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Let $A$ be a nonempty union of disjoint embedded circles in a closed oriented topological surface ${\Sigma }_g$, and let $c$ be the number of components after cutting along them. Excision and Poincare-Lefschetz duality identify the relative homology with the complementary-degree cohomology of the cut surface. Its components have boundary, so the relative groups are free, with the displayed ranks; $H_0$ and higher groups vanish. The rank of the circle classes in $H_1({\Sigma }_g)$ is $|{\pi }_0(A)|+1-c$. For a single circle these groups detect whether its class is zero, but for genus at least two cannot distinguish a disk boundary from an essential separating circle.

For a map of oriented disk pairs, the multiplier on top relative homology is its relative mapping degree. Naturality of the connecting homomorphism identifies it with the mapping degree of the boundary map.

If $L$ is a simplicial subcomplex of $K$, thenIts homology is the simplicial relative homology $H_k(K,L)$.

For a compact oriented three-manifold, the integral relative class $u$ has norm the minimum of ${\chi }_{-}(F)={\sum }_j\max (0,-\chi (F_j))$ over properly embedded oriented topological surfaces representing $u$. Homogeneity and continuity extend it to real relative homology. It is a seminorm, since spheres, disks, annuli and tori have zero cost.

A cobordism is trivial when it is diffeomorphic to a cylinder with the prescribed incoming boundary identification. Both boundary inclusions are homotopy equivalences and its relative homology with respect to either end vanishes. It can have critical points for a chosen cobordism Morse function; those must form an acyclic relative Morse chain complex.