## § Cap product [TODO ]

- https://www.youtube.com/watch?v=oxthuLI8PQk

- We need an ordered simplex, so there is a total ordering on the vertices. This is to split a chain apart at number $k$.
- Takes $i$ cocahins and $k$ chains to spit out a $k - i$ chain given by $\xi \frown \gamma \equiv \sum_a \gamma_a \xi (a_{\leq i}) a_{\geq i}$.
- The action of the boundary on a cap product will be $\partial (\xi \frown \gamma) \equiv (-1)^i [(\xi \frown \partial \gamma) - (\partial \gamma \frown \gamma)]$.
- Consequence: cocycle cap cycle is cycle.
- coboundary cap cycle is boundary.
- cocyle cap boundary is boundary.
- Cap product will be zero if the chain misses the cochain.
- Cap product will be nonzero if the chain
*must * always intersect the cochain. - This is why it's also called as the intersection product, since it somehow counts intersections.