10.2 Cellular homology

Cellular homology is a powerful tool for calculating homology groups of spaces built from simple pieces. It simplifies computations by focusing on the structure of cell attachments, making it easier to understand complex topological spaces.

This approach connects to the broader applications of algebraic topology by providing a concrete method to analyze spaces. It bridges the gap between abstract concepts and practical calculations, showcasing how algebraic tools can reveal geometric properties.

CW Complexes and Cell Attachment

Construction and Properties of CW Complexes

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• CW complexes are built by attaching cells of increasing dimension to a discrete set of points (0-cells)
• Cells are attached via continuous maps from the boundary of an n-dimensional ball to the (n-1)-skeleton
• The resulting space has the weak topology with respect to the cell decomposition
• CW complexes are Hausdorff, second-countable, and locally contractible
• The dimension of a CW complex is the highest dimension of its cells

Cell Attachment and Homotopy Type

• Cell attachment is the process of building a CW complex by attaching cells of increasing dimension
• The homotopy type of a CW complex is determined by the homotopy classes of the attaching maps
• Attaching an n-cell to an (n-1)-connected space results in an n-connected space
• Two CW complexes are homotopy equivalent if they have the same homotopy type
• Cell attachment can be used to construct spaces with desired homotopy properties (spheres, tori)

Cellular Chain Complexes and Boundary Operators

Construction of Cellular Chain Complexes

• The cellular chain complex $C_*(X)$ of a CW complex $X$ is a sequence of abelian groups $C_n(X)$ connected by boundary operators $\partial_n: C_n(X) \rightarrow C_{n-1}(X)$
• $C_n(X)$ is the free abelian group generated by the n-cells of $X$
• The boundary operator $\partial_n$ encodes the incidence relations between n-cells and (n-1)-cells
• The composition of two consecutive boundary operators is always zero: $\partial_{n-1} \circ \partial_n = 0$
• The cellular chain complex is a chain complex, satisfying the property $\partial \circ \partial = 0$

Properties of Cellular Boundary Operators

• The cellular boundary operator $\partial_n$ measures the degree of the attaching map of each n-cell
• The degree of an attaching map is the number of times the boundary of the n-cell wraps around the (n-1)-cell, with orientation taken into account
• The boundary of a sum of cells is the sum of their boundaries: $\partial(a+b) = \partial(a) + \partial(b)$
• The boundary operator satisfies the Leibniz rule: $\partial(a \cdot b) = \partial(a) \cdot b + (-1)^{\deg(a)} a \cdot \partial(b)$
• The cellular boundary operators completely determine the homology of the CW complex

Cellular Homology and Cohomology

Cellular Homology Groups and Their Properties

• The n-th cellular homology group $H_n(X)$ of a CW complex $X$ is the quotient group $\ker(\partial_n) / \operatorname{im}(\partial_{n+1})$
• Elements of $H_n(X)$ are equivalence classes of n-cycles (elements of $\ker(\partial_n)$) modulo n-boundaries (elements of $\operatorname{im}(\partial_{n+1})$)
• Cellular homology groups are topological invariants, independent of the cell decomposition
• Cellular homology satisfies the Eilenberg-Steenrod axioms for a homology theory
• The dimension of $H_n(X)$ as a vector space is the n-th Betti number $\beta_n(X)$, which counts the number of "n-dimensional holes" in $X$

Cellular Cohomology and the Universal Coefficient Theorem

• The n-th cellular cohomology group $H^n(X; G)$ of a CW complex $X$ with coefficients in an abelian group $G$ is the group of homomorphisms from $H_n(X)$ to $G$
• Cellular cohomology groups are related to cellular homology groups by the Universal Coefficient Theorem: $H^n(X; G) \cong \operatorname{Hom}(H_n(X), G) \oplus \operatorname{Ext}(H_{n-1}(X), G)$
• The Universal Coefficient Theorem allows for the computation of cohomology groups from homology groups
• Cellular cohomology satisfies the Eilenberg-Steenrod axioms for a cohomology theory
• The cup product gives cellular cohomology the structure of a graded ring, with the product $H^p(X; R) \otimes H^q(X; R) \rightarrow H^{p+q}(X; R)$ induced by the diagonal map $X \rightarrow X \times X$