My question is far more specific than the one in the title, and so, I urge you to not reply to the question in the title alone, and to try and reply to the questions I shall pose in this note.
I am a Master's student in an above-average European university, interested in geometry and topology, and will be applying to graduate schools in the U.S. this December. While I have done the standard graduate-level introductions to algebraic topology, differential topology, representation theory and differential geometry, I have no background in commutative algebra and algebraic geometry. I would like to work in low-dimensional topology, and my thesis will be in this area.
Will the fact that I know little to no algebraic geometry hinder my application? Do graduate committees specifically look for background in certain related areas, when candidates apply for a particular area? I ask because while people often say it doesn't matter much, it (possibly concidentally?) is also quite clear that most entry-level graduate students in top-tier graduate schools know substantial knowledge (at least a first course of all, and a second course of maybe two or three) of algebraic topology, differential geometry, representation theory and algebraic geometry.
P.S. In your answers, please consider that the graduate schools I am applying to in the United States will be part of the top-ten lists by USNews for topology schools or geometry schools.