Massey product

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In mathematics, particularly in algebraic topology but also in geometric topology and differential topology, the Massey product is a cohomology operation of higher order.

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[edit] Definition

On differential forms the triple product is formally defined as

MP(\omega_1,\omega_2,\omega_3) \ \stackrel{\mathrm{def}}{=}\ \omega_1\wedge d^{-1}(\omega_2\wedge\omega_3) + d^{-1}(\omega_1\wedge\omega_2)\wedge\omega_3

whenever ω1 ∧ω2 and ω2 ∧ω3 are exact forms.

[edit] The Massey product is a secondary operation

The exterior derivative is not in fact invertible in the space of differential forms. Instead the inverse is only well-defined modulo the addition of a closed form.

Therefore the Massey product

PMP(ω1,ω2,ω3)

on differential forms is only well-defined modulo products of ω1 and ω3 with closed forms. These closed forms may represent nontrivial cohomology classes, and so the Massey product in de Rham cohomology is only well-defined modulo elements which may be written as a product of the class of a linear combination of ω1 and ω3 with an arbitrary cohomology element. For triple Massey product to be in de-Rham cohomology group one should have ω1 and ω3 both closed.

It is for this reason that the Massey product is a secondary and not a primary cohomology operation.

[edit] The Massey product in the AHSS

Massey products appear in the Atiyah-Hirzebruch spectral sequence (AHSS), which computes twisted K-theory with twist given by a 3-class H. Michael Atiyah and Graeme Segal have shown, in Twisted K-theory and cohomology, that rationally the higher order differentials

d2p+1

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.

[edit] See also

[edit] References

  • 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, MR0238929, DOI 10.1016/0021-8693(69)90027-1 
  • 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