Doesn't cosmic microwave background radiation provide an absolute reference point for motion?

Physics 101 teaches that, if Bob is moving at speed X relative to Mary, both Bob and Mary are perfectly entitled to claim that they are standing still and the other is moving. Neither view point is right or wrong because there is no absolute space to measure motion against. However, doesn't cosmic microwave... show more Physics 101 teaches that, if Bob is moving at speed X relative to Mary, both Bob and Mary are perfectly entitled to claim that they are standing still and the other is moving. Neither view point is right or wrong because there is no absolute space to measure motion against. However, doesn't cosmic microwave background radiation provide this absolute space? If Bob wanted to confirm whether or not he was really at rest or not, couldn't he just check for dopplar effect of this radiation? If there wasn't any, then he could claim he was at rest and Mary was moving.
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