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Eilenberg–Steenrod axioms of homology

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The Eilenberg–Steenrod axioms apply to homology as a link between topology and algebra.

Suppose

  • ${\mathscr Top}$ is the category of all pairs of topological spaces with continuous maps as morphisms;
  • ${\mathscr Mod}$ is the category of $R$-modules (alternatively we can use ${\mathscr Ab}$ is the category of abelian groups with homomorphisms as morphisms).

Then

  • homology theory ${\mathscr Hom}$ is a sequence of functors $\{H_m\}$ from ${\mathscr Top}$ to ${\mathscr Ab}$.

Notation for the functor:

  • $(X,A) \leadsto H_m(X,A)$, with $H_{m}(A)=H_{m}(A,\emptyset)$;
  • $\big( f:(X, A) \rightarrow (Y,B) \big) \leadsto \big( f_m:H_m(X, A) \rightarrow H_m(Y,B) \big).$

Collectively:

  • $H(f):H(X, A) \rightarrow H(Y,B).$

The five axioms are:

1. Homotopy: Homotopic maps induce the same map in homology. That is, if two maps are homotopic: $$g \sim h:(X, A) \rightarrow (Y,B),$$ then their homology maps are the same: $$H(g) = H(h):H(X, A) \rightarrow H(Y,B).$$

2. Excision: Cutting out open sets doesn't change the homology. That is, if $(X,A)$ is a pair and $U$ is a subset of $X$ such that the closure of $U$ is contained in the interior of $A$, then the inclusion map $$i : (X-U, A-U) \to (X, A)$$ induces an isomorphism in homology: $$H(i) : H(X-U, A-U) \to H(X, A)$$

3. Dimension: The homology of the one-point space $P$ is acyclic. That is, $$H_n(P) = 0,n \neq 0.$$

4. Additivity: Homology is additive. That is, if space is the disjoint union of a family of topological spaces $\{X_{\alpha}\}$: $$X = \coprod_{\alpha}{X_{\alpha}},$$ then its homology is the direct sum of their homologies: $$H_m(X) \cong \bigoplus_{\alpha} H_m(X_{\alpha}).$$

5. Exactness: For each pair $(X, A)$ there are the connecting homomorphisms $$\partial _m: H_m (X,A) \rightarrow H_{m-1}(A),m=1,2,...,$$ with the following property: for any map of pairs $$f:(X,A) \rightarrow (Y,B)$$ the following commutative diagram has "long exact sequences" as rows: $$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{ccccccccccc} & ... & \ra{} & H_m(A) & \ra{i_m} & H_m(X) & \ra{j_m} & H_m(X,A) & \ra{\partial _m} & H_{m-1}(A) & \ra{} & ... \\ & & & \da{(f|_A)_m} & & \da{f_{m}} & & \da{f_{m}} & & \da{(f|_A)_{m-1}} &\\ & ... & \ra{} & H_m(B) & \ra{i_m} & H_m(Y) & \ra{j_m} & H_m(Y,B) & \ra{\partial _m} & H_{m-1}(B) & \ra{} & ... \\ \end{array} $$ where $i$ and $j$ are the inclusions.

In the cohomological analogue of these axioms, the arrows are reversed.