On homotopy groups of quandle spaces and the quandle homotopy invariant of links

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], , and the quandle homotopy invariant of links is defined in Z[π2(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13], . It is known that the cocycle invariants introduced in Carter et al. (2005) [3], , Carter et al. (2003) [5], , Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π2(BX) is finitely generated, and that, for a connected finite quandle X, π2(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π2(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.