For an oriented closed connected -manifold, the fundamental class is the unique generator of that restricts to the orientation generator in every local homology group.
If a closed oriented manifold is covered by two open sets and a positive-degree cohomology class restricts to zero on both, then its cap product with the fundamental class vanishes in every degree strictly below the top. A chain proof uses the small simplex theorem and replaces the two restricted cocycles by coboundaries.
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In the context of mathematics, particularly in algebraic topology, the **fundamental class** refers to a specific object associated with a homology class of a manifold or a topological space. It is particularly significant in the study of dimensional homology. Here's a more detailed explanation: 1. **Homology Theory**: Homology is a mathematical concept used to study topological spaces through algebraic invariants. It provides a way to classify spaces based on their shapes and features like holes.