Massey product

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In algebraic topology, the Massey product is a cohomology operation of higher order introduced in (Massey 1958).

Contents 1 Massey triple product 2 Higher order Massey products 3 Applications 4 See also 5 References

[edit] Massey triple product

In a differential graded algebra Γ with differential d, the cohomology H(Γ) is an algebra. Define to be (-1)^deg(u)+1u. The cohomology class of an element u of Γ will be denoted by [u]. The Massey triple product of three cohomology classes is defined by

The Massey product of 3 cohomology classes is not an element of H(Γ) but a set of elements of H(Γ), possibly empty and possibly containing more than one element.

The Massey product is nonempty if the products uv and vw are both exact, in which case all its elements are in the same element of the quotient group

So the Massey product can be regarded as a function defined on triples of classes such that the product of the first or last two is zero, taking values in the above quotient group.

[edit] Higher order Massey products

More generally the n-fold Massey product 〈a_1,1, a_2,2, ...,a_n,n〉 of n elements of H(Γ) is defined to be the set of elements of the form

for all solutions of the equations

, 1≤i≤j≤n, (i,j)≠(1,n).

In other words it can be thought of as the obstruction to solving the latter equations for all 1≤i≤j≤n, in the sense that it contains the 0 cohomology class if and only if these equations are solvable. This n-fold Massey product is an n−1 order cohomology operation, meaning that for it to be nonempty many lower order Massey operations have to contain 0, and moreover the cohomology classes it represents all differ by terms involving lower order operations. The 2-fold Massey product is just the usual cup product and is a first order cohomology operation, and the 3-fold Massey product is the same as the triple Massey product defined above and is a secondary cohomology operation.

May (1969) described a further generalization called Matric Massey products, which can be used to describe the differentials of the Eilenberg-Moore spectral sequence.

[edit] Applications

The complement of the Borromean rings gives an example where the triple Massey product is defined and non-zero. If u, v, and w are 1-cochains dual to the 3 rings, then the product of any two is a multiple of the corresponding linking number and is therefore zero, while the Massey product of all three elements is non-zero, showing that the Borromean rings are linked. The algebra reflects the geometry: the rings are pairwise unlinked, corresponding to the pairwise (2-fold) products vanishing, but are overall linked, corresponding to the 3-fold product not vanishing.

More generally, n-component Brunnian links – links such that any (n − 1)-component sublink is unlinked, but the overall n-component link is non-trivially linked – correspond to n-fold Massey products, with the unlinking of the (n − 1)-component sublink corresponding to the vanishing of the (n − 1)-fold Massey products, and the overall n-component linking corresponding to the non-vanishing of the n-fold Massey product.

Uehara & Massey (1957) used the Massey triple product to prove that the Whitehead product satisfies the Jacobi identity.

Massey products of higher order appear in the Atiyah–Hirzebruch spectral sequence (AHSS), which computes twisted K-theory with twist given by a 3-class H. Atiyah & Segal (2008) showed that rationally the higher order differentials

in the AHSS acting on a class x are given by the Massey product of p copies of H with a single copy of x.

Salvatore & Longoni (2005) use a Massey product to show that the homotopy type of the configuration space of two points in a lens space depends non-trivially on the simple homotopy type of the lens space.

[edit] See also

• Toda bracket

[edit] References

• Atiyah, Michael; Segal, Graeme (2006), "Twisted K-theory and cohomology", Inspired by S. S. Chern, Nankai Tracts Math., 11, World Sci. Publ., Hackensack, NJ, pp. 5–43, MR2307274, http://arxiv.org/abs/math.KT/0510674 
• Salvatore, Paolo; Longoni, Riccardo (2005), "Configuration spaces are not homotopy invariant", Topology 44: 375–380, doi:10.1016/j.top.2004.11.002 
• Massey, William. S. (1958), "Some higher order cohomology operations.", Symposium internacional de topología algebraica (International symposium on algebraic topology), Mexico City: Universidad Nacional Autónoma de México and UNESCO, pp. 145–154, MR0098366 
• May, J. Peter (1969), "Matric Massey products", J. Algebra 12: 533–568, doi:10.1016/0021-8693(69)90027-1, MR0238929 
• McCleary, John (2001), A User's Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, 58 (2nd ed.), Cambridge University Press, doi:10.2277/0521567599, MR1793722, ISBN 978-0-521-56759-6 
• Uehara, Hiroshi; Massey, W. S. (1957), "The Jacobi identity for Whitehead products", Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton, N. J.,: Princeton University Press, pp. pp. 361–377, MR0091473