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Eilenberg–Steenrod Axioms

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Homotopy Axiom

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If f,g:XYf, g: X \rightarrow Y are homotopic maps, then their induced homomorphisms on homology groups are equal: f=g:Hn(X)Hn(Y)f_* = g_*: H_n(X) \rightarrow H_n(Y) for all nn. This signifies that homology is a topological invariant insensitive to continuous deformations.

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Excision Axiom

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For a subset AXA \subseteq X whose closure is contained in the interior of another subset BXB \subseteq X, the inclusion (XA,BA)(X,B)(X \setminus A, B \setminus A) \hookrightarrow (X,B) induces an isomorphism in homology: Hn(XA,BA)Hn(X,B)H_n(X \setminus A, B \setminus A) \cong H_n(X,B) for all nn. This axiom shows that the homology depends only on the local structure, not the set AA itself.

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Exactness Axiom

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For any pair (X,A)(X,A), where AA is a subspace of XX, there is a long exact sequence

Hn(A)Hn(X)Hn(X,A)Hn1(A)\cdots \rightarrow H_n(A) \rightarrow H_n(X) \rightarrow H_n(X,A) \rightarrow H_{n-1}(A) \rightarrow \cdots
of homology groups, demonstrating how the inclusion of a subspace affects the homology.

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Dimension Axiom

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For a one-point space {pt}\{pt\}, the nnth homology group is trivial for all n>0n>0: Hn({pt})=0H_n(\{pt\}) = 0 for n>0n>0 and H0({pt})=ZH_0(\{pt\}) = \mathbb{Z}. This axiom establishes the initial case for building up the theory of homology groups.

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Additivity Axiom

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For a disjoint union iIXi\sqcup_{i \in I} X_i of spaces XiX_i, the nnth homology group is the direct sum of the homology groups of the pieces: Hn(iIXi)=iIHn(Xi)H_n(\sqcup_{i \in I} X_i) = \bigoplus_{i \in I} H_n(X_i) for all nn. This axiom allows us to compute homology for disjoint unions piecewise.

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Functoriality

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If f:(X,A)(Y,B)f: (X,A) \rightarrow (Y,B) is a continuous map, then it induces a homomorphism f:Hn(X,A)Hn(Y,B)f_*: H_n(X,A) \rightarrow H_n(Y,B) for each nn, which aligns with composition of functions ((gf)=gf (g \circ f)_* = g_* \circ f_*). This axiom ensures that homology behaves predictably under continuous maps.

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Homology of a Point

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For a one-point space {pt}\{pt\}, the only non-trivial homology group is H0({pt})=ZH_0(\{pt\}) = \mathbb{Z}, which indicates that we can 'begin' homology theory with a simple familiar object, the point.

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