TITLE: Singular complex = cohomology ring + Steenrod operations?
QUESTION [10 upvotes]: Fix a prime $p$ and consider everything mod $p$. Steenrod operations arise somehow from the loss of information passing from the singular complex of a space to its cohomology ring. Are they exactly this gap, i.e. can I get the singular complex back from the cohomology ring of a space and its structure as a module over the Steenrod algebra?

REPLY [4 votes]: It depends of what structure you want to consider on the complex of singular cochains. If you want to look at it just as a complex, then the cohomology groups are enough. If you want it as a differential graded algebra, then you would need the cohomology groups with an A-infinity algebra structure, etc. Steenrod operations are something which come from the E-infinity structure on cochains, but they are weaker.