  In this paper, we present that genotype-phenotype mapping can be
theoretically interpreted using the concept of quotient space in mathematics.
Quotient space can be considered as mathematically-defined phenotype space in
the evolutionary computation theory. The quotient geometric crossover has the
effect of reducing the search space actually searched by geometric crossover,
and it introduces problem knowledge in the search by using a distance better
tailored to the specific solution interpretation. Quotient geometric crossovers
are directly applied to the genotype space but they have the effect of the
crossovers performed on phenotype space. We give many example applications of
the quotient geometric crossover.
