# Computing Massey products via intersection theory

Let $$K$$ be an $$n$$-manifold with boundary and let $$x,y,z \in H^*(K)$$ be cohomology classes with $$x\cup y=y\cup z=0$$.

The Massey product $$\langle x,y,z \rangle$$ is defined as the set of cohomology classes $$[a\cup \tilde z - (-1)^{\deg (x)}\tilde x \cup b]$$, where $$a,b$$ are cochains with $$\partial a=\tilde x\cup\tilde y$$, $$\partial b=\tilde y \cup\tilde z$$ and $$\tilde x,\tilde y, \tilde z$$ are representatives of the cohomology classes $$x,y,z$$.

In his paper "higher order linking numbers" Massey uses another approach to compute elements of Massey products using the Poincaré-duality of cohomology+cup-products and homology+intersection of manifolds:

Let the fundamental classes $$[M],[N],[P] \in H_*(K,\partial K)$$ be the dual classes of the cohomology classes $$x,y,z$$, such that $$X\cap P$$ and $$M \cap Y$$ intersect transversally, for $$X,Y$$ manifolds with $$M⫛ N= \partial X$$ and $$N ⫛ P= \partial Y$$.

Then the sum $$X \cap P - (-1)^{n-\deg(x)}M\cap Y$$ obviously represents the Poincaré-dual of a triple product.

How exactly does this obvious duality work?

In my attempt I didnt get very far...

Using Bredon's intersection theory I get $$[M \cap N]= D(x \cup y)$$, for $$D:H^*(K,\partial K) \to H_{n-*}(K)$$ the duality isomorphism. This implies, that the cap product of a representative of $$[K]$$ and $$\tilde x \cup \tilde y$$ will be send to a representative of $$[M \cap N]$$, which we call $$r_{mn}$$. Since $$x\cup y = 0$$ it follows that $$[M\cap N]=0$$ and therefore $$r_{m,n}$$ is the boundary of a non-cyclic chain $$r$$.

Since this chain is non-cyclic, it is not a representative of any homology class. Thus I dont see any way to continue, using the duality $$[M \cap N]= D(x \cup y)$$.