Fundamental class - Knowlpedia

Let $M$ be a connected closed $n$ -manifold with an orientation over a commutative coefficient ring $R$ . A fundamental class is the unique class

[M]\in H_n(M;R)

whose image under

H_n(M;R)\longrightarrow H_n(M,M\setminus\{x\};R)

is the local orientation generator at every $x\in M$ . Thus $[M]$ is the global homology class that consistently assembles all local choices of orientation.

Smooth manifolds

For a smooth manifold, an orientation of the tangent bundle determines the corresponding integral fundamental class.

Existence and variants

A connected closed manifold has an integral fundamental class exactly when it is orientable. Every closed manifold has a canonical mod- $2$ fundamental class. If $M$ has boundary, the oriented class lies instead in relative homology $H_n(M,\partial M;R)$ ; if $M$ is noncompact, the corresponding object belongs to locally finite homology. These variants preserve the same local-generator condition.

Pairings and consequences

The evaluation pairing sends a top-degree cohomology class $a$ to

\langle a,[M]\rangle\in R.

Cap product with $[M]$ is the map underlying Poincaré duality. Products of characteristic classes evaluated in this way produce characteristic numbers, while the Euler class of $TM$ evaluates to the Euler characteristic under the usual hypotheses.

Examples and orientation dependence

The standard orientation of $S^n$ for $n>0$ selects one of the two generators of $H_n(S^n;\mathbb Z)\cong\mathbb Z$ . Reversing the orientation replaces $[M]$ by $-[M]$ . For a closed nonorientable manifold, no integral class satisfies the local-generator condition, although its mod- $2$ fundamental class still exists.

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Author-hosted book record. Relevant: §3.3, orientations, fundamental classes, and Poincaré duality.