I have read the Wikipedia definition of cap product. Here is the Wikipedia definition that I know. https://en.wikipedia.org/wiki/Cap_product

What I understand: For a good space, for example, an orientable closed $n$ manifold using the Poincare Duality and definition of cup product, I can convince myself that there should be such a map, but I don't understand what this bilinear map actually mean. Can anyone please clarify the definition? I just want to know what motivates us to define cap product, and what actually it means.

Thank you very much.

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    $\begingroup$ Do you want to know what a bilinear map is? $\endgroup$ Dec 6, 2019 at 5:15
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    $\begingroup$ What would it "actually" mean that isn't in the definition? $\endgroup$
    – anomaly
    Dec 6, 2019 at 5:20
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    $\begingroup$ @anomaly There are plenty of obfuscating definitions that have illuminating geometric interpretations. I would say that geometry is what the definition could “actually” mean. $\endgroup$ Dec 6, 2019 at 5:41
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    $\begingroup$ @MMM I’m not comfortable enough with this to write a cohesive answer, but there’s a sense in which the cap product describes a geometric intersection (using Poincaré duality). You may appreciate this discussion over at MO. $\endgroup$ Dec 6, 2019 at 6:15
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    $\begingroup$ It basically means integration. If you have a manifold than cap product can be interpreted in terms of deRham cohomology. Cohomological classes are represented by differential forms and you integrate these forms along certain cycles on a manifold. This approach could be extended to a general situation using en.wikipedia.org/wiki/Rational_homotopy_theory $\endgroup$ Dec 6, 2019 at 6:18


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