Fundamental class of a compact oriented manifold

Let $M$ be a compact boundaryless $n$-manifold with a specified $R$-orientation $\mu$, where $R$ is a commutative unital ring. Its fundamental class is $[M]=[M{]}_M\in H_n(M,M\setminus M;R)=H_n(M;R),$ using Compatible orientation classes over compact subsets with the compact support $K=M$. The final equality holds at the chain-complex level: $C_{\ast }(\mathrm{\varnothing };R)=0$, so quotienting by it leaves $C_{\ast }(M;R)$. The cited lemma proves existence and uniqueness of this class with point restriction ${\mu }_x$ at every $x\in M$. Thus the notation denotes a class determined by the supplied orientation, not an arbitrary generator chosen afterward.

The same lemma gives finitely many open-and-closed components $M_1,\dots ,M_k$ and the restriction isomorphism $H_n(M;R)⟶{⨁}_{j=1}^k{H}_n(M,M\setminus M_j;R)\mathrm{.}$ The summand on the right identifies with $H_n(M_j;R)$: apply Excision for singular homology to remove the closed set $M\setminus M_j$, which is the entire open relative subspace. The resulting pair is $(M_j,\mathrm{\varnothing })$. Under this identification $[M]$ corresponds to the finite tuple of the component fundamental classes.

If the orientation is negated on one component, its new fundamental-class summand is the negative of its old one. Indeed restriction is linear, so the negative class restricts to $-{\mu }_x$ at all points of that component, and uniqueness in the compact-set lemma identifies it with the new class. The other component summands are unchanged. In characteristic two this negation can leave the orientation and class unchanged; no distinction between $\mu$ and $-\mu$ is imposed when they are equal.

For empty $M$ the fundamental class is $0$ in the zero homology group, with an empty direct sum. For $R=0$ the unique orientation gives the zero class. A compact zero-manifold has finitely many singleton components, and its fundamental zero-cycle has at each point the coefficient prescribed by its local module generator. The definition requires a supplied orientation and uses no AC or simultaneous choice of one orientation per component.