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Quandle cocycles from invariant theory. (English) Zbl 1287.55002
For a group \(G\) and a right \(G\)-module \(M\), one has the structure of a quandle on \(X=G \times M\) by defining a binary operation \((a,g)\triangleleft(b,h) = ((a-b)h + b, h^{-1}gh)\), which is understood as the associated quandle of a structure of a \(G\)-family of quandles on \(M\), a family of quandle operations on \(M\) indexed by \(G\) [A. Ishii, M. Iwakiri, Y. Jang and K. Oshiro, “A \(G\)-family of quandles and handlebody-knots”, arXiv:1205.1855].
Let \(C_{*}^{gr}(M;\mathbb{Z})\) be the non-homogeneous group chain complex of \(M\), where \(M\) is simply regarded as an abelian group. The \(G\)-action on \(M\) induces an action on \(C_{*}^{gr}(M;\mathbb{Z})\), and let \(C_{*}^{gr}(M;\mathbb{Z})_{G}\) be the \(G\)-coinvariants. The author constructs a chain map from a non-homogenous coordinate version of the rack complex of \(X\) to \(C_{*}^{gr}(M;\mathbb{Z})_{G}\). Dually, this provides a way to obtain quandle cocycles from \(G\)-invariant group cocycles, and in particular, yields a cocycle of a \(G\)-family of quandles. Since group (co)invariant (co)cycles have been much studied and are easier to treat than quandle cocycles, the author succeeds in constructing various non trivial quandle cocycles explicitly.

55N35 Other homology theories in algebraic topology
20J06 Cohomology of groups
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI
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