Glossary

10.4 Eilenberg-Steenrod axioms

3 min read•august 7, 2024

The are the foundation of in algebraic topology. They define how homology groups behave under various operations, allowing us to calculate and understand the structure of topological spaces.

These axioms connect different aspects of topology, from to the decomposition of spaces. They provide a powerful framework for analyzing spaces by breaking them down into simpler parts and relating their homology groups.

Axioms of Homology Theory

Homotopy Invariance and Dimension

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  • axiom states that if two continuous maps between topological spaces are homotopic, then they induce the same homomorphism on homology groups
    • Implies that homology groups are invariants of (spaces that are homotopy equivalent have homology groups)
    • Allows for the calculation of homology groups of a space by considering a simpler space homotopy equivalent to it
  • Dimension axiom specifies the homology groups of a single point space
    • States that the Hn(pt)H_n(pt) is isomorphic to Z\mathbb{Z} for n=0n=0 and is trivial for n>0n>0
    • Provides a starting point for calculating homology groups using the other axioms

Excision and Additivity

  • relates the homology of a space XX to the homology of a subspace A⊂XA \subset X and its closure
    • States that if the closure of AA is contained in the interior of a subspace U⊂XU \subset X, then the inclusion (X∖A,U∖A)↪(X,U)(X \setminus A, U \setminus A) \hookrightarrow (X,U) induces isomorphisms on homology groups
    • Allows for the calculation of homology groups by decomposing a space into smaller, simpler pieces
  • states that the homology of a disjoint union of spaces is isomorphic to the direct sum of the homology of each space
    • Formally, if X=⨆αXαX = \bigsqcup_{\alpha} X_{\alpha}, then Hn(X)≅⨁αHn(Xα)H_n(X) \cong \bigoplus_{\alpha} H_n(X_{\alpha}) for all nn
    • Enables the computation of homology groups of a space by breaking it down into its

Exactness and Long Exact Sequence

  • relates the homology of a space, a subspace, and the corresponding quotient space
    • For a pair (X,A)(X,A) with A⊂XA \subset X, there is a of homology groups: ⋯→Hn(A)→Hn(X)→Hn(X,A)→Hn−1(A)→⋯\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \to \cdots
    • Connects the homology groups of different spaces and allows for their computation using the properties of exact sequences
    • The connecting homomorphism Hn(X,A)→Hn−1(A)H_n(X,A) \to H_{n-1}(A) is induced by the in the

Fundamental Results

Homology Theory and Uniqueness

  • Homology theory refers to a collection of functors from the category of topological spaces (or a suitable subcategory) to the category of abelian groups, satisfying the Eilenberg-Steenrod axioms
    • Different homology theories may arise from different choices of chain complexes or different methods of construction
    • Examples include , , and
  • states that any two homology theories satisfying the Eilenberg-Steenrod axioms and agreeing on the homology groups of a point are naturally isomorphic
    • Implies that the homology groups of a space are independent of the choice of homology theory, as long as the axioms are satisfied
    • Allows for the use of different homology theories depending on the context and the available data about the space (simplicial complexes, CW complexes, etc.)

Key Terms to Review (21)

Abelian group: An abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying four key properties: closure, associativity, identity, and invertibility, along with commutativity. The commutative property distinguishes abelian groups from general groups, meaning the order of operation does not affect the outcome. This concept is fundamental in understanding structures in algebra, particularly when discussing characterizations and examples as well as foundational axioms in topology.
Additivity Axiom: The additivity axiom is a fundamental principle in homological algebra, which states that a functor that satisfies this axiom preserves exact sequences. Essentially, if you have two short exact sequences, the functor applied to the sequences will yield another exact sequence. This property is crucial in the study of derived functors and cohomology, linking various algebraic structures and facilitating computations.
Boundary operator: The boundary operator is a fundamental concept in algebraic topology, specifically used to describe how chains in a chain complex relate to each other. It assigns a boundary to each chain, representing the 'edges' or 'faces' of a given geometric object, and plays a crucial role in defining homology and cohomology theories. The boundary operator helps formalize how structures change through the mapping of chains and allows us to derive important topological invariants.
Cellular Homology: Cellular homology is a method in algebraic topology that computes homology groups using a cell complex, which is built from cells of various dimensions. It provides a systematic way to study topological spaces by breaking them down into simpler pieces, allowing for the analysis of their structure and properties. This method connects deeply with various applications in algebra and topology, particularly in understanding how different spaces relate and behave under continuous transformations.
Chain Complex: A chain complex is a sequence of abelian groups or modules connected by homomorphisms, where the composition of any two consecutive homomorphisms is zero. This structure allows for the study of algebraic properties and relationships in various mathematical contexts, particularly in the fields of topology and algebra. Chain complexes are fundamental in defining homology and cohomology theories, which provide powerful tools for analyzing mathematical objects.
Cohomology: Cohomology is a mathematical concept that assigns algebraic invariants to topological spaces and chain complexes, capturing information about their structure and relationships. It provides a dual perspective to homology, focusing on the study of cochains and cocycles, which can reveal properties of spaces that homology alone might miss. This tool is essential in various areas of mathematics, connecting geometry, algebra, and topology.
Connected Components: Connected components refer to the maximal subsets of a topological space that are connected, meaning there exists a path between any two points within each subset. Understanding connected components is crucial for analyzing the structure of spaces and their relationships, particularly in algebraic topology and when applying the Eilenberg-Steenrod axioms, which formalize the axiomatic framework for homology and cohomology theories.
Eilenberg-Steenrod Axioms: The Eilenberg-Steenrod axioms are a set of properties that define the category of singular homology in algebraic topology. They serve as a foundation for homological algebra by establishing how functors behave with respect to topological spaces, ensuring consistency and a systematic approach to deriving properties of topological invariants. These axioms connect to various concepts in topology and homological algebra, providing essential tools for understanding cellular homology and derived functors.
Exactness Axiom: The exactness axiom is a fundamental principle in homological algebra that ensures the preservation of exact sequences under certain operations, such as taking products or coproducts. This axiom plays a crucial role in the formulation of functorial properties and the behavior of derived functors, which are essential for understanding the relationships between different algebraic structures.
Excision Axiom: The excision axiom is a fundamental principle in algebraic topology that states if a space can be decomposed into two parts, then the homology groups of the entire space can be computed from those of the subspaces by 'removing' a smaller subspace. This axiom emphasizes the importance of local properties in determining global behavior and supports the idea that certain topological features can be ignored when computing homological invariants.
Homology Group: A homology group is an algebraic structure that captures the topological features of a space by associating a sequence of abelian groups or modules to a topological space. These groups help in understanding the shape and structure of the space by measuring its holes in different dimensions, which are critical for applying the Eilenberg-Steenrod axioms that provide a foundational framework for homology theories.
Homology Theory: Homology theory is a mathematical framework that studies topological spaces and algebraic structures through the use of chains, cycles, and boundaries to quantify their shape and structure. This approach allows mathematicians to derive invariants that classify spaces up to homotopy equivalence, revealing deeper properties of those spaces. It serves as a cornerstone for connecting algebraic topology with other branches of mathematics, providing insights into complex concepts such as cellular structures and axiomatic foundations.
Homotopy: Homotopy is a concept in algebraic topology that describes a continuous deformation between two continuous functions or paths. In simple terms, if you can continuously transform one function into another without breaking or tearing, they are said to be homotopic. This idea is fundamental in understanding shapes and spaces, providing a way to classify them based on their structure and properties.
Homotopy equivalence: Homotopy equivalence is a relationship between two topological spaces that indicates they can be transformed into one another through continuous deformations, known as homotopies. This concept implies that the two spaces share the same topological properties and allows one to transfer information between them, particularly in the study of homology and cohomology, where it helps establish isomorphisms of their respective algebraic structures. Additionally, homotopy equivalence serves as a crucial foundation for understanding more complex relationships in homological algebra and the application of the Eilenberg-Steenrod axioms.
Homotopy type: Homotopy type refers to a classification of topological spaces based on their homotopy equivalence, meaning that spaces that can be continuously transformed into one another through deformation belong to the same homotopy type. This concept plays a crucial role in understanding the shape and structure of spaces, providing a framework for comparing and relating different topological constructs in both algebra and topology.
Isomorphic: Isomorphic refers to a concept in mathematics where two structures can be mapped onto each other in such a way that the operations and relationships of the structures are preserved. In the context of algebraic structures like groups, rings, or topological spaces, isomorphism indicates that the two structures are essentially the same, even if they may appear different at first glance.
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.
Natural Isomorphism: A natural isomorphism is a type of isomorphism between functors that preserves the structure in a coherent way, meaning that the isomorphisms can be chosen 'naturally' with respect to the morphisms of the categories involved. This concept connects deeply with how functors relate to each other and allows for the transfer of properties across categories while maintaining their relationships. Understanding natural isomorphisms is crucial for comprehending transformations, adjoint functors, derived functors, and axioms in homological algebra.
Simplicial Homology: Simplicial homology is a mathematical tool used to study topological spaces by associating sequences of abelian groups or modules to simplicial complexes, allowing for the calculation of various algebraic invariants. This method captures information about the shape and connectivity of a space through its simplicial structure and provides a way to derive homological properties. The relationship between simplicial homology and chain complexes is essential for establishing algebraic structures that satisfy the Eilenberg-Steenrod axioms, which formalize the foundational properties of homology theories.
Singular homology: Singular homology is an algebraic topology concept that assigns a sequence of abelian groups or modules to a topological space, capturing its shape and structure through singular simplices. This tool is essential for understanding various properties of spaces, including their connectivity and the presence of holes, and plays a crucial role in linking algebraic and topological concepts.
Uniqueness Theorem: The uniqueness theorem in the context of the Eilenberg-Steenrod axioms states that a homology theory satisfying these axioms is uniquely determined by its values on a certain class of spaces, typically the CW complexes. This theorem emphasizes that if two homology theories agree on a collection of spaces, they must coincide on all spaces that can be derived from those by taking limits. The result highlights the powerful implications of the Eilenberg-Steenrod axioms in categorizing homology theories.
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