Milnor’s 𝜇̄-invariants and Massey products

Porter, Richard D.

doi:10.1090/s0002-9947-1980-0549154-9

Cited by 45 publications

(46 citation statements)

References 18 publications

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“…Stallings [13] conjectured that there is a relationship between the 71-invariants and Massey products. Porter [10] and Turaev [16] were the first to prove this. The proof of Theorem 3.1 utilizes a refined expression of ~ and several lemmas.…”

Section: Each Of the 71(i) Compares With A A(/')mentioning

confidence: 91%

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Massey Products in the Cohomology of Groups with Applications to Link Theory

Stein¹

1990

Transactions of the American Mathematical Society

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Section: Each Of the 71(i) Compares With A A(/')mentioning

confidence: 91%

“…The link invariants are compared with Milnor's II-invariants [8,9]. Porter [10] and Turaev [16] were the first to prove Stallings' conjecture [13] linking the II-invariants and Massey products. For two component links, an infinite family of the based invariants is independent of the basing.…”

Section: Introductionmentioning

confidence: 99%

Massey Products in the Cohomology of Groups with Applications to Link Theory

Stein¹

1990

Transactions of the American Mathematical Society

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“…For some background on this notion, see Milnor 1,4 and, for example, Massey, 16 Casson, 17 Turaev, 18 Porter, 19 Fenn, 20 Orr, 21 Cochran, 22 and Habegger and Lin. 23 Configuration spaces come into the picture as follows.…”

Section: Configuration Spacesmentioning

confidence: 99%

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

DeTurck

Gluck

Komendarczyk

et al. 2013

Journal of Mathematical Physics

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To each three-component link in the 3-sphere we associate a generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientationpreserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space. C 2013 American Institute of Physics.

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“…Clearly, if ω has central curvature, then Ω ∧ ω = ω ∧ Ω = 0, which implies dΩ = 0 by Bianchi's identity (2.6). In particular, dΩ 12 …”

Section: Nilpotent Connectionsmentioning

confidence: 99%

“…This article applies the theory of connections from differential geometry to prove de Rham cohomology versions of the Porter-Turaev Theorem which correlate the Milnor and Massey invariants of a link in the 3-sphere [12,18]. We have borrowed several ideas from [11], specifically how to realize Massey products as the curvature of certain nilpotent connections and how the Milnor numbers figure into the holonomy of a longitude.…”

Section: Introductionmentioning

confidence: 99%