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?



Singular complex = cohomology ring + Steenrod operations?
No. For instance, Massey products on the cohomology are extra information that neither the ring structure nor the Steenrod operations see. The complement of the Borromean rings, for example, and the complement of three unlinked circles in $R^3$ have the same cohomology ring and Steenrod operations, but cannot be chain equivalent because of nonzero Massey products in the former.


