H0ab

H0ab is a notation occasionally used in informal algebraic topology to denote the zeroth homology group with coefficients in an abelian group. In formal texts this object is written H_0(X; A), where X is a topological space and A is an abelian group. The abbreviation "H0ab" is not standard in rigorous references but can appear in lecture notes or slide decks to emphasize the coefficient group.

Definition and computation: For a space X and abelian group A, H_0(X; A) can be described via singular homology. It is a free module over A with a basis element for each path-connected component of X. Consequently, if X has c connected components, H_0(X; A) ≅ A^c. In particular, with integer coefficients H_0(X; Z) ≅ Z^c. When X is empty, H_0(∅; A) = 0.

Functoriality and interpretation: Continuous maps f: X → Y induce homomorphisms H_0(f; A): H_0(X; A) → H_0(Y; A). Intuitively, H_0 counts connected components; the rank of H_0(X; A) equals the number of components of X, and the induced map tracks how components map under f.

Examples: If X is two disjoint intervals, H_0(X; Z) ≅ Z ⊕ Z. If X is connected, H_0(X; Z) ≅ Z.

Notes: Reduced homology versions adjust H_0 by quotienting out a copy of A in the connected case. For precise usage, prefer H_0(X; A) notation in formal writing.

See also: H_0, singular homology, homology with coefficients, reduced homology.