🔢Algebraic Topology Unit 3 – CW Complexes and Cellular Homology

Key Concepts and Definitions

• CW complexes are topological spaces constructed by attaching cells of increasing dimension
• Cells are homeomorphic images of open balls $B^n = \{x \in \mathbb{R}^n : |x| < 1\}$
• The boundary of an $n$-cell is attached to the $(n-1)$-skeleton by a continuous map called the attaching map
• The $n$-skeleton $X^n$ is the union of all cells of dimension $\leq n$
• A CW complex is a Hausdorff space $X$ with a filtration $X^0 \subset X^1 \subset \cdots \subset X$ such that $X = \bigcup_{n=0}^\infty X^n$
  • Each $X^n$ is obtained from $X^{n-1}$ by attaching $n$-cells via attaching maps
• Cellular homology is a method for computing the homology groups of a CW complex using its cellular structure
• The cellular chain complex $C_*(X)$ is a sequence of abelian groups $C_n(X)$ connected by boundary operators $\partial_n: C_n(X) \to C_{n-1}(X)$

CW Complex Construction

• Begin with a discrete set of points $X^0$ called the 0-skeleton
• Attach 1-cells (intervals) to $X^0$ via continuous maps $\varphi_\alpha: \partial B^1 \to X^0$ to obtain the 1-skeleton $X^1$
• Attach 2-cells (disks) to $X^1$ via continuous maps $\varphi_\beta: \partial B^2 \to X^1$ to obtain the 2-skeleton $X^2$
• Continue this process, attaching $n$-cells to the $(n-1)$-skeleton $X^{n-1}$ via continuous maps $\varphi_\gamma: \partial B^n \to X^{n-1}$ to obtain the $n$-skeleton $X^n$
• The resulting space $X = \bigcup_{n=0}^\infty X^n$ is a CW complex
• The topology on $X$ is the weak topology, where a subset $A \subset X$ is closed if and only if $A \cap X^n$ is closed in $X^n$ for all $n$
• Examples of CW complexes include simplicial complexes, polytopes, and smooth manifolds with a cell decomposition

Cellular Homology Basics

• For each $n$, the $n$-th cellular chain group $C_n(X)$ is the free abelian group generated by the $n$-cells of $X$
• The boundary operator $\partial_n: C_n(X) \to C_{n-1}(X)$ is defined by $\partial_n(e_\alpha^n) = \sum_\beta d_{\alpha\beta} e_\beta^{n-1}$, where $d_{\alpha\beta}$ is the degree of the attaching map $\varphi_\alpha: \partial B^n \to X^{n-1}$ restricted to the boundary of the $n$-cell $e_\alpha^n$
• The cellular chain complex $C_*(X)$ is the sequence of abelian groups and boundary operators: $\cdots \xrightarrow{\partial_{n+1}} C_n(X) \xrightarrow{\partial_n} C_{n-1}(X) \xrightarrow{\partial_{n-1}} \cdots$
• The $n$-th cellular homology group $H_n(X)$ is defined as the quotient $\ker(\partial_n) / \operatorname{im}(\partial_{n+1})$
• Cellular homology satisfies the Eilenberg-Steenrod axioms for a homology theory

Boundary Operators and Chain Complexes

• The boundary operator $\partial_n: C_n(X) \to C_{n-1}(X)$ encodes the incidence relations between cells of adjacent dimensions
• For an $n$-cell $e_\alpha^n$, the coefficient $d_{\alpha\beta}$ in $\partial_n(e_\alpha^n) = \sum_\beta d_{\alpha\beta} e_\beta^{n-1}$ represents the number of times the boundary of $e_\alpha^n$ wraps around the $(n-1)$-cell $e_\beta^{n-1}$, with orientation taken into account
• The composition of two consecutive boundary operators is always zero: $\partial_{n-1} \circ \partial_n = 0$ for all $n$
  • This property ensures that $\operatorname{im}(\partial_{n+1}) \subset \ker(\partial_n)$, allowing the definition of homology groups
• The cellular chain complex $C_*(X)$ is a algebraic object that encodes the cellular structure of $X$ and allows for the computation of homology groups
• Chain complexes can be studied independently of their topological origins, leading to the development of homological algebra

Calculating Homology Groups

• To calculate the $n$-th homology group $H_n(X)$, first determine the cellular chain groups $C_n(X)$ and the boundary operators $\partial_n$
• Compute the kernel $\ker(\partial_n)$, which consists of $n$-chains $c \in C_n(X)$ such that $\partial_n(c) = 0$ (called $n$-cycles)
• Compute the image $\operatorname{im}(\partial_{n+1})$, which consists of $n$-chains $c \in C_n(X)$ such that $c = \partial_{n+1}(d)$ for some $(n+1)$-chain $d$ (called $n$-boundaries)
• The $n$-th homology group $H_n(X)$ is the quotient group $\ker(\partial_n) / \operatorname{im}(\partial_{n+1})$
  • Elements of $H_n(X)$ are equivalence classes of $n$-cycles, where two $n$-cycles are equivalent if their difference is an $n$-boundary
• The rank of $H_n(X)$ is called the $n$-th Betti number $\beta_n(X)$ and provides information about the number of "holes" of dimension $n$ in the space $X$
• Homology groups are topological invariants and can be used to distinguish between non-homeomorphic spaces

Applications in Topology

• Cellular homology is a powerful tool for studying the topological properties of spaces that admit a CW complex structure
• Homology groups can be used to detect the presence of "holes" or "voids" in a space
  • For example, a non-trivial first homology group $H_1(X)$ indicates the presence of 1-dimensional holes (loops) in $X$
• The Euler characteristic of a CW complex $X$ can be computed using the alternating sum of Betti numbers: $\chi(X) = \sum_{n=0}^\infty (-1)^n \beta_n(X)$
• Homology groups are functorial, meaning that continuous maps between spaces induce homomorphisms between their homology groups
  • This allows for the study of maps and their effects on topological invariants
• The homology groups of a product space $X \times Y$ can be computed using the Künneth formula, which relates them to the homology groups of $X$ and $Y$
• Cellular homology can be used to study the topology of manifolds, as every compact manifold admits a CW complex structure

Examples and Problem-Solving Techniques

• When computing cellular homology, it is often helpful to start by determining the cellular chain groups $C_n(X)$ and writing out the boundary operators $\partial_n$ as matrices with respect to a chosen basis of cells
• For simple spaces like the torus or the Klein bottle, the cellular chain complex can be computed directly from their standard CW complex structures
• For more complex spaces, it may be necessary to first construct a CW complex structure by attaching cells incrementally
  • This process can be guided by the space's topological properties or by decomposing it into simpler pieces (e.g., using the CW complex structure of a simplicial complex)
• When working with chain complexes, techniques from linear algebra such as row reduction and the rank-nullity theorem can be used to compute kernels and images of boundary operators
• The snake lemma and the five lemma are powerful tools for studying the relationship between homology groups in short exact sequences of chain complexes
• Spectral sequences, such as the Serre spectral sequence and the Atiyah-Hirzebruch spectral sequence, can be used to compute homology groups of more complicated spaces like fiber bundles and CW complexes with a filtration

Connections to Other Algebraic Topology Concepts

• Cellular homology is closely related to singular homology, another homology theory defined using singular simplices
  • For CW complexes, cellular homology and singular homology yield isomorphic homology groups
• The cellular chain complex $C_*(X)$ can be seen as a special case of a chain complex of free abelian groups, which is the main object of study in homological algebra
• Cohomology, a contravariant functor from the category of topological spaces to the category of abelian groups, can be defined using the dual of the cellular chain complex (called the cellular cochain complex)
  • Cohomology groups often carry additional structure, such as a ring structure given by the cup product
• The cap product is a bilinear pairing between homology and cohomology groups that generalizes the evaluation of cochains on chains
• Poincaré duality relates the homology and cohomology groups of orientable compact manifolds, stating that $H^k(M) \cong H_{n-k}(M)$ for an $n$-dimensional manifold $M$
• The Hurewicz theorem relates the homotopy groups of a space to its homology groups, providing a connection between two important invariants in algebraic topology
• Spectral sequences, which are algebraic tools for computing homology groups of chain complexes with a filtration, have numerous applications in algebraic topology beyond cellular homology (e.g., in the study of fiber bundles and the Adams spectral sequence in stable homotopy theory)