In a stainless or glass mixing bowl, combine the jicama and vinegar and refrigerate for several hours.
In another mixing bowl add the lime juice and slowly drizzle in the oil while whisking. Salt to taste and reserve.
In another mixing bowl, combine the crab, red peppers, yellow peppers, tomato, jalapeño, onion, avocado, basil, cilantro and mango. Drain the reserved jicama and add to the bowl. Dress mixture with lime vinaigrette; toss and season to taste with salt. Set aside.
Pour enough corn oil in a medium saucepan to come 1/2” up the side and heat to 350°F., testing with a thermometer. Fry the corn tortillas one at a time until crispy, about one minute, and drain on paper towels.
Place a tostada on four serving plates and spoon some of the corn sauce around the plate. Place a heaping tablespoon of the crab mixture on each tostada and cover top of crab mixture with another tostada. Repeat the crabmeat layer, then top with one more tostada.