# algebraic topology – Relative cap product

I’d like to understand relative cap product based on my notes. We want a map $$H^i(X,A) otimes H_n(X,A cup B) longmapsto H_{n-i}(X,B)$$

In order to understand the construction, we want to give a map $$C_bullet(X,A cup B) overset{tilde{Delta}}{longmapsto} C_bullet(X,A) otimes C_bullet(X,B)$$.

In fact, we can consider $$0 longmapsto C_bullet(A) longmapsto C_bullet(X) longmapsto C_bullet(X,A) longmapsto 0$$ and tensor with $$C_bullet(X,B)$$ to get

$$0 longmapsto C_bullet(A) otimes C_bullet(X,B) longmapsto C_bullet(X)otimes C(X,B) longmapsto C_bullet(X,A) otimes C_bullet(X,B) longmapsto 0$$

Now consider the following commutative diagram

$$begin{array}{ccccccccc} C_bullet(A) & to & C_bullet(X,B) & to & C_bullet(X,A cup B) & to & 0 \ downarrow{Delta} & & downarrow{Delta} & & downarrow{Delta} \ C_bullet(A times (X,B)) & & C_bullet(A times(X,B)) & & C_bullet((X,A)times (X,B)) \ downarrow{EZ} & & downarrow{EZ} & & downarrow{EZ} \ 0 to C_bullet(A) otimes C_bullet(X,B) &to & C_bullet(X)otimes C(X,B) & to & C_bullet(X,A) otimes C_bullet(X,B) & to 0 end{array}$$

I have the following questions:

$$0$$. Why having the map $$C_bullet(X,A cup B) overset{tilde{Delta}}{longmapsto} C_bullet(X,A) otimes C_bullet(X,B)$$ we end up with the cap product $$H^i(X,A) otimes H_n(X,A cup B) longmapsto H_{n-i}(X,B)$$ ?

$$1$$. The sequence on the first line how is it build? I think the map $$C_bullet(X,B) longmapsto C_bullet(X,A cup B)$$ is surjective, is this enough in order to get the sequence?

$$2$$. Why the diagram is commutative and how this could help to prove $$0$$?

$$3$$. The $$EZ$$ in the diagram, seem the relative Eilenberg Zielber maps (reference here).

If so, are $$C_bullet((X,A) times (Y,B)) = C_bullet(X times Y) / (C_bullet(A times Y)+ Cbullet(Xtimes B))$$ and $$C_bullet(A times (X,B))$$ defined as the previous with $$A = (A,varnothing)$$?

$$4$$. Are the vertical $$Delta$$s in reality the diagonal map composed with a projection?

Any help or reference would be appreciated.