ are free abelian groups with basis in one-to-one correspondence with the n-cells of X
  • For instance, if X has three 2-cells, then $C₂(X)$ is isomorphic to $ℤ³$
• The cellular boundary map $∂ₙ$ is determined by the degrees of the attaching maps of the n-cells, which can be computed using the cellular boundary formula
  • The degree of an attaching map $φ: S^{n-1} → X^{(n-1)}$ counts the number of times the image of $φ$ wraps around each (n-1)-cell in X^{(n-1)}
• The boundary maps $∂ₙ$ form a chain complex, meaning that $∂ₙ₋₁ ∘ ∂ₙ = 0$ for all n, which is crucial for the definition of homology groups

Cellular Homology Axioms

Dimension and Degree Axioms

• The dimension axiom states that if X is a single point, then $Hₙ(X) = 0$ for all $n ≠ 0$, and $H₀(X) ≅ ℤ$
  • This axiom sets the base case for homology computations
• The degree axiom relates the degree of a map between spheres to the induced homomorphism on homology groups
  • For a map $f: S^n → S^n$ of degree d, the induced homomorphism $f_: Hₙ(S^n) → Hₙ(S^n)$ is multiplication by d

Excision and Additivity Axioms

• The excision axiom allows the computation of homology groups of a space by decomposing it into simpler pieces
  • If $(X, A)$ is a CW pair and U is a subcomplex of A such that the closure of U is contained in the interior of A, then the inclusion $(X - U, A - U) ↪ (X, A)$ induces isomorphisms $Hₙ(X - U, A - U) ≅ Hₙ(X, A)$ for all n
• The additivity axiom states that the homology of a disjoint union of spaces is the direct sum of the homology of each space
  • For a collection of spaces $\{Xᵢ\}_{i∈I}$, there is an isomorphism $Hₙ(⨆ᵢ Xᵢ) ≅ ⨁ᵢ Hₙ(Xᵢ)$ for all n, where $⨆$ denotes disjoint union and $⨁$ denotes direct sum

Homotopy Invariance and Exactness Axioms

• The homotopy invariance axiom implies that homotopy equivalent spaces have isomorphic homology groups
  • If $f, g: X → Y$ are homotopic maps, then the induced homomorphisms $f_*, g_*: Hₙ(X) → Hₙ(Y)$ are equal for all n
• The exactness axiom relates the homology of a space, a subspace, and the quotient space through a long exact sequence
  • For a CW pair $(X, A)$, there is a long exact sequence of homology groups: $... → Hₙ(A) → Hₙ(X) → Hₙ(X/A) → Hₙ₋₁(A) → ...$ where the maps are induced by the inclusions $A ↪ X$ and $X ↪ X/A$

Cellular vs Simplicial Homology

Efficiency of Cellular Homology

• Cellular homology is often more efficient than simplicial homology because CW complexes typically have fewer cells than simplices in a triangulation
  • For example, a torus can be constructed with just one 0-cell, two 1-cells, and one 2-cell, while a simplicial complex requires many more simplices
• The cellular chain complex is usually smaller than the simplicial chain complex, leading to simpler computations of homology groups
  • The number of generators in each dimension is determined by the number of cells, which is often fewer than the number of simplices

Flexibility of Cellular Homology

• Cellular homology allows for more flexibility in the choice of cell structure, which can be adapted to the specific space being studied
  • Different CW structures on the same space can lead to different cellular chain complexes, but the homology groups will be isomorphic
• The cellular boundary maps are determined by the attaching maps of the cells, which can be easier to compute than the simplicial boundary maps
  • Attaching maps are defined on spheres, which have a simpler structure than the boundaries of simplices

Simplifying Homology Calculations

Exploiting CW Structure

• Identify the n-cells of the CW complex and their attaching maps to determine the cellular chain groups $Cₙ(X)$ and the cellular boundary maps $∂ₙ$
  • The chain groups are generated by the cells, and the boundary maps are determined by the degrees of the attaching maps
• Use the dimension of the cells and the connectivity of the attaching maps to infer properties of the kernel and image of the boundary maps
  • For example, if there are no (n+1)-cells, then $im(∂ₙ₊₁) = 0$, simplifying the computation of $Hₙ(X)$

Decomposition and Excision

• Exploit the presence of contractible subcomplexes or subspaces with known homology to simplify the computation using the excision axiom
  • If A is a contractible subcomplex of X, then $Hₙ(X, A) ≅ Hₙ(X/A)$ for all n, reducing the computation to the quotient space
• Decompose the CW complex into simpler pieces, such as wedge sums or product spaces, and use the additivity and Künneth formula to compute the homology of the whole space
  • For a wedge sum $X ∨ Y$, there is an isomorphism $Hₙ(X ∨ Y) ≅ Hₙ(X) ⊕ Hₙ(Y)$ for all $n > 0$, and $H₀(X ∨ Y) ≅ H₀(X) ≅ H₀(Y) ≅ ℤ$

Strategic Application of Axioms

• Apply the cellular homology axioms strategically to minimize the number of direct computations needed, leveraging the relationships between different spaces and their homology groups
  • Use the dimension axiom for base cases, the degree axiom for maps between spheres, and the homotopy invariance axiom for homotopy equivalent spaces
• Combine the excision and additivity axioms to break down a complex space into simpler components, compute their homology groups separately, and then reassemble the results using the Mayer-Vietoris sequence or other algebraic tools
  • The Mayer-Vietoris sequence relates the homology of a space X to the homology of two subspaces A and B such that $X = A ∪ B$, providing a powerful tool for computing homology by decomposition