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Greg Friedman
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Maybe this is a little too on the nose, but: obstruction theory to extension and lifting problems in the sense of, e.g., Whitehead Chapter 6 or Davis-Kirk Chapter 7. Here vanishing of the obstruction classes (which can sometimes be guaranteed by properties of the spaces) suffices to demonstrate the existence of extensions of maps or lifts of maps from the base space to the total space of a fibration.

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