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Given a platform that is floating independent of Earth's gravity, would it be possible to put a platform over the equator and over the day, would the Earth rotate from under it allowing you to travel around the world?

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Well, there is a partial yes that is a direct result of the Coriolis force:

If you go up in a hot air balloon, you will be subject to various winds which will move you. And these winds are a result of the Earth spinning. In principle you should be able to navigate to most places on the globe by choosing height etc. in reality it is much too complex to do that with any degree of reliability.

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Right. – anna v Jun 28 '13 at 13:42

Nope, you will move with the earth due to your inertia. Your whole life you've been moving approximately with the earth's speed of rotation (about 1000 mph), and going out into the ocean on a platform won't change that.

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And don't forget that that platform would still be subject to momentum imparted on it by flowing air and/or water as well, iow wind, waves, currents. – jwenting Jun 28 '13 at 8:17

Whatever is in the atmosphere is moving / rotating together with the earth itself - anything on the surface, as well as the air, clouds & everything else.

Once you get off the atmosphere and stop yourself ( as initially you have a speed of the earth surface rotation ) then you could see the earth turning and "land wherever you like" :)

Another example would be - imagine you're standing in a moving bus. At that time your traveling speed is the same the speed of the bus. So if you jump inside a bus - you will not change your position relative to the bus floor when you land, because you're moving at the same speed; OR - not moving if we compare relatively to the bus.

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