Not every integral homology class is represented by an embedded submanifold

The geometric pairing and the self-intersection statements of this page apply to closed oriented embedded submanifolds, not to arbitrary homology classes. Not every integral homology class of a closed oriented manifold is the image of the fundamental class of a closed oriented manifold under a continuous map. Steenrod operations give obstructions to such integral realization, as in Thom's Chapter III, Section 4 and its lens-space product example (printed pp. 59-62). With ${\mathbb{F}}_2$ coefficients every class is represented by the mod-two fundamental class of a closed manifold under a map (Thom, Theorem III.2; Cohen's remark after Theorem 9.5) (The geometric intersection number is the Poincare-dual cup pairing requires the geometric representatives to exist before it can be applied). The algebraic pairing of the cup product remains the general object, defined for all classes; where the design uses the geometric model it must either supply embedded representatives or pass to the algebraic statement. Cohen's remark after Theorem 9.5 records the integral and mod-two representability caveat; the Steenrod-operation obstruction is separately sourced to Thom. Realization by a map does not assert realization by an embedding.