Loop Space Models and Integral Homology of Connected Quandles

We construct a marked integral delooping of a connected quandle's covering monoid, realizing its rack space as a homotopy fiber. For finite connected quandles at primes $p$ not dividing a point-stabilizer order, we determine the local $A_\infty$ type and the ordinary rack cohomology shuffle algebra over fields of characteristic $p$. We compute integral quandle homology in all positive degrees for finite connected regular cyclic Alexander quandles and the tetrahedral quandle. At such primes dividing the quandle order, rack-space homotopy groups have exponentially many cyclic direct summands of fixed $p$-power order in each sufficiently large degree.