1.4 Homology groups

Homology groups are algebraic tools that capture the "holes" in topological spaces. They provide a way to distinguish between spaces with different topological properties, even when those spaces are homotopy equivalent.

Homology groups are defined using various theories, including singular, simplicial, and cellular homology. These theories assign abelian groups to topological spaces, allowing mathematicians to study the spaces' properties through algebraic means.

Definition of homology groups

• Homology groups are algebraic invariants associated to a topological space that capture information about the space's "holes" or "voids" in various dimensions
• They provide a way to distinguish between spaces that have different topological properties but may be homotopy equivalent
• Homology groups are defined using various theories, including singular homology, simplicial homology, and cellular homology

Singular homology theory

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• Assigns to each topological space $X$ a sequence of abelian groups $H_n(X)$, where $n$ is a non-negative integer
• Constructed by considering singular n-simplices, which are continuous maps from the standard n-simplex to the space $X$
  • The standard n-simplex is the convex hull of the standard basis vectors in $\mathbb{R}^{n+1}$
• The boundary of a singular n-simplex is a formal sum of its (n-1)-dimensional faces with alternating signs
• The singular n-chains are formal sums of singular n-simplices, and the boundary operator extends linearly to these chains
• The n-th singular homology group is defined as the quotient of the kernel of the boundary operator on n-chains by the image of the boundary operator on (n+1)-chains

Simplicial homology theory

• Applicable to simplicial complexes, which are spaces built from simplices glued together along their faces
• The simplicial n-chains are formal sums of oriented n-simplices in the complex
• The boundary of an n-simplex is the alternating sum of its (n-1)-dimensional faces
• The n-th simplicial homology group is defined similarly to the singular case, using the simplicial boundary operator
• Simplicial homology is more combinatorial in nature and easier to compute than singular homology

Cellular homology theory

• Applicable to CW complexes, which are spaces built by attaching cells of increasing dimension
• The cellular n-chains are formal sums of n-cells in the complex
• The boundary of an n-cell is a linear combination of (n-1)-cells determined by the attaching maps
• The n-th cellular homology group is defined using the cellular boundary operator
• Cellular homology is even more efficient to compute than simplicial homology, as there are typically fewer cells than simplices in a given space

Properties of homology groups

• Homology groups satisfy several important properties that make them useful tools in algebraic topology
• These properties allow for the computation of homology groups in various situations and the derivation of topological invariants

Homotopy invariance

• If two spaces $X$ and $Y$ are homotopy equivalent, then their homology groups are isomorphic
  • Homotopy equivalence is a weaker notion than homeomorphism, allowing for continuous deformations between spaces
• This property implies that homology groups are topological invariants, depending only on the topology of the space and not on its particular representation
• Homotopy invariance allows for the computation of homology groups using simpler, homotopy equivalent spaces

Functoriality

• Homology groups are functorial, meaning that continuous maps between spaces induce homomorphisms between their homology groups
• Given a continuous map $f: X \to Y$, there are induced homomorphisms $f_*: H_n(X) \to H_n(Y)$ for each $n$
• These induced homomorphisms are compatible with composition of maps and preserve the group structure
• Functoriality allows for the study of maps between spaces using the induced maps on homology groups

Excision theorem

• The excision theorem states that the homology groups of a space are determined locally
• If $U \subset X$ is an open set whose closure is contained in the interior of another subspace $A \subset X$, then the inclusion $(X \setminus U, A \setminus U) \to (X, A)$ induces an isomorphism on homology groups
• This theorem allows for the computation of homology groups by decomposing a space into simpler pieces and studying their local behavior

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a long exact sequence that relates the homology groups of a space $X$ to those of two subspaces $A$ and $B$ whose union covers $X$
• The sequence has the form: $\cdots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to H_{n-1}(A \cap B) \to \cdots$
• This sequence allows for the computation of homology groups by breaking a space down into simpler subspaces and studying their intersections
• The Mayer-Vietoris sequence is a powerful tool for computing homology groups of spaces that can be decomposed into simpler pieces

Computation of homology groups

• Computing homology groups is a central task in algebraic topology, as it allows for the extraction of topological information from a space
• There are several techniques for computing homology groups, depending on the type of space and the homology theory being used

Homology of spheres

• The n-dimensional sphere $S^n$ has a simple homology structure
  • $H_0(S^n) = \mathbb{Z}$, as the sphere is connected
  • $H_n(S^n) = \mathbb{Z}$, as the sphere has a single "hole" in dimension $n$
  • $H_k(S^n) = 0$ for all other values of $k$
• This computation can be done using any of the homology theories (singular, simplicial, or cellular)
• The homology of spheres serves as a building block for computing the homology of more complex spaces

Homology of tori

• The n-dimensional torus $T^n$ is the product of $n$ circles, $T^n = S^1 \times \cdots \times S^1$
• The homology groups of $T^n$ can be computed using the Künneth formula, which relates the homology of a product space to the homology of its factors
  • $H_k(T^n) = \bigoplus_{i_1 + \cdots + i_n = k} H_{i_1}(S^1) \otimes \cdots \otimes H_{i_n}(S^1)$
• In particular, the homology groups of the 2-dimensional torus $T^2$ are:
  • $H_0(T^2) = \mathbb{Z}$
  • $H_1(T^2) = \mathbb{Z} \oplus \mathbb{Z}$
  • $H_2(T^2) = \mathbb{Z}$
  • $H_k(T^2) = 0$ for $k > 2$

Homology of surfaces

• Surfaces are 2-dimensional manifolds, which are locally homeomorphic to $\mathbb{R}^2$
• The homology groups of a surface depend on its genus $g$, which is the number of "handles" attached to a sphere
  • $H_0(S_g) = \mathbb{Z}$, as the surface is connected
  • $H_1(S_g) = \mathbb{Z}^{2g}$, as each handle contributes two generators to the first homology group
  • $H_2(S_g) = \mathbb{Z}$, as the surface is orientable and has a single 2-dimensional "hole"
  • $H_k(S_g) = 0$ for $k > 2$
• The homology of surfaces can be computed using triangulations and simplicial homology

Homology of CW complexes

• CW complexes are spaces built by attaching cells of increasing dimension, generalizing the notion of a simplicial complex
• The homology groups of a CW complex can be computed using cellular homology, which is often more efficient than singular or simplicial homology
• The cellular chain complex is determined by the attaching maps of the cells, and the homology groups are the quotients of the kernels and images of the cellular boundary operators
• The homology of CW complexes can be used to study more general topological spaces, as many spaces admit CW structures

Applications of homology groups

• Homology groups are powerful tools in algebraic topology, with numerous applications in mathematics and related fields
• They can be used to prove theorems, derive invariants, and study the properties of topological spaces

Brouwer fixed point theorem

• The Brouwer fixed point theorem states that any continuous function from a n-dimensional ball to itself has a fixed point
• This theorem can be proved using homology groups, by showing that the induced map on homology groups has a non-zero degree
• The idea is to use the fact that the homology groups of the ball and its boundary sphere are related by the long exact sequence of a pair
• The Brouwer fixed point theorem has applications in various fields, including economics and game theory

Borsuk-Ulam theorem

• The Borsuk-Ulam theorem states that any continuous function from the n-sphere to n-dimensional Euclidean space maps some pair of antipodal points to the same point
• This theorem can be proved using the homology groups of the sphere and the concept of the degree of a map
• The key idea is to show that the induced map on homology groups has odd degree, which implies the existence of antipodal points with the same image
• The Borsuk-Ulam theorem has applications in combinatorics, algebraic topology, and the study of symmetries

Lefschetz fixed point theorem

• The Lefschetz fixed point theorem is a generalization of the Brouwer fixed point theorem to more general spaces
• It states that if a continuous map from a compact triangulable space to itself has a non-zero Lefschetz number, then it has a fixed point
• The Lefschetz number is defined as the alternating sum of the traces of the induced maps on homology groups
• The theorem relies on the relationship between the homology groups of the space and the induced maps on these groups
• The Lefschetz fixed point theorem has applications in dynamical systems and the study of periodic points

Degree of a map

• The degree of a continuous map between oriented n-dimensional manifolds is an integer that measures the "winding number" or "multiplicity" of the map
• It can be defined using the induced map on n-th homology groups, as the image of a fundamental class of the domain manifold
• The degree is a homotopy invariant and satisfies several properties, such as additivity under disjoint unions and multiplicativity under composition
• The concept of degree is used in various applications, such as the hairy ball theorem in vector field analysis and the study of covering spaces in algebraic topology

Relationship to other invariants

• Homology groups are one of several algebraic invariants used to study topological spaces
• They are related to other invariants, such as homotopy groups, cohomology groups, and the Euler characteristic, each capturing different aspects of the space's structure

Homology vs homotopy groups

• Homotopy groups $\pi_n(X)$ are another set of algebraic invariants associated to a topological space $X$
• While homology groups measure the "holes" in a space, homotopy groups measure the distinct ways in which spheres can be mapped into the space
• Homotopy groups are generally more difficult to compute than homology groups, as they have a more complex algebraic structure (non-abelian for $n > 1$)
• There are relationships between homology and homotopy groups, such as the Hurewicz theorem, which states that the first non-trivial homotopy group is isomorphic to the corresponding homology group

Homology vs cohomology groups

• Cohomology groups $H^n(X)$ are another set of algebraic invariants, dual to homology groups
• While homology groups are constructed using chains (formal sums of simplices or cells), cohomology groups are constructed using cochains (functions from chains to a coefficient group)
• Cohomology groups have a natural ring structure, given by the cup product, which is not present in homology groups
• Homology and cohomology groups are related by the universal coefficient theorem, which expresses cohomology groups in terms of homology groups and the coefficient group

Homology vs Euler characteristic

• The Euler characteristic $\chi(X)$ is a topological invariant that can be defined for spaces admitting a finite cell decomposition
• It is defined as the alternating sum of the number of cells in each dimension: $\chi(X) = \sum_{n} (-1)^n c_n$, where $c_n$ is the number of n-cells
• The Euler characteristic is related to homology groups by the Euler-Poincaré formula: $\chi(X) = \sum_{n} (-1)^n \text{rank}(H_n(X))$
• While the Euler characteristic is easier to compute than homology groups, it captures less information about the space's structure

Advanced topics in homology

• Beyond the basic definitions and properties of homology groups, there are several advanced topics that extend and generalize the theory
• These topics include relative homology, homology with coefficients, the Künneth formula, and the universal coefficient theorem

Relative homology groups

• Relative homology groups $H_n(X, A)$ are defined for a pair of spaces $(X, A)$, where $A$ is a subspace of $X$
• They measure the homology of the quotient space $X/A$, or equivalently, the homology of $X$ "relative to" the homology of $A$
• Relative homology groups fit into a long exact sequence that relates them to the homology groups of $X$ and $A$: $\cdots \to H_n(A) \to H_n(X) \to H_n(X, A) \to H_{n-1}(A) \to \cdots$
• Relative homology is useful in the study of excision and the Mayer-Vietoris sequence, as well as in the definition of the relative cup product in cohomology

Homology with coefficients

• Homology groups can be defined with coefficients in any abelian group $G$, denoted as $H_n(X; G)$
• The construction of homology with coefficients involves tensoring the chain complex with the coefficient group $G$
• Homology with coefficients allows for the detection of torsion in homology groups, which is not visible with integer coefficients
• The universal coefficient theorem relates homology with different coefficients, expressing $H_n(X; G)$ in terms of $H_n(X; \mathbb{Z})$ and the tensor product and Tor functors

Künneth formula

• The Künneth formula is a theorem that relates the homology groups of a product space $X \times Y$ to the homology groups of its factors $X$ and $Y$
• It states that there is a split short exact sequence: $0 \to \bigoplus_{i+j=n} H_i(X) \otimes H_j(Y) \to H_n(X \times Y) \to \bigoplus_{i+j=n-1} \text{Tor}(H_i(X), H_j(Y)) \to 0$
• The Künneth formula is particularly useful when the homology groups of the factors are torsion-free, in which case the Tor term vanishes and the homology of the product is simply the tensor product of the homology of the factors
• The Künneth formula has applications in the computation of homology groups of product spaces, such as tori and product manifolds

Universal coefficient theorem

• The universal coefficient theorem relates homology and cohomology groups with different coefficients
• It states that for any abelian group $G$, there is a split short exact sequence: $0 \to \text{Ext}(H_{n-1}(X), G) \to H^n(X; G) \to \text{Hom}(H_n(X), G) \to 0$
• The Ext and Hom terms are derived functors that measure the torsion and free parts of the homology groups, respectively
• The universal coefficient theorem is a powerful tool in the computation of cohomology groups, as it allows for the deduction of cohomology with arbitrary coefficients from homology with integer coefficients
• It also highlights the relationship between homology and cohomology, and the role of torsion in these theories.