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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The term "Cap product" can refer to different concepts depending on the context. Here are a few interpretations: 1. **In Finance**: "Cap" often refers to a limit or ceiling, especially in terms of investments or financial instruments. For example, a "cap rate" is a term used in real estate to indicate the rate of return on an investment property.