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Cap product with the locally finite fundamental class of an $n$-manifold without boundary gives the displayed duality, including a possibly nontrivial orientation local system ${\mathcal{O}}_M$. It is the noncompact version of Poincare duality; for a compact manifold Borel-Moore homology is ordinary homology. Locally it is the relative fundamental-class calculation in an oriented coordinate ball; compatibility on overlaps and the Mayer–Vietoris sequence give the global statement. Since the chain complex of locally finite homology is zero in negative degrees, this duality implies $H^k(M;\mathbb{Z})=0$ for $k>n$.

A simplicial cycle is a simplicial chain with zero boundary. Its homology class is unchanged on addition of a boundary; in the Mayer–Vietoris sequence, splitting a cycle between two subcomplexes constructs the connecting map.

For an open cover, take the subcomplex generated by singular simplices whose images lie in single cover members. Repeated barycentric subdivision makes each finite singular chain small, and the standard subdivision chain homotopy shows that inclusion induces homology isomorphisms. For a two-set cover, the short exact sequence of these small chains gives the Mayer–Vietoris sequence; excising the complement of one open set from the pair with the other set gives its equivalent relative-homology formulation.

For an oriented rank-four real vector bundle $E\to S^4$, its sphere bundle has fibre $S^3$. Let $m=⟨e(E),[S^4]⟩$ be its Euler class evaluated on the orientation class. The Gysin sequence gives integral homology $\mathbb{Z}$ in degrees $0,7$, $\mathbb{Z}/m\mathbb{Z}$ in degree $3$, and a copy of $\mathbb{Z}$ in degree $4$ exactly when $m=0$, with all other groups zero. Here $\mathbb{Z}/0\mathbb{Z}=\mathbb{Z}$. A transition function $v\mapsto q^{k}v{q}^l$ on the unit quaternions has Euler number $k+l$, up to an overall orientation sign: evaluating it at a fixed unit vector gives the power map $q\mapsto q^{k+l}$, whose mapping degree is $k+l$. The same homology calculation follows by applying the Mayer–Vietoris sequence to the two trivializations over the hemispheres.

For a connected n-manifold without boundary, a finite good cover presents its top compactly supported cohomology by one generator per disc and relations $u_i=\pm u_j$ on overlaps. This follows from the compact-support Mayer–Vietoris sequence and vanishing above dimension n. The overlap graph is connected, so the group is cyclic. Coherent local orientations give $\mathbb{Z}$; an orientation-reversing loop gives $\mathbb{Z}/2$.