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Geometric Topology Seminar
December 17, 2018 15:30–17:30, Moscow, Math Department of the HSE (Usachyova, 6), room 212

Brunnian link maps in the 4-sphere and the Chinese Rings puzzle

S. A. Melikhov

Abstract: A link map is a map $X_1\sqcup…\sqcup X_k \to Y$ such that the images of the $X_i$'s are pairwise disjoint. A link homotopy is a homotopy whose every time instant is a link map. For example, link maps $S^p\sqcup S^q \to S^{p+q+1}$ are classified (up to link homotopy) by the linking number, and link maps $S^1\sqcup S^1\sqcup S^1 \to S^3$ are classified by Milnor's triple $\bar\mu$-invariant. In a 2017 preprint R. Schneiderman and P. Teichner proved a long-standing conjecture that link maps $S^2\sqcup S^2 \to S^4$ are classified by Kirk's invariant, which takes values in the infinitely generated abelian group $\mathbb Z[x]\oplus\mathbb Z[y]$.
In this talk I will compute the image of the Kirk–Koschorke invariant of link maps $S^2\sqcup S^2\sqcup S^2 \to S^4$, which takes values in $\mathbb Z[(\mathbb Z\oplus\mathbb Z)/-]^3$. The main step is a new elementary construction of Brunnian link maps $S^2\sqcup…\sqcup S^2 \to S^4$. (“Brunnian” means that all proper sublinks are trivial up to link homotopy.) This construction is closely related to the minimal solution of the Chinese Rings puzzle. I will also show that the Kirk–Koschorke invariant is incomplete, using a new non-abelian invariant of link maps $S^2\sqcup…\sqcup S^2\to S^4$ with values in $\mathbb Z[RF_{m-1}/t]^m$, where $RF_k$ is the Milnor free group ($RF_2$ is also known as the discrete Heisenberg group) and $t(g)=g^{-1}$.
As explained in https://arxiv.org/abs/1711.03514 , computation of the images of invariants of link maps in $S^4$ is actually motivated by the study of classical links. In particular, the computation of the image of the Kirk–Koschorke invariant has the following application, which will be discussed if time permits: Two links $S^1\sqcup S^1\sqcup S^1\to S^3$ that are link homotopic to the unlink are related by $C_2^{xxx}$-moves and $C_3^{xx,yz}$-moves if and only if they have equal $\bar\mu$-invariants (with possibly repeating indices) of length at most 4.