Quandle homology groups, their Betti numbers, and virtual knots

Lower bounds for the Betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The results can be applied to knot theory as the first step towards evaluating the state-sum invariants defined from quandle cohomology.