On 6-connected finite H-spaces with two torsion
When realizing a Hopf algebra over the mod 2 Steenrod Algebra as the cohomology of an H-space, there are many restrictions. Nevertheless, for several Hopf algebras it is unknown if they can be realized as the cohomology of a finite mod 2 H-space. The Hopf algebra \(A=({\mathbb{Z}}_ 2[x_ 7]/(x_ 7)^ 4)\otimes \wedge (x_{11},x_{13})\) is perhaps the most natural candidate to be realized as the cohomology of a finite H-space. This work shows that there is no H-space X such that \(H^*(X,{\mathbb{Z}}_ 2)\approx A\). The authors study the cohomology of \(\Omega\) X and use that \(\Omega\) X is a homotopy commutative H-space. A tertiary cohomology operation is used in order to detect a class in \(H^*(\Omega X,{\mathbb{Z}}_ 2)\) which leads to a contradiction for the existence of the H-space X. This work reinforces the idea that there is no new mod 2 finite H-space. The known examples of mod 2 finite H-spaces are just a product of seven spheres by compact Lie groups.
realizing a Hopf algebra over the mod 2 Steenrod Algebra
scientific article; zbMATH DE number 4115529