

Geometric Topology Seminar
March 15, 2018 14:00, Moscow, Math Department of the Higher School of Economics, Room 108






An unlinking theorem for link maps in the 4sphere
A. C. Lightfoot^{} 
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Abstract:
In this talk, having placed one component $f_1$ of a link map $f_1\sqcup f_2: S^2_1\sqcup S^2_2\to S^4$ into a standard form, we construct 2spheres representing generators for the second homotopy group of its complement $S^4\setminus f_1$. These 2spheres are constructed using accessory disks and Whitney disks for the immersion $f_1$. Our first application of this construction is to compute the image of Kirk's invariant (which was first proved by Kirk in his foundational work). We then establish criteria, in terms of Wall intersections, for a link map to be link homotopically trivial. These criteria will be seen to be relatively weak; indeed, the proof of their sufficiency will require an application of Freedman's embedding theorem.
This is the fourth in a series of talks in which we give a careful exposition of a recent groundbreaking paper of Rob Schneiderman and Peter Teichner, The Group of Disjoint 2Spheres in 4Space. arXiv:1708.00358.
A link map $f_1\sqcup f_2:S^2_1\sqcup S^2_2\to S^4$ is a map of two 2spheres into the 4sphere such that $f(S^2_1)\cap f(S^2_2)=\emptyset$, and a link homotopy is a homotopy through link maps. That is, throughout the homotopy each component may selfintersect, but the two components must stay disjoint. Schneiderman and Teichner resolved a longstanding problem by proving that such link maps, modulo link homotopy, are classified by a certain invariant due to Paul Kirk. (This is a higherdimensional analogue of the classical result in knot theory that the linking number classifies links $S^1\sqcup S^1\to S^3$ up to link homotopy.) The goal of these talks is to obtain a complete understanding of the proof of this result.
Website:
https://arxiv.org/abs/1708.00358

