A uniqueness result for orthogonal groups as 2-compact groups
Given two connected compact Lie groups it is well known that they are isomorphic if and only if their maximal torus normalizers are isomorphic. Recent work showed that a corresponding statement also holds for \(p\)-compact groups in case \(p\) is an odd prime, and it is conjectured that the analogous statement is also true for \(p=2\). In the paper under review the author provides a partial result towards affirming this conjecture. More precisely he proves the following: let \(G\) be a \(2\)-compact group which corresponds to a product of orthogonal groups and let \(X\) be a \(2\)-compact group; then \(G\) and \(X\) are isomorphic if and only if they are so-called \(N\)-isomorphic. As a corollary to this the author obtains the corresponding statements for the \(2\)-compact groups which correspond to \(Spin(2k+1)\) and \(SO(2k+1)\) respectively. To obtain the main result the author uses a theorem of \textit{H. Morgenroth} proved in [Homotopy uniqueness of products of orthogonal groups, Mathematica Gottingensis 6 (1996; Zbl 0896.55007)] which says that a \(2\)-compact group \(X\) is isomorphic to \(G\) if and only if the corresponding cohomology rings \(H^*(BX;\mathbb Z/2)\) and \(H^*(BG;\mathbb Z/2)\) are abstractly isomorphic as modules over the mod-2-Steenrod algebra. Given a \(2\)-compact group \(X\) which is \(N\)-isomorphic to \(G\) it therefore suffices to verify the cohomological condition of the theorem. To see the latter the author proceeds as follows. Let \(E_G\subset G\) be the inclusion of the maximal abelian \(2\)-group. Since \(X\) is \(N\)-isomorphic to \(G\) it follows that there is a corresponding inclusion \(E_G\subset X\). The Weyl groups \(W_G\) and \(W_X\) which belong to the two embeddings then both act on \(H^*(BE_G;\mathbb Z/2)\). The author then obtains the result by showing that the two Weyl groups and their actions are isomorphic and that \(H^*(BX;\mathbb Z/2)\) can be computed (as in the case where \(X=G\)) by \(H^*(BX;\mathbb Z/2)\cong H^*(BE_G;\mathbb Z/2)^{W_X}\).
scientific article; zbMATH DE number 1742825