10.3 Cellular homology

Cellular homology is a powerful tool for calculating homology groups of spaces built from cells. It simplifies computations by using the structure of CW complexes, making it more efficient than simplicial homology for many spaces.

This method connects to the broader study of homology groups by providing a practical way to compute these important topological invariants. It demonstrates how different approaches to homology can yield equivalent results, reinforcing the fundamental nature of these algebraic structures in topology.

Cellular Homology

Definition and Structure

Top images from around the web for Definition and Structure

• 

  Frontiers | Transcriptional Regulation of NK Cell Development by mTOR Complexes View original

  Is this image relevant?

• 

  Figures and data in Resolving homology in the face of shifting germ layer origins: Lessons from ... View original

  Is this image relevant?

• 

  Frontiers | Mechanism and Control of Meiotic DNA Double-Strand Break Formation in S. cerevisiae View original

  Is this image relevant?

• 

  Frontiers | Transcriptional Regulation of NK Cell Development by mTOR Complexes View original

  Is this image relevant?

• 

  Figures and data in Resolving homology in the face of shifting germ layer origins: Lessons from ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Structure

• 

  Frontiers | Transcriptional Regulation of NK Cell Development by mTOR Complexes View original

  Is this image relevant?

• 

  Figures and data in Resolving homology in the face of shifting germ layer origins: Lessons from ... View original

  Is this image relevant?

• 

  Frontiers | Mechanism and Control of Meiotic DNA Double-Strand Break Formation in S. cerevisiae View original

  Is this image relevant?

• 

  Frontiers | Transcriptional Regulation of NK Cell Development by mTOR Complexes View original

  Is this image relevant?

• 

  Figures and data in Resolving homology in the face of shifting germ layer origins: Lessons from ... View original

  Is this image relevant?

1 of 3

• Cellular homology computes homology groups of CW complexes built from cells of varying dimensions
• Constructs cellular chain complex using CW complex structure
  • Chain groups generated by cells
  • Boundary maps defined by attaching maps
• Generalizes simplicial homology for more efficient computations on broader class of spaces
• Cellular chain complex typically smaller and more manageable than simplicial chain complex for same space
• Preserves fundamental homology properties
  • Homotopy invariance
  • Long exact sequence of a pair

Relation to Simplicial Homology

• Established through CW approximation
  • Any space approximated by CW complex up to homotopy equivalence
• Cellular homology more efficient for spaces with natural cell decompositions (manifolds, algebraic varieties)
• Simplicial homology limited to simplicial complexes
• Both methods yield isomorphic homology groups for homotopy equivalent spaces

Cellular Homology Groups

Constructing the Chain Complex

• Cellular chain complex (C_(X), ∂_) uses skeletal filtration of CW complex X
• n-th chain group C_n(X) free abelian group generated by n-cells of X
• Boundary map ∂n: C_n(X) → C(n-1)(X) defined by degrees of attaching maps
  • Maps n-cells to (n-1)-cells
• Example: 2-sphere S^2
  • C_0(S^2) generated by single 0-cell
  • C_1(S^2) = 0 (no 1-cells)
  • C_2(S^2) generated by single 2-cell

Computing Homology Groups

• Determine kernels and images of boundary maps
• n-th cellular homology group H_n(X) = ker(∂n) / im(∂(n+1))
• Techniques for simplifying computations
  • Exact sequences
  • Cellular approximations
  • Cellular collapses
• Examples of cellular homology computations
  • Spheres: H_n(S^n) = Z for n=0 and n, 0 otherwise
  • Torus: H_0(T^2) = Z, H_1(T^2) = Z ⊕ Z, H_2(T^2) = Z
  • Real projective space RP^n: H_k(RP^n) = Z for k=0, Z_2 for odd k<n, 0 otherwise

Cellular vs Singular Homology

Isomorphism Proof

• Construct chain map between cellular and singular chain complexes
• Use skeletal filtration of CW complex and relative homology of pairs
• Apply long exact sequence of a pair and excision theorem for singular homology
• Show relative singular homology of (X^n, X^(n-1)) isomorphic to free abelian group generated by n-cells
• Use five lemma to prove induced map on homology is isomorphism
• Demonstrates consistency of different homology theories
• Validates use of cellular methods for topological invariants

Practical Implications

• Allows interchangeable use of cellular and singular homology in computations
• Cellular homology often more efficient for spaces with natural cell decompositions
• Singular homology applicable to wider class of topological spaces
• Both methods yield same topological information for CW complexes
• Choice between methods depends on specific problem and space structure

Applications of Cellular Homology

Topological Computations

• Compute homology groups of complex spaces by decomposing into simpler CW structures
• Calculate Euler characteristics and Betti numbers of CW complexes
  • Euler characteristic: alternating sum of ranks of homology groups
  • Betti numbers: ranks of homology groups
• Study topology of configuration spaces (robotics, motion planning)
• Distinguish spaces with similar appearances but different topological properties
  • Example: torus vs sphere (different H_1 groups)

Advanced Techniques

• Combine with spectral sequences for more complex problems
• Automate computations for efficient algorithms
  • Topological data analysis
  • Persistent homology
• Apply to algebraic varieties and quotient spaces
  • Example: compute homology of complex projective spaces
• Use in conjunction with cohomology theories for additional insights
  • Example: compute cup products in cellular cohomology