Eilenbergg

A lightweight algebraic topology application. Eilenberg-Steenrod Axioms Tool** brings core ideas from axiomatic homology theory into a focused mobile workflow. From the home screen you can switch between Singular, Cellular, Generalized, and Axiomatic homology perspectives, then dive into dedicated modules for each topic.  The Axiom Verification module lets you enter a space pair (X, A), pick coefficients (Z, Z/2, Q), set the working dimension with a stepper, toggle individual axioms, and run batch checks with pass/check feedback.  Space Pair Input combines text fields, a space catalog picker, and switches for excision hypotheses to inspect centralized chain complexes and relative groups H_n(X,A).  Long Exact Sequence builds the fragment … → H_n(A) → H_n(X) → H_n(X,A) → H_{n-1}(A) → … with live updates as you change n or coefficients. The CW Complex Builder uses steppers for cell counts c_0…c_4 and computes cellular homology; Singular vs Cellular compares both functors on standard spaces. Additional sections cover generalized homology (K-theory, bordism, BP) and a filterable axiomatic reference.