Orientability and the Fundamental Class

Poincare duality requires the notion of orientation for manifolds, formalized through the concept of a fundamental class in homology. We develop these concepts from the perspective of local homology.

Local Homology and Orientation

Definition

Let $M$ be a topological $n$ -manifold and $x \in M$ . The local homology group at $x$ is $H_n(M, M \setminus \{x\}; \mathbb{Z}) \cong \mathbb{Z}$ A local orientation at $x$ is a choice of generator $\mu_x$ of this group. An orientation of $M$ is a continuous choice of local orientations $\{\mu_x\}_{x \in M}$ , meaning that for each $x$ there exists an open neighborhood $U$ and a class $\mu_U \in H_n(M, M \setminus U)$ whose image in $H_n(M, M \setminus \{y\})$ is $\mu_y$ for all $y \in U$ .

The continuity condition distinguishes global orientations from arbitrary local choices and ensures coherence across the manifold.

Definition

A connected $n$ -manifold $M$ is orientable if it admits an orientation. Equivalently, $M$ is orientable if and only if the orientation double cover $\tilde{M} \to M$ is disconnected. A manifold together with a chosen orientation is called an oriented manifold.

The Fundamental Class

Theorem8.1Existence of the Fundamental Class

Let $M$ be a closed (compact, without boundary), connected, oriented $n$ -manifold. Then $H_n(M; \mathbb{Z}) \cong \mathbb{Z}$ , and there exists a unique generator $[M] \in H_n(M; \mathbb{Z})$ , called the fundamental class, whose image in each local homology group $H_n(M, M \setminus \{x\})$ agrees with the chosen local orientation $\mu_x$ .

ExampleFundamental classes of spheres

The sphere $S^n$ is orientable for all $n$ , with $H_n(S^n) \cong \mathbb{Z}$ . The standard orientation gives a fundamental class $[S^n]$ that can be represented by the sum of all top-dimensional simplices in a triangulation, with signs determined by the orientation.

Non-Orientable Manifolds

Remark$\mathbb{Z}/2$ coefficients

Every closed connected $n$ -manifold, orientable or not, possesses a fundamental class $[M] \in H_n(M; \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$ . Poincare duality with $\mathbb{Z}/2$ coefficients therefore holds for all closed manifolds, not just orientable ones. The real projective space $\mathbb{RP}^n$ demonstrates this: it is non-orientable for even $n$ but always carries a $\mathbb{Z}/2$ fundamental class.

The fundamental class is the starting point for Poincare duality, serving as the element against which cohomology classes are "capped" to produce the duality isomorphism.