The theme of the paper can roughly be described as follows: Let \(G\) be a compact (connected) Lie group and \(A_G\) the category of non trivial elementary abelian subgroups \(V\) with morphisms the injections induced by conjugation with elements of \(G\). Sending \(V\) to the classifying space \(BG_G(V)\) of it’s centralizer in \(G\) one obtains a functor \(a_G: A_G^{\text{op}}\to \text{Top}\) whose homotopy colimit (in a \(p\)-complete setting) retrieves the classifying space of \(G\) [for details see W. G. Dwyer and C. W. Wilkerson, Ann. Math., II. Ser. 139, No. 2, 395-442 (1994; Zbl 0801.55007), Contemp. Math. 181, 119-157 (1995; Zbl 0828.55009)]. It is natural to ask which properties characterize \(a_G\). The authors in the paper proceed in two steps:
1. Descend to the homotopy category and define the notion of \(H^*\)-clone. An \(H^*\)-clone of two functors \(F,G\) is a pointwise equivalence, which becomes a functor in cohomology. A result is, that there is no clone for the 2 completion of the exceptional Lie group \(G_2\).
2. Consider the space of realizations, i.e. the space of functors to spaces over a functor to the (cohomology quotient) of the homotopy category.
Using methods of W. G. Dwyer and C. W. Wilkerson [J. Am. Math. Soc. 6, No. 1, 37-64 (1993; Zbl 0769.55007)] and the previous result the space of realizations for the 2-completed \(G_2\) is shown as connected. The authors announce, that in a sequel it will be shown, that the 2-completed \(G_2\) is actually determined by \(H^*\) as a module over the Steenrod algebra.