Glossary

2.2 Homology groups and their computation

2 min read•august 7, 2024

are a key concept in algebraic topology, measuring the "holes" in a topological space. They're defined as quotient modules of cycles and boundaries, capturing essential structural information about the space.

Computing homology groups involves tools like free resolutions and long exact sequences. These techniques allow us to break down complex structures into simpler components, making it easier to calculate and understand the homology of various mathematical objects.

Homology Groups and Their Properties

Definition and Components of Homology Groups

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  • Homology group Hn(C∗)H_n(C_*) defined as the ker⁡dn/im dn+1\ker d_n / \text{im } d_{n+1}
  • ker⁡dn\ker d_n consists of , elements of CnC_n that are mapped to zero by the boundary map dnd_n
  • im dn+1\text{im } d_{n+1} consists of , elements of CnC_n that are the image of the boundary map dn+1d_{n+1} from Cn+1C_{n+1}
  • Quotient module ker⁡dn/im dn+1\ker d_n / \text{im } d_{n+1} identifies n-cycles that differ by an n-boundary, capturing the "essential" n-dimensional holes in the complex

Invariants of Homology Groups

  • βn\beta_n represent the rank of the free part of the n-th homology group Hn(C∗)H_n(C_*)
    • β0\beta_0 counts the number of connected components
    • β1\beta_1 counts the number of 1-dimensional holes or "loops"
    • β2\beta_2 counts the number of 2-dimensional voids or "cavities"
  • of Hn(C∗)H_n(C_*) captures the non-free part of the homology group
    • Torsion elements have finite order, i.e., they become zero when multiplied by some non-zero scalar
    • Torsion can be represented by a list of the orders of the torsion subgroups (e.g., Z2,Z3\mathbb{Z}_2, \mathbb{Z}_3)

Computational Tools for Homology

Free Resolutions

  • of a module MM is an of free modules F∗F_* with a map ε:F0→M\varepsilon: F_0 \to M such that the sequence ⋯→F2→F1→F0→εM→0\cdots \to F_2 \to F_1 \to F_0 \xrightarrow{\varepsilon} M \to 0 is exact
  • Free resolutions can be used to compute homology groups by applying the H∗H_* to the resolution
    • Hn(M)≅Hn(F∗)H_n(M) \cong H_n(F_*) for all n≥0n \geq 0
  • Constructing a free resolution involves finding a generating set for the module and relations among the generators

Long Exact Sequences

  • is a tool for computing homology groups of a chain complex from the homology groups of simpler complexes
  • Given a short exact sequence of chain complexes 0→A∗→B∗→C∗→00 \to A_* \to B_* \to C_* \to 0 there is an associated long exact sequence in homology ⋯→Hn(A∗)→Hn(B∗)→Hn(C∗)→∂Hn−1(A∗)→⋯\cdots \to H_n(A_*) \to H_n(B_*) \to H_n(C_*) \xrightarrow{\partial} H_{n-1}(A_*) \to \cdots
  • The ∂\partial relates the homology groups of the complexes A∗,B∗,A_*, B_*, and C∗C_*
  • Long exact sequences can be used to compute unknown homology groups from known ones by exploiting the exactness property
    • Exactness means that the kernel of each map is equal to the image of the previous map

Key Terms to Review (14)

Betti numbers: Betti numbers are topological invariants that describe the number of independent cycles in a given space or object, providing crucial information about its shape and structure. They serve as key tools in algebraic topology and homological algebra, reflecting how many dimensions of holes exist in a space and linking to various concepts like cohomology, homology groups, and their applications in different mathematical contexts.
Connecting Homomorphism: A connecting homomorphism is a morphism that arises in the context of exact sequences, specifically serving as a bridge between different chain complexes. It captures the relationship between the homology of the chain complexes and helps facilitate the transfer of algebraic information across these complexes. This concept plays a crucial role in linking different algebraic structures, especially when analyzing how they interact under sequences and derived functors.
Exact Sequence: An exact sequence is a sequence of algebraic objects and morphisms between them, where the image of one morphism is equal to the kernel of the next. This concept is central in homological algebra as it allows us to study the relationships between different algebraic structures and provides insight into their properties through the notions of homology and cohomology.
Free resolution: A free resolution is an exact sequence of modules and homomorphisms that begins with a free module and ultimately maps onto a given module. This structure is fundamental in homological algebra as it helps in studying properties of modules, particularly through the computation of homology groups. The process involves constructing a sequence that captures essential information about the module's structure, leading to the derivation of useful invariants.
Homology Functor: The homology functor is a mathematical tool that assigns a sequence of algebraic structures, called homology groups, to a topological space or a chain complex. This functor captures essential topological features, allowing for the computation of homology groups that provide insight into the shape and structure of spaces. By transforming spaces into algebraic objects, the homology functor enables the analysis and classification of their properties.
Homology group h_n(c_*): The homology group h_n(c_*) is a fundamental concept in algebraic topology, representing the nth homology group associated with a chain complex c_*. It provides a way to classify topological spaces based on their shapes and connectivity by associating algebraic structures to them. These groups serve as powerful tools for understanding the properties of spaces, allowing mathematicians to compute invariants that reveal important characteristics of the underlying topological spaces.
Homology groups: Homology groups are algebraic structures that arise from chain complexes and serve to classify topological spaces by measuring their 'holes' in various dimensions. They provide crucial insights into the properties of spaces and are integral to understanding concepts in algebra, geometry, and topology.
Image: In mathematics, particularly in the study of algebraic structures and homomorphisms, the image of a function is the set of all outputs it can produce. This concept is fundamental for understanding how elements from one set map to another, revealing properties of homomorphisms and chain complexes, as well as providing insight into exact sequences and homology groups.
Kernel: The kernel of a homomorphism is the set of elements in the domain that map to the identity element of the codomain. It acts as a measure of how much a homomorphism fails to be injective and plays a crucial role in understanding algebraic structures and their relationships through exact sequences, chain complexes, and homology groups.
Long exact sequence: A long exact sequence is a sequence of abelian groups or modules connected by homomorphisms, which satisfies exactness at every point, indicating that the image of each homomorphism equals the kernel of the next. This concept is crucial in understanding the behavior of homology and cohomology theories, allowing one to relate different algebraic structures through their exact sequences and facilitating computations in various contexts.
N-boundaries: N-boundaries refer to the elements in the n-th homology group that can be represented as boundaries of (n+1)-chains. They play a crucial role in the computation of homology groups, as they help identify which cycles are not truly distinct, allowing for the simplification of complex structures. N-boundaries provide insights into the topological features of spaces by revealing how certain dimensions collapse or combine in higher dimensions.
N-cycles: An n-cycle is a specific type of chain in homological algebra that consists of a sequence of n elements which cyclically connect to form a closed loop. These cycles are essential for the computation of homology groups, as they represent the elements whose boundaries vanish, indicating that they are 'closed' and thus contribute to the structure of the associated homology group.
Quotient Module: A quotient module is formed by taking a module and dividing it by a submodule, resulting in a new module that represents the equivalence classes of the original module. This construction is important for understanding the structure of modules, as it allows for simplification and analysis of their properties, particularly in relation to homology groups and their computations.
Torsion: In algebra, torsion refers to elements of a module or group that have a finite order, meaning that when multiplied by some integer, the result is the identity element. This concept plays a crucial role in the study of homology groups, where torsion elements can indicate specific features of the underlying algebraic structure and can affect the computations and properties of those groups.
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