Reduced cohomology of spheres shifts down by one under desuspension.Suggest a slogan
For \(n\geq 1\) and all \(i\), the reduced cohomology groups of spheres satisfy \[ \tilde H^i(S^n) \cong \tilde H^{i-1}(S^{n-1}) \] [@Hat02].
The statement fails for unreduced cohomology: for \(n\geq 2\), \(H^1(S^n) = 0\) while \(H^0(S^{n-1})\cong\ZZ\).