3.4 Comparison of simplicial and cellular homology

Simplicial and cellular homology are two ways to measure a space's "holes" and topological features. While they use different building blocks (simplices vs. cells), they give the same results. This equivalence lets us choose the most efficient method for each situation.

Cellular homology often wins in efficiency, using fewer components to describe a space. However, finding the right cell structure can be tricky. The choice between methods depends on the space and available tools.

Simplicial vs Cellular Homology

Definition and Topological Invariants

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• Simplicial homology is defined using simplicial complexes, while cellular homology is defined using CW complexes
• Both simplicial and cellular homology capture topological invariants of a space, specifically the "holes" in different dimensions (Betti numbers, Euler characteristic)
• The simplicial chain complex is generated by oriented simplices (vertices, edges, triangles, tetrahedra), while the cellular chain complex is generated by oriented cells (0-cells, 1-cells, 2-cells)
• The boundary operators in simplicial and cellular homology are defined differently but serve an analogous purpose of encoding the incidence relations between simplices or cells of consecutive dimensions

Isomorphism of Homology Groups

• Simplicial and cellular homology groups are isomorphic for a given space, meaning they capture the same topological information
• The isomorphism arises from the fact that a CW complex can be constructed from a simplicial complex by taking its geometric realization and giving it a CW structure
• Conversely, a CW complex can be subdivided into a simplicial complex without changing its topology
• This equivalence allows for the choice of the most convenient or efficient homology computation method based on the available representation of the space

Computational Efficiency: Simplicial vs Cellular

Advantages of Cellular Homology

• Cellular homology is often more computationally efficient than simplicial homology, especially for spaces with a small CW complex structure
• The number of cells in a CW complex is typically much smaller than the number of simplices in a triangulation, leading to smaller chain complexes in cellular homology (fewer generators and boundary relations)
• Computing boundary operators in cellular homology involves fewer calculations compared to simplicial homology due to the reduced number of cells
• The sparsity of the cellular boundary matrices can be exploited for efficient computation, whereas simplicial boundary matrices tend to be denser

Challenges and Trade-offs

• In some cases, finding an optimal CW complex structure for a space may be challenging, potentially offsetting the computational advantage of cellular homology
• The process of converting a simplicial complex to a CW complex or vice versa may introduce additional computational overhead
• The choice between simplicial and cellular homology depends on the specific characteristics of the space, the available representation, and the computational resources at hand
• In practice, a combination of both methods may be used, leveraging their strengths in different stages of the homology computation pipeline

Isomorphism of Homology Groups

Chain Map Construction

• The proof relies on constructing a chain map between the simplicial and cellular chain complexes that induces an isomorphism on homology
• Given a simplicial complex K, one can construct a CW complex X by taking the geometric realization of K and giving it a CW complex structure
• A chain map f: C(K) → C(X) is defined by sending each simplex to the sum of the cells it intersects, with appropriate orientations (e.g., a triangle is sent to the sum of the 2-cells it overlaps)
• The map f commutes with the boundary operators, i.e., f ∘ ∂_simplicial = ∂_cellular ∘ f, making it a chain map

Induced Homomorphism and Isomorphism

• By the algebraic theorem that chain maps induce homomorphisms on homology, f induces a homomorphism f_*: H_n(K) → H_n(X) for each dimension n
• Using the geometric properties of the map f and the construction of X from K, one can show that f_* is an isomorphism (bijective homomorphism)
• The injectivity of f_* follows from the fact that if a cycle in K is mapped to a boundary in X, then it must have been a boundary in K
• The surjectivity of f_* is proven by showing that every cycle in X can be approximated by a cycle in K, using the cellular approximation theorem
• Consequently, the simplicial homology groups of K are isomorphic to the cellular homology groups of X

Homology Comparisons in Different Settings

Comparison Theorem

• The comparison theorem states that if two chain complexes are connected by a chain map inducing an isomorphism on homology, then their homology groups are isomorphic
• This theorem allows for the comparison of homology computations performed using different methods or in different settings (simplicial, cellular, singular)
• When computing homology, one can choose the most convenient or efficient method based on the available information about the space

Flexibility in Homology Computations

• If a space is given as a simplicial complex, one can compute its simplicial homology directly or convert it to a CW complex and compute cellular homology
• Conversely, if a space is given as a CW complex, one can compute its cellular homology directly or subdivide it into a simplicial complex and compute simplicial homology
• The comparison theorem ensures that the resulting homology groups will be isomorphic, regardless of the chosen method
• This flexibility allows for the selection of the most suitable approach based on the specific characteristics of the space and the available computational tools (software packages, algorithms)

Applications and Examples

• In topological data analysis, simplicial complexes are often used to represent high-dimensional datasets, and simplicial homology is computed to extract topological features (persistent homology)
• In algebraic topology, CW complexes are frequently used to construct spaces with desired properties, and cellular homology is employed to study their topological invariants (homotopy groups, cohomology rings)
• The comparison theorem enables the transfer of results between different homology theories and facilitates the study of topological spaces from multiple perspectives (Morse theory, sheaf theory, spectral sequences)