R-orientation of a topological manifold

Let $R$ be a commutative ring as in Commutative ring, with specified multiplicative identity $1$, and let $M$ be a boundaryless $n$-manifold. Its local $R$-homology system has fiber ${\mathcal{O}}_{R,x}=H_n(M,M\setminus \{x\};R)\mathrm{.}$ Construct its topology by the rule of Orientation local system and orientation cover, now using all classes in $H_n(M,M\setminus K;R)$ for closed coordinate balls $K$. The restriction isomorphisms needed for this rule hold over $R$ by Coordinate-ball classes identify local homology stalks. Since $M$ is boundaryless, Local homology detects manifold dimension, interior, and boundary computes every displayed fiber as $R$ in degree $n$; in particular each fiber is a free rank-one $R$-module.

For completeness, two such ball sections that agree at a point agree on a smaller ball: restrict both classes to a closed coordinate ball inside the intersection, then use injectivity of its restriction to that point and functoriality for all other points in its interior. Consequently the sheet images form a basis, and over a ball the system is the product of its interior with the discrete module of ball classes. This is the same explicit basis verification as for the integral system. It also shows that the group and $R$-module operations are compatible with these local charts.

An $R$-orientation is a continuous section $\mu :x\mapsto {\mu }_x$ of this system such that ${\mu }_x$ generates ${\mathcal{O}}_{R,x}$ as an $R$-module at every point. In a trivialization with a chosen module generator $e$, a candidate generator is $ue$. It generates exactly when $u$ is a unit: if it generates, $e=v(ue)$ for some $v\in R$, so $vu=1$; conversely if $vu=1$, then $e=v(ue)$ and it generates. Thus an orientation is locally a constant generator class; more precisely every point has a neighborhood on which the section is induced by a single generator in one ball group. Continuity implies this description because the module coordinate is discrete, and such local descriptions imply continuity by the product charts. There is no restriction to just two possible generators for a general coefficient ring.

For a manifold with boundary, an $R$-orientation means an $R$-orientation of its interior. The interior is an open boundaryless $n$-manifold: chart independence from the local-homology construction identifies it in each chart with the open strict half-space, and its open subspace inherits a countable basis and Hausdorff separation. This definition does not yet assign an orientation to the boundary; its induced sign belongs to the relative fundamental-class construction.

The empty manifold has the unique empty section. When $n=0$, the manifold is discrete, and orientation data specify a generator at each point. If $R$ is the zero ring, the zero module has its unique element as generator and $1=0$; there is a unique section for every $M$. For a nonzero ring the zero element is never a local generator. Orientation data can be restricted to components, and any supplied family of component orientations will be glued in the following proposition. No selection from a family of nonempty orientation sets is assumed here, and no AC is used.