Foundations of topological racks and quandles (Q2799076)

A quandle is a set \(Q\) with a binary operation \(\ast: Q\times Q\rightarrow Q\) such thatNEWLINENEWLINE(1) for any \(a\in Q\), we have \(a\ast a=a\).NEWLINENEWLINE(2) for any \(b, c\in Q\), there exists a unique \(a\in Q\) such that \(a\ast b=c\).NEWLINENEWLINE(3) for any \(a, b, c\in Q\), we have \((a\ast b)\ast c=(a\ast c)\ast(b\ast c)\).NEWLINENEWLINEThese axioms correspond to the three Reidemeister moves in knot theory. In particular, if the operation only satisfies the second and the third condition then we name it a rack. A topological rack/quandle is a topological space with a rack/quandle structure such that the operation \(\ast\) is continuous. Similar to finite quandles, one can also define the ``coloring invariant'' of knots with a fixed topological quandle, which are topological spaces in general.NEWLINENEWLINEIn this paper, the authors give a foundational account of topological quandles which focuses on the algebraic structure of topological quandles. For example, the notions of ideal, kernel and unit are introduced. Besides these, the authors also define topological rack modules and extensions of topological racks. This would be helpful to introduce the (co)homology theory of topological racks/quandles.