Modulo odd prime homotopy normality for \(H\)-spaces
An inclusion \(i:Y\to X\) of simply connected homotopy associative \(H\)-spaces is called homotopy normal, if the image of \(i_*:[Z,Y] \to[Z,X]\) is a normal subgroup for any finite complex \(Z\). This is equivalent to the fact that the map \(c_2:X\times Y\to X\), \((x,y) \mapsto(x\cdot y)\cdot (i(x)\cdot i(y))\) can be deformed into \(Y\), where \(i:X\to X\) is homotopy inversion. There are many known non-homotopy normality results for classical Lie groups; in particular, the inclusions \(G_2\subset F_2\subset E_6\subset E_7\subset E_8\) are not homotopy normal. The authors study the mod \(p\)-analogue. An \(H\)-map \(i:Y\to X\) between simply connected homotopy associate \(H\)-spaces is called mod~\(p\) normal if \(c_2(X_{(p)} \times i(Y_{(p)}))\) can be deformed into \(i(Y)_{(p)}\). They prove that the inclusions \(F_4\subset E_6\subset E_7\) and \(H\subset E_8\) for \(H\in\{G_2,\text{Spin}(7),\text{Spin}(8),\text{Spin}(9)\), \(F_4,E_6,E_7\}\) are not mod~3 normal. This is done by giving necessary conditions for a map \(Y \to X\) with \(X\in\{F_4,E_6,E_7,E_8\}\) to be mod~3 normal in terms of its behavior on \(H^*(X;\mathbb{Z}/3)\) or the Morava homologies \(K(2)_*(X)\) or \(K(3)_* (X)\).
scientific article; zbMATH DE number 1319567