Enhanced cohomology and obstruction theory

For any n ≥ 4 and a finite regular CW-complex K we define an abelian group called the enhanced n-th cohomology group of K, denoted EHⁿ(K). These groups have (contravariant) functorial properties; in fact, if ƒ ≃ g:K→L then ƒ^* = g⁸:EHⁿ(L)→EHⁿ(K). Thus they are invariant on the homotopy type. There is an exact sequence (sequence in PDF) The enhanced cohomology groups give a well-defined obstruction to extend maps to some n-2-connected spaces from a subcomplex A of K to K^[n+1]∪A. We thus have the primary and secondary obstructions encompassed in a single, well-defined step. We also have a well-defined obstruction to embedding an n-dimensional simplicial complex in R^2n-1 The classical Steenrod paper (Ann. of Math., 48, 290-320) that introduced the squares produces a 2-step approach to solving the extension problem mentioned above, without realizing the existence of EH^.