Eilenberg–Steenrod axioms

The Eilenberg–Steenrod axioms apply to a sequence of functors ${\displaystyle H_{n}}$ from the category of pairs ${\displaystyle (X,A)}$ of topological spaces to the category of abelian groups, together with a natural transformation ${\displaystyle \partial \colon H_{i}(X,A)\to H_{i-1}(A)}$ called the boundary map (here ${\displaystyle H_{i-1}(A)}$ is a shorthand for ${\displaystyle H_{i-1}(A,\emptyset )}$ . The axioms are:

1. Homotopy: Homotopic maps induce the same map in homology. That is, if ${\displaystyle g\colon (X,A)\rightarrow (Y,B)}$ is homotopic to ${\displaystyle h\colon (X,A)\rightarrow (Y,B)}$, then their induced homomorphisms are the same.
2. Excision: If ${\displaystyle (X,A)}$ is a pair and U is a subset of A such that the closure of U is contained in the interior of A, then the inclusion map ${\displaystyle i\colon (X\setminus U,A\setminus U)\to (X,A)}$ induces an isomorphism in homology.
3. Dimension: Let P be the one-point space; then ${\displaystyle H_{n}(P)=0}$ for all ${\displaystyle n\neq 0}$.
4. Additivity: If ${\displaystyle X=\coprod _{\alpha }{X_{\alpha }}}$, the disjoint union of a family of topological spaces ${\displaystyle X_{\alpha }}$, then ${\displaystyle H_{n}(X)\cong \bigoplus _{\alpha }H_{n}(X_{\alpha }).}$
5. Exactness: Each pair (X, A) induces a long exact sequence in homology, via the inclusions ${\displaystyle i\colon A\to X}$ and ${\displaystyle j\colon X\to (X,A)}$:

${\displaystyle \cdots \to H_{n}(A)\,{\xrightarrow {i_{*}}}\,H_{n}(X)\,{\xrightarrow {j_{*}}}\,H_{n}(X,A)\,{\xrightarrow {\partial }}\,H_{n-1}(A)\to \cdots .}$

If P is the one point space, then ${\displaystyle H_{0}(P)}$ is called the coefficient group. For example, singular homology (taken with integer coefficients, as is most common) has as coefficients the integers.

Some facts about homology groups can be derived directly from the axioms, such as the fact that homotopically equivalent spaces have isomorphic homology groups.

The homology of some relatively simple spaces, such as n-spheres, can be calculated directly from the axioms. From this it can be easily shown that the (n − 1)-sphere is not a retract of the n-disk. This is used in a proof of the Brouwer fixed point theorem.

A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an extraordinary homology theory (dually, extraordinary cohomology theory). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory, which are extraordinary cohomology theories, and come with homology theories dual to them.

• Zig-zag lemma

• Eilenberg, Samuel; Steenrod, Norman E. (1945). "Axiomatic approach to homology theory". Proceedings of the National Academy of Sciences of the United States of America. 31 (4): 117–120. Bibcode:1945PNAS...31..117E. doi:10.1073/pnas.31.4.117. MR 0012228. PMC 1078770. PMID 16578143.
• Eilenberg, Samuel; Steenrod, Norman E. (1952). Foundations of algebraic topology. Princeton, New Jersey: Princeton University Press. MR 0050886.
• Bredon, Glen (1993). Topology and Geometry. Graduate Texts in Mathematics. Vol. 139. New York: Springer-Verlag. doi:10.1007/978-1-4757-6848-0. ISBN 0-387-97926-3. MR 1224675.