Subject: Question for Algebraic Topology List From: Gregory Landweber Date: Fri, 13 Aug 2004 19:50:41 -0700 I would like to post the following question to the list: Subject: Classification of Central Extensions I am looking for a reference for the following fact, which I assume has been known for several decades: If G is a topological group (I am thinking Lie group, loop group, or more generally some sort of Banach-Lie group), then its central extensions of the form S1 --> \tilde G --> G are classified (up to homotopy via group extensions) by H3(BG;Z). Here is a sketch of a proof: Topologically, such extensions are circle bundles, classified by c_1(\tilde G) \in H2(G;Z). However, not every circle bundle admits a group structure, and the ones that do will induce fibrations of classifying spaces BS1 --> B\tilde G --> BG classified by the transgression d_3 c_1(\tilde G) \in H3(BG;Z). Alternatively, a circle bundle is given by a map G --> BS1. Since S1 is abelian, its classifying space BS1 admits a group structure (I believe it can be thought of as PGL(H) where H is a separable Hilbert space), and ES1 (viewed as GL(H) ) is a central extension. To pull back this group structure to \tilde G, we require the map G --> BS1 to be a group homomorphism, which induces a map of classifying spaces BG --> BBS1 = K(Z;3) which determines a class in H3(BG;Z). The other direction is more complicated, showing that elements of H3(BG;Z) directly correspond to central extensions of G. Looping gives a map from H3(BG;Z) --> H2(G;Z), i.e., given a map BG --> K(Z;3), we loop it to get a map G --> K(Z;2) = BS1, which determines a circle bundle. But how do we obtain a group structure on such a bundle (up to homotopy, of course)? We may be able to argue (as suggested by Dan Dugger) that clases in [BG, BBS1] correspond to those classes in [G,BS1] which contain group homomorphisms. That would prove the result provided that we consider our isomorphism classes of extensions up to homotopy (so that the case where G = R the real numbers works). What I would really like is a reference for the whole result, but I would settle for a reference for the homotopy theory details of the proof. Thanks for your help! -- Greg Landweber University of Oregon P.S. The standard cases are pretty well known. For G compact, the central extensions correspond to the torsion part of H2(G), which transgresses to all of H3(BG). (An interesting case is if G = T2, in which case we have H2(T2) = Z, but there are no non- trivial extensions, since H3(BT2) = 0. In that case, there are extensions of the Lie algebra which correspond to integral cohomology classes in H2(G;Z), but which do not extend to group extensions.) If we consider central extensions of the loop group LG, then H2(LG) = H3(BLG), and every circle bundle corresponds to a central extension. P.P.S. An equivalent way to state this result is that the obstruction to a circle bundle admitting the structure of a group extension is the class d_2 c_1(\tilde G) \in H2( BG; H1(G;Z) ) in the spectral sequence of the universal fibration G --> EG --> BG. The transgression d_3 then gives an isomorphism ker d_2 = H3(BG;Z). The inverse of this transgression is the looping map described above, giving us an exact sequence: 0 --> H3(BG;Z) --d_3^(-1)--> H2(G;Z) --d_2--> H2(BG;H1(G;Z)). Perhaps this description of the problem rings a bell for someone?