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The following result is analogous to (a 'dual' of) the (first two parts of) the [[The Thom isomorphism theorem]] for [[orientation of a vector bundle|oriented]] [[vector bundle|vector bundles]].
$R$ is a (say, [[commutative ring|commutative]]) [[ring]].
> [!theorem] Theorem. ([[The Thom Theorem for oriented manifolds]])
> Suppose $M$ is an [[manifold]] of dimension $d$ with $R$-[[(homological) orientation of a manifold|orientation]] $\{ \mu_{x} \}_{x \in M}$. Suppose $A \subset M$ is a [[compact]] [[subspace topology|subspace]]. Then:
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>1. There is a unique class $\mu_{A} \in H_{d}(M | A; R)$ which restricts[^1] to $\mu_{x} \in H_{d}(M | x;R)$ for each $x \in A$.
>2. $H_{i}(M | A; R)=0$ for all $i > d$.[^2]
^theorem
> [!proposition] Corollary. (Fundamental class)
> If $M$ is a [[compact]] $d$-[[manifold]] with $R$-[[(homological) orientation of a manifold|orientation]] $\{ \mu_{x} \}_{x \in M}$, then there exists a unique $[M]:=\mu_{M} \in H_{d}(M | M; R)=H_{d}(M ; R)$ that restricts to $\mu_{x}$ at every $x \in M$. This is called the **fundamental class** of $M$.
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> > [!note] Note. (maybe)
> > Should $M$ moreover be [[connected]], so $H^{0}(M;R)=R$, then by [[Poincare duality]] $H_{d}(M;R) \cong H^{0}(M;R) \cong R$. In this case, the fundamental class $[M] \in H_{d}(M)$ may be viewed as an 'orientation-canonical generator for top-homology $H_{d}(M;R)