Eilenberg–Steenrod axioms

In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are important rules that many homology theories of topological spaces follow. These rules help mathematicians understand the shape and structure of spaces by looking at their holes and loops in a more organized way.

The most famous example of a homology theory that follows these axioms is singular homology, created by mathematicians Samuel Eilenberg and Norman Steenrod. These axioms were developed in 1945 and give a clear framework for studying homology theories.

Using these axioms, mathematicians can prove important results that apply to all homology theories that follow the rules, like the Mayer–Vietoris sequence. If we ignore one of the rules, called the dimension axiom, we get something called an extraordinary homology theory. These special theories first appeared in areas like K-theory and cobordism.

Formal definition

The Eilenberg–Steenrod axioms describe important rules for homology theories in mathematics. These rules help us understand the shape and structure of spaces by looking at their properties in different dimensions.

The axioms include ideas like homotopy, which says that maps that can be continuously deformed into each other give the same results. There is also exactness, which creates a chain of connections between different homology groups. These rules help mathematicians study complex shapes in a systematic way.

Consequences

From the Eilenberg–Steenrod axioms, we can learn important facts about homology groups. For example, spaces that are homotopically equivalent have the same homology groups.

We can also use these axioms to find the homology of simple shapes like n-spheres. This helps prove that an (n − 1)-sphere cannot be a retract of an n-disk, which is a key step in proving the Brouwer fixed point theorem.

Dimension axiom

Some special types of homology theories are called "extraordinary homology theories." These are theories that follow most of the Eilenberg–Steenrod axioms but not all of them. Important examples from the 1950s include topological K-theory and cobordism theory. These are types of cohomology theories, which have related homology theories.