1.3 Singular homology

Singular homology assigns homology groups to topological spaces, capturing essential features like connectivity and holes. It's a powerful tool that provides invariants preserved under homeomorphisms, allowing us to study spaces' topological properties.

The chapter explores singular simplices, chain complexes, and boundary maps. It then delves into computations for specific spaces, homotopy invariance, and extensions like homology with coefficients and relative homology. These concepts form the foundation of algebraic topology.

Singular homology of topological spaces

• Singular homology is a powerful tool in algebraic topology that assigns homology groups to any topological space
• It captures essential topological features of spaces, such as connectivity, holes, and higher-dimensional voids
• Singular homology groups are invariants of topological spaces, meaning they are preserved under homeomorphisms

Singular n-simplices in topological spaces

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• A singular n-simplex in a topological space $X$ is a continuous map $\sigma: \Delta^n \rightarrow X$ from the standard n-simplex $\Delta^n$ to $X$
• Singular 0-simplices are points, singular 1-simplices are paths, singular 2-simplices are triangular surfaces, and so on
• The set of all singular n-simplices in $X$ is denoted by $S_n(X)$
• Examples of singular n-simplices:
  • A constant map from $\Delta^n$ to a single point in $X$ (degenerate simplex)
  • A homeomorphism from $\Delta^n$ to a subset of $X$ (non-degenerate simplex)

Singular chain complex of a space

• The singular chain complex of a topological space $X$ is a sequence of abelian groups $C_n(X)$ connected by boundary maps $\partial_n$
• $C_n(X)$ is the free abelian group generated by the set of singular n-simplices $S_n(X)$
• Elements of $C_n(X)$ are called singular n-chains and are formal linear combinations of singular n-simplices with integer coefficients
• The boundary maps $\partial_n: C_n(X) \rightarrow C_{n-1}(X)$ connect the chain groups and satisfy the property $\partial_{n-1} \circ \partial_n = 0$
• The singular chain complex encodes the combinatorial structure of the space $X$ and is denoted by $(C_*(X), \partial_*)$

Boundary maps in singular homology

• The boundary map $\partial_n: C_n(X) \rightarrow C_{n-1}(X)$ assigns to each singular n-simplex its oriented boundary
• For a singular n-simplex $\sigma: \Delta^n \rightarrow X$, the boundary $\partial_n(\sigma)$ is defined as the alternating sum of its faces
• The i-th face of $\sigma$ is obtained by restricting $\sigma$ to the i-th face of the standard n-simplex $\Delta^n$
• The boundary maps satisfy the fundamental property $\partial_{n-1} \circ \partial_n = 0$, which means the composition of two consecutive boundary maps is always zero
• This property allows the construction of homology groups from the singular chain complex

Singular homology groups of a space

• The n-th singular homology group of a topological space $X$ is defined as the quotient group $H_n(X) = \ker(\partial_n) / \operatorname{im}(\partial_{n+1})$
• Elements of $\ker(\partial_n)$ are called n-cycles, and elements of $\operatorname{im}(\partial_{n+1})$ are called n-boundaries
• Two n-cycles are homologous if their difference is an n-boundary, and the homology classes form the elements of $H_n(X)$
• Intuitively, $H_n(X)$ measures the n-dimensional holes in the space $X$ that are not filled by (n+1)-dimensional boundaries
• The rank of $H_n(X)$ is called the n-th Betti number of $X$ and provides a numerical invariant of the space

Computations of singular homology

• Computing singular homology groups is a central task in algebraic topology, as it reveals important topological properties of spaces
• Various techniques and tools are employed to calculate the homology groups of specific spaces, such as simplicial complexes, CW complexes, and manifolds
• The computations often involve the use of exact sequences, spectral sequences, and comparison theorems

Singular homology of a point

• The singular homology groups of a single point space $\{*\}$ are given by:
  • $H_0(\{*\}) \cong \mathbb{Z}$
  • $H_n(\{*\}) \cong 0$ for all $n \geq 1$
• This reflects the fact that a point has no higher-dimensional holes or non-trivial cycles
• The homology of a point serves as a building block for the homology of more complex spaces via the use of exact sequences and homotopy invariance

Singular homology of spheres

• The singular homology groups of the n-dimensional sphere $S^n$ are:
  • $H_0(S^n) \cong \mathbb{Z}$
  • $H_n(S^n) \cong \mathbb{Z}$
  • $H_k(S^n) \cong 0$ for all $k \neq 0, n$
• This captures the fact that an n-sphere has a single connected component and a single n-dimensional void
• The generator of $H_n(S^n)$ is represented by the fundamental class $[S^n]$, which corresponds to the orientation of the sphere
• Examples:
  • $H_*(S^1) \cong (\mathbb{Z}, \mathbb{Z}, 0, 0, \ldots)$
  • $H_*(S^2) \cong (\mathbb{Z}, 0, \mathbb{Z}, 0, 0, \ldots)$

Singular homology of projective spaces

• Real projective n-space $\mathbb{RP}^n$ is the quotient of the n-sphere $S^n$ by identifying antipodal points
• The singular homology groups of $\mathbb{RP}^n$ are:
  • $H_k(\mathbb{RP}^n) \cong \mathbb{Z}$ for $k = 0$ and $k = n$ odd
  • $H_k(\mathbb{RP}^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $0 < k < n$ odd
  • $H_k(\mathbb{RP}^n) \cong 0$ for $k$ even, $0 < k \leq n$
• The homology of projective spaces reflects their non-orientability and the presence of torsion in odd dimensions
• Similarly, complex projective n-space $\mathbb{CP}^n$ has homology groups $H_{2k}(\mathbb{CP}^n) \cong \mathbb{Z}$ for $0 \leq k \leq n$ and $H_{odd}(\mathbb{CP}^n) \cong 0$

Mayer-Vietoris sequence for singular homology

• The Mayer-Vietoris sequence is a powerful tool for computing the homology of a space by decomposing it into simpler subspaces
• Given a topological space $X$ and an open cover $X = U \cup V$, there is a long exact sequence of homology groups: $\ldots \rightarrow H_n(U \cap V) \xrightarrow{(\phi_n, \psi_n)} H_n(U) \oplus H_n(V) \xrightarrow{\eta_n} H_n(X) \xrightarrow{\partial_n} H_{n-1}(U \cap V) \rightarrow \ldots$
• The maps $\phi_n, \psi_n, \eta_n, \partial_n$ are induced by inclusions and connecting homomorphisms
• The Mayer-Vietoris sequence allows the computation of $H_n(X)$ from the homology of the subspaces $U, V$, and their intersection $U \cap V$
• It is particularly useful when $U, V$, and $U \cap V$ have simpler homology groups than $X$ itself

Homotopy invariance of singular homology

• A key property of singular homology is its homotopy invariance, which means that homotopic maps induce the same homomorphisms on homology groups
• This property allows the use of homotopy-theoretic techniques in the study of homology and the comparison of homology groups of homotopy equivalent spaces
• Homotopy invariance is a consequence of the existence of chain homotopies between chain complexes induced by homotopic maps

Chain homotopies and chain homotopy equivalences

• A chain homotopy between two chain maps $f, g: C_* \rightarrow D_*$ is a sequence of homomorphisms $h_n: C_n \rightarrow D_{n+1}$ satisfying $\partial_{n+1} \circ h_n + h_{n-1} \circ \partial_n = f_n - g_n$
• If there exists a chain homotopy between $f$ and $g$, they are called chain homotopic, and they induce the same homomorphisms on homology groups
• A chain homotopy equivalence between chain complexes $C_*$ and $D_*$ is a pair of chain maps $f: C_* \rightarrow D_*$ and $g: D_* \rightarrow C_*$ such that $f \circ g$ and $g \circ f$ are chain homotopic to the respective identity maps
• Chain homotopy equivalent complexes have isomorphic homology groups

Homotopy invariance theorem for singular homology

• The homotopy invariance theorem states that if two continuous maps $f, g: X \rightarrow Y$ are homotopic, then they induce the same homomorphisms $f_*, g_*: H_n(X) \rightarrow H_n(Y)$ on singular homology groups for all $n$
• Consequently, homotopy equivalent spaces have isomorphic singular homology groups
• The proof of the homotopy invariance theorem relies on the construction of a chain homotopy between the chain maps induced by homotopic maps
• Examples:
  • The homology groups of a space are invariant under deformation retracts
  • Contractible spaces have trivial homology groups in all dimensions except $H_0$

Excision theorem for singular homology

• The excision theorem is another powerful tool in singular homology that allows the computation of homology groups by excising subspaces
• It states that if $(X, A)$ is a pair of spaces and $U \subset A$ is a subspace such that the closure of $U$ is contained in the interior of $A$, then the inclusion $(X \setminus U, A \setminus U) \hookrightarrow (X, A)$ induces isomorphisms on relative homology groups
• In other words, the homology of $(X, A)$ is not affected by removing a "small" subspace $U$ from both $X$ and $A$
• The excision theorem is a key ingredient in the proof of the Mayer-Vietoris sequence and the long exact sequence of a pair

Applications of homotopy invariance in computations

• Homotopy invariance and the excision theorem are powerful tools in the computation of singular homology groups
• They allow the reduction of complex spaces to simpler ones with the same homology, such as CW complexes or simplicial complexes
• Examples of applications:
  • Computing the homology of a space by deforming it to a homotopy equivalent CW complex
  • Using the excision theorem to decompose a space into simpler pieces and computing the homology via the Mayer-Vietoris sequence
  • Proving the topological invariance of homology by showing that homeomorphic spaces are homotopy equivalent

Singular homology with coefficients

• Singular homology can be generalized by using coefficients in an arbitrary abelian group instead of integers
• This allows for the detection of more subtle topological features and the study of torsion phenomena in homology
• Homology with coefficients is particularly useful in the study of manifolds and in algebraic topology

Singular chain complex with coefficients

• Given a topological space $X$ and an abelian group $G$, the singular chain complex with coefficients in $G$ is defined as $C_n(X; G) = C_n(X) \otimes_{\mathbb{Z}} G$
• The boundary maps $\partial_n: C_n(X; G) \rightarrow C_{n-1}(X; G)$ are induced by the boundary maps of the integral chain complex $C_*(X)$
• The resulting chain complex $(C_*(X; G), \partial_*)$ is called the singular chain complex of $X$ with coefficients in $G$
• Examples of coefficient groups:
  • Finite cyclic groups $\mathbb{Z}/n\mathbb{Z}$
  • Field coefficients, such as $\mathbb{Q}, \mathbb{R}$, or $\mathbb{F}_p$

Singular homology groups with coefficients

• The singular homology groups of $X$ with coefficients in $G$ are defined as the homology groups of the chain complex $C_*(X; G)$
• They are denoted by $H_n(X; G) = \ker(\partial_n) / \operatorname{im}(\partial_{n+1})$, where the kernel and image are taken in $C_*(X; G)$
• Homology with coefficients satisfies the same formal properties as integral homology, such as functoriality, homotopy invariance, and the existence of long exact sequences
• The choice of coefficients can reveal torsion information and simplify computations, especially when using field coefficients

Universal coefficient theorem for homology

• The universal coefficient theorem relates the integral homology groups $H_n(X)$ to the homology groups with coefficients $H_n(X; G)$
• It states that there is a natural short exact sequence: $0 \rightarrow H_n(X) \otimes G \rightarrow H_n(X; G) \rightarrow \operatorname{Tor}(H_{n-1}(X), G) \rightarrow 0$
• The tensor product term $H_n(X) \otimes G$ captures the free part of $H_n(X; G)$, while the torsion product term $\operatorname{Tor}(H_{n-1}(X), G)$ captures the torsion part
• The universal coefficient theorem allows the computation of homology with coefficients from integral homology and provides a way to study the torsion subgroups of homology

Künneth formula for singular homology

• The Künneth formula is a theorem that relates the homology of a product space to the homology of its factors
• For topological spaces $X$ and $Y$, and a principal ideal domain $R$, there is a natural short exact sequence: $0 \rightarrow \bigoplus_{i+j=n} H_i(X; R) \otimes_R H_j(Y; R) \rightarrow H_n(X \times Y; R) \rightarrow \bigoplus_{i+j=n-1} \operatorname{Tor}_R^1(H_i(X; R), H_j(Y; R)) \rightarrow 0$
• When $R$ is a field or when the homology groups are torsion-free, the Tor term vanishes, and the Künneth formula provides an isomorphism: $H_n(X \times Y; R) \cong \bigoplus_{i+j=n} H_i(X; R) \otimes_R H_j(Y; R)$
• The Künneth formula is a powerful tool for computing the homology of product spaces and for studying the algebraic structure of homology

Relative singular homology

• Relative singular homology is an extension of singular homology that considers pairs of spaces $(X, A)$, where $A$ is a subspace of $X$
• It captures the homological information of the space $X$ relative to its subspace $A$ and is particularly useful in the study of excision and long exact sequences
• Relative homology is a key ingredient in the development of cohomology theories and in the study of manifolds with boundary

Relative singular chain complex

• Given a pair of spaces $(X, A)$, the relative singular chain complex $C_*(X, A)$ is defined as the quotient chain complex $C_*(X) / C_*(A)$
• The boundary maps $\partial_n: C_n(X, A) \rightarrow C_{n-1}(X, A)$ are induced by the boundary maps of $C_*(X)$ and satisfy $\partial_{n-1} \circ \partial_n = 0$
• The relative