Massey Product and Its Applications

This is the note for my talk in Graduate Student Seminar NEU. W. Massey defined Massey product as a higher order cohomology operation, which is a generalization of cup product. A first application of Massey product is in Knot theory showing that Borromean rings are linked with zero linking numbers. As another application, people can use Massey product describe the differentials in spectral sequences. There are many other important applications, such as obstruction to the formality of a space, which may be omitted in this talk. 1. Massey Products 1.1. Cup Products review. Given a topological space X and a commutative ring R, we can define the cohomology groups H∗(X;R). There is also a product structure on H∗(X;R), called cup product ∪ : H(X;R) ∧H(X;R)→ H(X;R). Example 1.1. Let T = S1 × S1 be the torus. Then H0(T ;R) = R, H1(T ;R) = R2 generated by {a, b}, H2(T ;R) = R generated by {γ}. Then, we have a ∪ b = γ = −b ∪ a and the other cup products are zeros. In fact, the cohomology ring H∗(T ;R) of torus T is the exterior algebra ∧ R[a, b]. In general, H ∗(Tn;R) = ∧ R[a1, · · · , an]. Cup product is not easy to compute. However, many methods can be used to compute cup product, definition combined simplicial homology, method in Hatcher’s book, intersection method combined Poincare duality, product formula, even (Leray-Serre) spectral sequence etc.. Remark 1.2. Cup products are great because rings have more properties, and cup products can be used to distinguish between spaces that might have the same cohomology groups. In above example, the torus T = S1 × S1 can be distinguished from the the wedge sum of two circles and one spheres S1 ∨ S1 ∨ S2, even though they have the same homology groups, since all cup products in S1 ∨ S1 ∨ S2 are zeros. 1.2. Triple Massey Products. If x ∈ Ci(X;R), the symbol x will denote (−1)1+ix. We first define the Massey triple product. Let x1, x2 and x3 be cocycle of degrees r1, r2 and r3 with cohomology classes [x1] ∪ [x2] = 0 and [x2] ∪ [x3] = 0 . Thus, there are cochains x12 of degree r1 + r2 − 1 and x23 of degree r2 + r3 − 1 such that dx12 = x̄1 ∪ x2 and dx23 = x̄2 ∪ x3. Define the cochain ω of degree (r1 + r2 + r3 − 1) by ω = x12 ∪ x3 + x1 ∪ x23, Date: Oct. 4, 2012. 1