Step 1Peel the grapefruit and cut it into sections: Using a very sharp knife, cut off the top and the bottom of the grapefruit, so it will sit flat on the cutting board. Starting where you see the pink grapefruit separate from the white pith, cut away one section of peel and pith, following the line of the fruit. This will expose the underlying fruit. Continue cutting away sections of the peel and pith until only fruit remains. When you're done, go back over the fruit, removing any traces of pith.
Step 2Working over a small bowl to catch the juice, slice the fruit into sections -- make a cut between the fruit and membrane, then do the same on the other side, freeing pure fruit. Repeat until you have removed all the fruit from the membrane. Squeeze any juice from the remaining membrane into the bowl. You will need about 2 tablespoons
Step 3Quarter the fennel lengthwise and remove the triangular core at the center. Use a mandolin or a very sharp knife to slice the fennel as thin as possible, about one-eighth inch; you'll have about 3 cups, lightly packed. Do the same with the red onion; you'll have about one-half cup. Combine the fennel and red onion in a large bowl.
Step 4Add one-half teaspoon salt and the red pepper flakes to the reserved grapefruit juice and whisk in the olive oil.
Step 5Toss the fennel with just enough of the dressing to lightly moisten it. Arrange the fennel on a platter in a low mound.
Step 6Add the arugula to the same bowl and toss to moisten it; you may need to add another teaspoon of vinaigrette. Arrange the arugula on top of the fennel.
Step 7Add the grapefruit and crabmeat to the same bowl and toss very gently to avoid breaking up the citrus or the crab. You want it barely moistened; if you need, add another teaspoon of vinaigrette. Arrange this on top of the arugula and serve immediately.