Non-abelian homology

Non-abelian homology is a branch of mathematics that studies shapes and spaces using structures where operations do not necessarily commute—meaning the order of operations matters. Unlike traditional homology, which uses abelian (commutative) groups to analyze shapes, non-abelian homology employs more complex, non-commutative groups to capture richer, more nuanced symmetries and properties of geometric objects. This approach allows mathematicians to explore spaces with intricate twisting and linking behaviors that cannot be fully understood using simpler, commutative methods alone.