EilenbergSteenrodin

EilenbergSteenrodin is a concept in algebraic topology, specifically related to homology theories. It refers to a set of axioms that define a generalized homology theory. These axioms were formulated by Samuel Eilenberg and Norman Steenrod in their seminal work "Foundations of Algebraic Topology."

The EilenbergSteenrod axioms are fundamental to understanding the structure of homology. They typically include the dimension axiom, which states that the homology of a point is trivial in all dimensions except zero, where it is a copy of the ground ring. The exactness axiom describes the behavior of homology under a short exact sequence of spaces. The homotopy axiom asserts that homotopic maps induce the same homomorphism in homology. Finally, the additivity axiom (or wedge sum axiom) relates the homology of a disjoint union of spaces to the direct sum of their individual homologies.

These axioms provide a powerful framework for constructing and analyzing various homology theories, such as singular homology, cellular homology, and simplicial homology. By satisfying these axioms, a homology theory is guaranteed to have certain desirable properties. The EilenbergSteenrod theorem states that any homology theory satisfying these axioms is uniquely determined by its value on a single point, up to natural transformations. This theorem highlights the strength and universality of the axiomatic approach in algebraic topology.