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, and the quandle homotopy invariant of links
is defined in its homotopy group $\Z[\pi_2(BX)]$. It is known that the
cocycle invariants can be derived from the quandle homotopy invariant
and that $\pi_2(BX)$ is compatible with the X-colored link bordism group.

The main theorem is that for a finite quandle X, $\pi_2(BX)$ is finitely
generated. It follows that the space spanned by cocycle invariants for a
finite quandle is finitely generated. Further, for some concrete quandles
we calculate $\pi_2(BX)$ with generators represented by some links. From
the calculation, all cocycle invariants for those quandles are concretely
presented.

In this talk, I will review the homotopy invariant and
present how to calculate $\pi_2(BX)$. I will present some applications of
the above calculations.