🧬Homological Algebra Unit 2 – Chain Complexes and Homology

Key Concepts and Definitions

• Chain complexes consist of a sequence of abelian groups or modules connected by homomorphisms called boundary operators or differentials
• Homology groups measure the failure of the composition of two consecutive boundary operators to be zero, capturing "holes" or "cycles" in the chain complex
• Exact sequences are sequences of homomorphisms between abelian groups or modules where the image of each homomorphism equals the kernel of the next
• Short exact sequences are exact sequences with only three non-zero terms, providing a way to relate the homology of different chain complexes
• The snake lemma is a powerful tool for studying the relationships between homology groups in a commutative diagram of short exact sequences
• The connecting homomorphism is a map between homology groups that arises from the snake lemma, allowing the computation of homology in certain situations
• The long exact sequence in homology is an important tool for computing homology groups, relating the homology of a chain complex to that of its subcomplexes and quotient complexes

Chain Complexes: Structure and Properties

• A chain complex is a sequence of abelian groups or modules $C_n$ connected by homomorphisms $d_n: C_n \to C_{n-1}$ called boundary operators or differentials
• The main property of a chain complex is that the composition of any two consecutive boundary operators is zero: $d_{n-1} \circ d_n = 0$ for all $n$
  • This property allows the definition of homology groups, which measure the failure of this composition to be zero
• The elements of $C_n$ are called n-chains, and the elements of the kernel of $d_n$ are called n-cycles (denoted $Z_n$)
• The elements of the image of $d_{n+1}$ are called n-boundaries (denoted $B_n$)
  • The property $d_{n-1} \circ d_n = 0$ implies that every n-boundary is also an n-cycle, i.e., $B_n \subseteq Z_n$
• A chain complex is exact if the image of each boundary operator equals the kernel of the next, i.e., $\operatorname{im} d_{n+1} = \ker d_n$ for all $n$
• A chain map between two chain complexes is a sequence of homomorphisms that commutes with the boundary operators, preserving the structure of the chain complexes

Homology Groups: Construction and Meaning

• The n-th homology group of a chain complex $C_*$ is defined as the quotient group $H_n(C_*) = Z_n(C_*) / B_n(C_*)$, where $Z_n(C_*)$ is the group of n-cycles and $B_n(C_*)$ is the group of n-boundaries
• Homology groups measure the failure of the composition of two consecutive boundary operators to be zero, capturing "holes" or "cycles" in the chain complex
  • Elements of the homology group are equivalence classes of cycles that differ by a boundary
• If the homology group $H_n(C_*)$ is trivial (i.e., consists only of the zero element), it means that every n-cycle is an n-boundary, and there are no "holes" in dimension $n$
• Non-trivial homology groups indicate the presence of cycles that are not boundaries, representing non-trivial topological features or algebraic structures
• Homology groups are invariants of the chain complex, meaning that they are preserved by chain maps that induce isomorphisms on homology
• The rank of the homology group $H_n(C_*)$, called the n-th Betti number, provides information about the number of independent "holes" or "cycles" in dimension $n$

Exact Sequences and Short Exact Sequences

• An exact sequence is a sequence of homomorphisms between abelian groups or modules, $\cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_n} A_{n-1} \to \cdots$, such that the image of each homomorphism equals the kernel of the next, i.e., $\operatorname{im} f_{n+1} = \ker f_n$ for all $n$
• Short exact sequences are exact sequences with only three non-zero terms, of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$, where $f$ is injective, $g$ is surjective, and $\operatorname{im} f = \ker g$
  • Short exact sequences provide a way to relate the objects $A$, $B$, and $C$, often allowing the computation of one in terms of the others
• The splitting lemma states that a short exact sequence splits (i.e., $B \cong A \oplus C$) if and only if there exists a homomorphism $h: C \to B$ such that $g \circ h = \operatorname{id}_C$
• The five lemma is a tool for proving that a homomorphism between two short exact sequences is an isomorphism, based on the commutativity of the diagram and the isomorphism of certain objects
• The snake lemma is a powerful result that relates the homology of the objects in a commutative diagram of short exact sequences, providing a long exact sequence connecting their homology groups

Computational Techniques for Homology

• The computation of homology groups often involves finding the kernel and image of the boundary operators, which can be done using linear algebra techniques when the chain complex is finitely generated
• For simplicial complexes, the boundary operators can be represented by matrices, and the homology groups can be computed using Smith normal form or other matrix reduction techniques
  • This process involves finding bases for the cycle and boundary spaces and computing their quotient
• The Mayer-Vietoris sequence is a long exact sequence that relates the homology of a space to the homology of its subspaces, providing a way to compute homology by decomposing a space into simpler pieces
• Spectral sequences are a powerful computational tool that arise from filtered chain complexes, allowing the computation of homology by successively approximating it with simpler homology groups
  • The Leray-Serre spectral sequence is a specific example that relates the homology of a fiber bundle to the homology of its base and fiber
• Künneth formulas provide a way to compute the homology of a tensor product of chain complexes in terms of the homology of the individual chain complexes, which is useful for computing the homology of product spaces
• The universal coefficient theorem relates the homology of a chain complex with coefficients in a different group to the original homology and the Ext and Tor functors, providing a way to change coefficients in homology computations

Applications in Topology and Algebra

• In topology, homology groups are used to classify and distinguish topological spaces, providing invariants that capture essential features of the space
  • For example, the homology groups of a surface determine its genus, and the homology groups of a knot complement determine its knot type
• Homology groups can be used to prove the existence of fixed points or the non-existence of certain mappings between spaces, using techniques such as the Lefschetz fixed-point theorem and the Brouwer fixed-point theorem
• In algebra, homology theories can be constructed for various algebraic structures, such as groups, Lie algebras, and associative algebras, providing invariants and tools for their study
  • Group homology, for example, is related to the classification of group extensions and the cohomology of groups, which has applications in representation theory and algebraic number theory
• Homological algebra provides a framework for studying derived functors, which arise naturally in many algebraic contexts, such as the Ext and Tor functors in module theory and the sheaf cohomology in algebraic geometry
• The study of homological dimensions, such as projective and injective dimensions, provides information about the complexity and structure of modules and algebras
  • For example, the global dimension of a ring is related to its homological properties and has implications for the category of modules over the ring

Common Examples and Exercises

• Computing the homology groups of simple topological spaces, such as spheres, tori, and projective spaces, using simplicial or cellular homology
• Proving that the fundamental group of the circle is isomorphic to the integers using the Hurewicz theorem and the homology of the circle
• Showing that the torus and the Klein bottle are not homotopy equivalent by computing their homology groups with different coefficients
• Using the Mayer-Vietoris sequence to compute the homology of a space decomposed into simpler subspaces, such as a union of two spaces or a space with a contractible subspace removed
• Applying the Künneth formula to compute the homology of a product space, such as the torus as a product of two circles or the projective plane as a product of a circle and an interval
• Computing the homology of a chain complex of vector spaces using matrix reduction techniques and the Smith normal form
• Using the snake lemma to prove the exactness of a sequence or to compute the homology of a quotient complex
• Constructing a spectral sequence from a filtered chain complex and using it to compute the homology of the total complex
• Computing the group homology of a cyclic group or a dihedral group using a resolution by free modules
• Proving that the category of chain complexes is an abelian category and studying its homological properties, such as the existence of enough projectives and injectives

Connections to Other Mathematical Areas

• Algebraic topology: Homology is a fundamental tool in algebraic topology, used to study topological spaces and their invariants, such as homotopy groups and cohomology rings
  • The singular homology of a topological space is defined using the chain complex of singular simplices, and its computation often involves techniques from homological algebra
• Differential geometry: The de Rham cohomology of a smooth manifold is defined using the chain complex of differential forms, and its relationship to singular cohomology is established by the de Rham theorem
  • Hodge theory studies the harmonic forms on a Riemannian manifold, which are related to the de Rham cohomology and have applications in complex geometry and mathematical physics
• Algebraic geometry: Sheaf cohomology is a fundamental tool in algebraic geometry, used to study the properties of algebraic varieties and their morphisms
  • The derived category of coherent sheaves on a variety is a homological invariant that captures important geometric information, and its study involves techniques from homological algebra
• Representation theory: Homological methods, such as the computation of Ext and Tor groups, are used to study the structure and properties of representations of algebras and groups
  • The cohomology of groups, defined using the chain complex of group cochains, has applications in the classification of group extensions and the study of group actions
• Mathematical physics: Homological techniques appear in various areas of mathematical physics, such as the BRST formalism in quantum field theory and the study of topological quantum field theories
  • The Floer homology of a symplectic manifold, defined using the chain complex of pseudo-holomorphic curves, has applications in the study of knots and 3-manifolds, as well as in mirror symmetry and string theory