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Geometric Topology Seminar
July 20, 2017 15:00, Moscow, Math Department of the Higher School of Economics, Room 108
 


$\Delta$-link homotopy of links in $S^3$ and invariants of link maps in $S^4$

S. A. Melikhov
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Abstract: In 2003, Nakanishi and Ohyama obtained a classification of $2$-component links up to $\Delta$-link homotopy. Namely, they are classified by the linking number and the generalized Sato-Levine invariant. Using Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variation due to Koschorke, we obtain a simple proof of the Nakanishi–Ohyama theorem, and also its version for string links. We also prove that $3$-component links that are trivial up to link homotopy are classified up to weak $\Delta$-link homotopy by $\bar\mu$-invariants of length $\le 4$. The proof uses a computation of the image of Koschorke's $\tilde\beta$-invariant of link maps $S^2\sqcup S^2\sqcup S^2\to S^4$ (which is strictly stronger than Gui-Song Li's version of Kirk's invariant). This computation in is turn based on Yasuhara's results about $\Delta$-link homotopy. This talk is based on joint work with Yuka Kotorii.

Materials: extended_abstract.pdf (108.9 Kb)

Language: English

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