Relative fundamental class and boundary orientation

Let $M$ be a compact $R$-oriented $n$-manifold with boundary $A$, where $R$ is a commutative unital ring and the orientation is supplied on $M\setminus A$. Its relative fundamental class is the unique class $[M,A]\in H_n(M,A;R)$ whose restriction at every interior point is the prescribed local generator. Existence and uniqueness, including manifolds with closed components, are proved in A collar constructs the relative orientation class and its boundary class.

The induced boundary orientation is the orientation whose local generator at $x\in A$ is the restriction of $\mathrm{\partial }[M,A]$, where $\mathrm{\partial }$ is the homology pair connector. The same lemma proves that these restrictions are generators and form a continuous section. Thus $\mathrm{\partial }[M,A]=[A]$ for this orientation. In tangent-then-inward coordinates its generator is $(-1{)}^n$ times the tangent generator for the product orientation: this is the outward-normal-first convention. Each component inherits its sign from the supplied interior orientation; there is no further independent selection of signs.

If $A$ is empty, this is the absolute fundamental class and the boundary class is zero. In dimension zero the boundary is empty. Empty $M$ and the zero coefficient ring give the unique zero class. These definitions and their well-definedness use no AC.