The Milnor triple linking number of string links by cut-and-paste topology. (English) Zbl 1297.55016
Let \(\mathcal{L}_k\) be the space of \(k\)-component long links in \(\mathbb{R}^3\), i.e. embeddings of \(k\) disjoint copies of \(\mathbb{R}^3\) that are fixed outside a compact set. The paper under review exhibits the Milnor triple linking number, an integer-valued invariant of 3-component string links, via a homotopy-theoretic Pontrjagin-Thom construction.
More precisely, the author takes as the starting point the results from [R. Koytcheff et al., J. Knot Theory Ramifications 22, No. 11, Article ID 1350061, 73 p. (2013; Zbl 1307.57015)] where it is shown that an adaptation of Bott-Taubes, or configuration space, integration to string links yields all finite type invariants of these spaces. Since Milnor invariants are finite type (for string links), it follows that some combination of these integrals gives the Milnor triple linking. This is precisely what the author first proves – the sum of configuration space integrals over four particular diagrams is the triple linking.
Next, the author applies a Pontrjagin-Thom approach that he developed in a previous work [R. Koytcheff, Algebr. Geom. Topol. 9, No. 3, 1467–1501 (2009; Zbl 1175.57012)] to this situation. Rather than collapsing the boundaries of certain manifolds with corners that appear in Bott-Taubes integration, as he had previously done, he glues this boundary in order for its contribution to integration to cancel out. This part is inspired by Kuperberg and Thurston’s techniques from “Perturbative 3-manifold invariants by cut-and-paste topology”. There is then a map from \(S^6\) to the quotient of these bundles modulo the glued boundary and further from this space to \(S^2\times S^2\times S^2\). The composite is the map that defines a certain 6-dimensional cohomology class on \(S^6\), and the main result is that pairing this class with the fundamental class gives precisely the triple linking.
This interpretation of the Milnor triple linking number is nice, and the reviewer looks forward to seeing the promised work of realizing all the Bott-Taubes classes of string links via such gluing brought to fruition.
More precisely, the author takes as the starting point the results from [R. Koytcheff et al., J. Knot Theory Ramifications 22, No. 11, Article ID 1350061, 73 p. (2013; Zbl 1307.57015)] where it is shown that an adaptation of Bott-Taubes, or configuration space, integration to string links yields all finite type invariants of these spaces. Since Milnor invariants are finite type (for string links), it follows that some combination of these integrals gives the Milnor triple linking. This is precisely what the author first proves – the sum of configuration space integrals over four particular diagrams is the triple linking.
Next, the author applies a Pontrjagin-Thom approach that he developed in a previous work [R. Koytcheff, Algebr. Geom. Topol. 9, No. 3, 1467–1501 (2009; Zbl 1175.57012)] to this situation. Rather than collapsing the boundaries of certain manifolds with corners that appear in Bott-Taubes integration, as he had previously done, he glues this boundary in order for its contribution to integration to cancel out. This part is inspired by Kuperberg and Thurston’s techniques from “Perturbative 3-manifold invariants by cut-and-paste topology”. There is then a map from \(S^6\) to the quotient of these bundles modulo the glued boundary and further from this space to \(S^2\times S^2\times S^2\). The composite is the map that defines a certain 6-dimensional cohomology class on \(S^6\), and the main result is that pairing this class with the fundamental class gives precisely the triple linking.
This interpretation of the Milnor triple linking number is nice, and the reviewer looks forward to seeing the promised work of realizing all the Bott-Taubes classes of string links via such gluing brought to fruition.
Reviewer: Ismar Volic (Wellesley)
MSC:
| 55R80 | Discriminantal varieties and configuration spaces in algebraic topology |
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 55R12 | Transfer for fiber spaces and bundles in algebraic topology |
| 57R40 | Embeddings in differential topology |
Keywords:
Bott-Taubes integrals; configuration space integrals; string links; triple linking number; finite type invariantsReferences:
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