10.4 Eilenberg-Steenrod axioms

The Eilenberg-Steenrod axioms are the foundation of homology theory 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 homotopy equivalence 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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• Homotopy 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 homotopy type (spaces that are homotopy equivalent have isomorphic 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 homology group $H_n(pt)$ is isomorphic to $\mathbb{Z}$ for $n=0$ and is trivial for $n>0$
  • Provides a starting point for calculating homology groups using the other axioms

Excision and Additivity

• Excision axiom relates the homology of a space $X$ to the homology of a subspace $A \subset X$ and its closure
  • States that if the closure of $A$ is contained in the interior of a subspace $U \subset X$, then the inclusion $(X \setminus A, U \setminus A) \hookrightarrow (X,U)$ induces isomorphisms on homology groups
  • Allows for the calculation of homology groups by decomposing a space into smaller, simpler pieces
• Additivity axiom states that the homology of a disjoint union of spaces is isomorphic to the direct sum of the homology of each space
  • Formally, if $X = \bigsqcup_{\alpha} X_{\alpha}$, then $H_n(X) \cong \bigoplus_{\alpha} H_n(X_{\alpha})$ for all $n$
  • Enables the computation of homology groups of a space by breaking it down into its connected components

Exactness and Long Exact Sequence

• Exactness axiom relates the homology of a space, a subspace, and the corresponding quotient space
  • For a pair $(X,A)$ with $A \subset X$, there is a long exact sequence of homology groups: $\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \to \cdots$
  • Connects the homology groups of different spaces and allows for their computation using the properties of exact sequences
  • The connecting homomorphism $H_n(X,A) \to H_{n-1}(A)$ is induced by the boundary operator in the chain complex

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 simplicial homology, singular homology, and cellular homology
• Uniqueness theorem 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.)