🔢Elementary Algebraic Topology Unit 9 – Singular Homology

What's Singular Homology?

• Singular homology is a powerful algebraic tool used to study the topological properties of spaces
• Assigns a sequence of abelian groups, called homology groups, to a topological space
• Captures essential features of a space, such as the number of connected components, holes, and voids
• Provides a way to distinguish between spaces that cannot be continuously deformed into one another
• Singular homology is constructed using singular simplices, which are continuous maps from standard simplices to the space
• The homology groups are obtained by considering formal linear combinations of singular simplices, called singular chains, and their boundary relations
• Singular homology is a foundational concept in algebraic topology and has numerous applications in mathematics and related fields

Building Blocks: Simplices and Chains

• The building blocks of singular homology are simplices and chains
• A simplex is a generalization of a triangle or tetrahedron to arbitrary dimensions
  • A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron
• Singular simplices are continuous maps from the standard n-simplex $\Delta^n$ to a topological space $X$
  • Singular simplices capture how the standard simplex can be "placed" or "embedded" in the space
• Singular n-chains are formal linear combinations of singular n-simplices with coefficients in an abelian group (usually $\mathbb{Z}$ or a field)
  • The set of all singular n-chains forms an abelian group denoted by $C_n(X)$
• The boundary of an n-simplex is the alternating sum of its (n-1)-dimensional faces
  • The boundary of a 1-simplex (line segment) is the difference of its endpoints
  • The boundary of a 2-simplex (triangle) is the sum of its edges with appropriate orientations
• The boundary operator $\partial_n: C_n(X) \to C_{n-1}(X)$ extends the notion of boundary to singular chains
  • It satisfies the important property $\partial_{n-1} \circ \partial_n = 0$, which means the boundary of a boundary is always zero

The Boundary Operator

• The boundary operator $\partial_n: C_n(X) \to C_{n-1}(X)$ is a linear map that assigns to each singular n-chain its boundary, which is a singular (n-1)-chain
• The boundary operator satisfies the fundamental property $\partial_{n-1} \circ \partial_n = 0$, known as the "square zero" property
  • This property ensures that the boundary of a boundary is always zero, which is crucial for defining homology groups
• The boundary operator connects the chain groups $C_n(X)$ into a sequence called a chain complex
  • $\cdots \xrightarrow{\partial_{n+1}} C_n(X) \xrightarrow{\partial_n} C_{n-1}(X) \xrightarrow{\partial_{n-1}} \cdots$
• The kernel of $\partial_n$, denoted by $\ker(\partial_n)$ or $Z_n(X)$, consists of n-chains whose boundary is zero, called n-cycles
• The image of $\partial_{n+1}$, denoted by $\operatorname{im}(\partial_{n+1})$ or $B_n(X)$, consists of n-chains that are the boundary of some (n+1)-chain, called n-boundaries
• The "square zero" property implies that every n-boundary is also an n-cycle, i.e., $B_n(X) \subseteq Z_n(X)$

Homology Groups: Definition and Intuition

• The n-th homology group of a topological space $X$, denoted by $H_n(X)$, is defined as the quotient group $H_n(X) = Z_n(X) / B_n(X)$
  • It is the group of n-cycles modulo n-boundaries
• Intuitively, homology groups measure the "holes" or "voids" in a space that cannot be detected by the boundary operator
  • Elements of $H_n(X)$ represent n-dimensional holes or cycles that are not boundaries of (n+1)-dimensional objects
• The rank of $H_0(X)$ counts the number of connected components of the space $X$
• The rank of $H_1(X)$ counts the number of 1-dimensional holes or "loops" in $X$ that cannot be contracted to a point
• The rank of $H_2(X)$ counts the number of 2-dimensional voids or "cavities" in $X$ that cannot be filled by a 3-dimensional object
• Homology groups are topological invariants, meaning they are preserved under continuous deformations (homeomorphisms) of the space
  • If two spaces have different homology groups, they cannot be homeomorphic
• Homology groups provide a powerful algebraic tool to distinguish and classify topological spaces

Computing Homology: Examples and Techniques

• Computing homology groups involves finding the kernel and image of the boundary operators and taking their quotient
• For simplicial complexes, the computation can be done using linear algebra techniques on the matrices representing the boundary operators
  • Smith normal form and row reduction are often used to simplify the matrices and read off the homology groups
• For CW complexes, the cellular chain complex and cellular boundary operators can be used to compute homology more efficiently
• The Mayer-Vietoris sequence is a powerful tool for computing homology of spaces that can be decomposed into simpler pieces
  • It relates the homology of the whole space to the homology of its parts and their intersection
• Examples:
  • The homology groups of a point are $H_0(\text{point}) = \mathbb{Z}$ and $H_n(\text{point}) = 0$ for $n > 0$
  • The homology groups of a circle $S^1$ are $H_0(S^1) = \mathbb{Z}$, $H_1(S^1) = \mathbb{Z}$, and $H_n(S^1) = 0$ for $n > 1$
  • The homology groups of a torus $T$ are $H_0(T) = \mathbb{Z}$, $H_1(T) = \mathbb{Z} \oplus \mathbb{Z}$, $H_2(T) = \mathbb{Z}$, and $H_n(T) = 0$ for $n > 2$
• Computing homology of more complex spaces often involves a combination of various techniques and tools from algebraic topology

Key Properties of Singular Homology

• Homotopy invariance: Singular homology is invariant under homotopy equivalence
  • If two spaces $X$ and $Y$ are homotopy equivalent, then $H_n(X) \cong H_n(Y)$ for all $n$
• Excision: If $A$ is a subspace of $X$ such that the closure of $A$ is contained in the interior of a larger subspace $U$, then $H_n(X, A) \cong H_n(X \setminus A, U \setminus A)$ for all $n$
  • This property allows for local computations of homology
• Long exact sequence of a pair: For a pair of spaces $(X, A)$, there is a long exact sequence relating the homology of $X$, $A$, and the relative homology $H_n(X, A)$
  • $\cdots \to H_n(A) \to H_n(X) \to H_n(X, A) \to H_{n-1}(A) \to \cdots$
• Mayer-Vietoris sequence: For a space $X$ that can be decomposed into two open subspaces $U$ and $V$, there is a long exact sequence relating the homology of $X$, $U$, $V$, and $U \cap V$
  • $\cdots \to H_n(U \cap V) \to H_n(U) \oplus H_n(V) \to H_n(X) \to H_{n-1}(U \cap V) \to \cdots$
• Künneth formula: For two spaces $X$ and $Y$, there is a relationship between the homology of $X \times Y$ and the tensor product of the homology of $X$ and $Y$
  • $H_n(X \times Y) \cong \bigoplus_{p+q=n} H_p(X) \otimes H_q(Y)$
• Universal coefficient theorem: Relates homology with coefficients in different abelian groups
  • $H_n(X; G) \cong H_n(X) \otimes G \oplus \operatorname{Tor}(H_{n-1}(X), G)$

Applications in Topology

• Singular homology is a fundamental tool in algebraic topology and has numerous applications
• Distinguishing spaces: Homology groups can be used to show that two spaces are not homeomorphic or homotopy equivalent
  • If two spaces have different homology groups, they cannot be homeomorphic or homotopy equivalent
• Classifying manifolds: Homology groups play a crucial role in the classification of manifolds
  • The homology groups of a closed, connected, orientable n-manifold satisfy Poincaré duality: $H_k(M) \cong H_{n-k}(M)$
  • The Euler characteristic of a closed, triangulable n-manifold can be computed using homology groups: $\chi(M) = \sum_{k=0}^n (-1)^k \operatorname{rank}(H_k(M))$
• Intersection theory: Homology classes can be used to define and study the intersection of submanifolds
  • The intersection product of homology classes corresponds to the geometric intersection of submanifolds
• Obstruction theory: Homology groups can be used to study the existence and properties of continuous maps between spaces
  • The vanishing of certain homology groups can provide obstructions to the existence of certain types of maps
• Topological data analysis: Homology groups and their persistent versions are used to study the shape and structure of data sets in high-dimensional spaces

Connecting to Other Concepts

• Singular homology is closely related to other homology theories in algebraic topology
• Simplicial homology: Defined for simplicial complexes using simplicial chains and boundary operators
  • Singular homology agrees with simplicial homology for triangulable spaces
• Cellular homology: Defined for CW complexes using cellular chains and boundary operators
  • Singular homology agrees with cellular homology for CW complexes
• de Rham cohomology: Defined for smooth manifolds using differential forms and the exterior derivative
  • The de Rham theorem states that de Rham cohomology is isomorphic to singular cohomology with real coefficients for smooth manifolds
• Sheaf cohomology: A general cohomology theory defined using sheaves and their resolutions
  • Singular cohomology is a special case of sheaf cohomology for the constant sheaf
• K-theory: A generalization of homology theory that studies vector bundles and their stable equivalence classes
  • Singular homology can be viewed as a special case of K-theory for the trivial bundle
• Bordism theory: Studies manifolds and their bordisms (cobordisms) using homology-like invariants
  • Singular homology is related to oriented bordism theory via the Atiyah-Hirzebruch spectral sequence
• Homotopy theory: Studies spaces and continuous maps up to homotopy equivalence
  • Singular homology is a homotopy invariant and can be used to define and study homotopy groups and homotopy types of spaces