§ 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.