From: Robin Chapman Subject: Re: what are the topological building blocks for finite dim. compact Lie groups? Date: Mon, 15 May 2000 10:44:30 GMT Newsgroups: sci.math Summary: [missing] In article <8fojgd$ota$1@nnrp1.deja.com>, David Bernier wrote: > Suppose G is a compact subgroup of GL(n,R) or GL(n,C) for some > finite n. How can G be decomposed as combination of topological sums > and products of "simpler" topological spaces? > > For example, O(2) is homeomorphic to S^1 + S^1; RxR/(ZxZ) is > homeomorphic to S^1 x S^1 . Possible blocks would be > points and the S^k spheres for k=>1. Are there any others? > > What if G is closed but not necessarily compact? Well, a compact semisimple Lie group is "almost" a product of simple compact Lie groups, but the topology of these might be quite complicated. For example we have SU(2) which is a quaternion group and an S^3. Next we get SU(3). Is the topology of this 8-dimensional manifold "reducible" in some meaningful way? -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: Boudewijn Moonen Subject: Re: what are the topological building blocks for finite dim. compact Lie Date: Tue, 16 May 2000 14:37:04 +0200 Newsgroups: sci.math To: David Bernier David Bernier wrote: > > Suppose G is a compact subgroup of GL(n,R) or GL(n,C) for some > finite n. How can G be decomposed as combination of topological sums > and products of "simpler" topological spaces? > > For example, O(2) is homeomorphic to S^1 + S^1; RxR/(ZxZ) is > homeomorphic to S^1 x S^1 . Possible blocks would be > points and the S^k spheres for k=>1. Are there any others? > First, any compact Lie group can be realized as a compact subgroup of some G(n,C), so we may as well just talk about compact Lie groups, no further specification. To begin with, the set C of connected components of a compact Lie group G is a finite group in a natural way, and, if we denote the connected component of the identity of G by G^o, there is an exact sequence 1 --> G^o --> G --> C --> 1 , so that G appears as an extension of a connected compact Lie group by a finite group. Conversely, extending a connected compact Lie group by any finite group gives a compact Lie group. Thus we restrict ourselves to connected compact Lie groups (ccLg) G. Any Lie group G has its adjoint representation Ad : G --> GL(Lie(G)). The image Ad(G) \subseteq GL(Lie(G)) is called the *adjoint group* of G. The connected component of the centre of G acts trivially in Ad. For a ccLg G, it then can be shown that Ad(G) is semisimple. From these facts, in turn, it follows that G is the quotient of the product of a torus T and a semisimple ccLg S by a finite central (hence abelian) subgroup Z of T x S: G isom T x S / Z where Z might be "diagonally" embedded. So the topological type of G is determined by the topological type of T x S and the action of Z. The topological type of T x S is as follows. T itself is topologically just the product of finitely many circles S^1 = U(1) and so known. A semisimple ccLs is a finite product of simple ccLg's, which are of the type SU(n), SO(n), Sp(n), and the exceptional groups E_6, E_7, E_8, F_4, G_2. About the topology of these groups, I think an awful lot is known, if not (almost) anything one would like to ask. E.g., if I remember well, their rational cohomology is that of a product of odd spheres (this holds for any ccLg), and (almost?) all torsion in integral cohomology is known. > > What if G is closed but not necessarily compact? > To answer this needs an expert, I think, which I am not. Conveivably, a lot of noncompact semisimple and reductive Lie groups appear here, the topology of which should, IMO, reduce to that of ccLg's, since they should deformation retract onto their maximal compact subgroups. Regards, -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712 ============================================================================== From: rusin@olympus.math.niu.edu (Dave Rusin) Subject: Re: what are the topological building blocks for finite dim. compact Lie groups? Date: 15 May 2000 14:04:26 GMT Newsgroups: sci.math In article <8fojgd$ota$1@nnrp1.deja.com>, David Bernier wrote: >Suppose G is a compact subgroup of GL(n,R) or GL(n,C) for some >finite n. How can G be decomposed as combination of topological sums >and products of "simpler" topological spaces? > >For example, O(2) is homeomorphic to S^1 + S^1; RxR/(ZxZ) is >homeomorphic to S^1 x S^1 . Possible blocks would be >points and the S^k spheres for k=>1. Are there any others? I'm guessing this reply is in the "more than you wanted to know" category, but here goes: Topological sums are not really very interesting here; the connected component G0 of the identity is a subgroup, and G is a sum of copies of G. Products are also a little restrictive. What you might consider instead is fibrations, which include products as a special case. An example of a fibration (and I _think_ it's true that these are equivalent in the context of finite-dimensional manifolds) is a fibre bundle, that is, a mapping of spaces p: Y -> Z which has the property that around each point z of Z there is a neighborhood U of z for which the inverse image p^{-1}(U) is a topological product U x F; one adds a consistency condition on these decompositions which is enough to force F to be independent of z and U. An extension F -> Y -> Z of Lie groups is then an example of a fibration, so your search for "building blocks" need go no further than an examination of the simple Lie groups. Moreover, there are natural decompositions of some of these, too. For example, if you let SO(n) act on the unit vectors in R^n, then by observing the action on any single vector you obtain a fibration SO(n-1) -> SO(n) -> S^{n-1}, the fibre being the stabilizer of the one fiduciary vector. This is perhaps the origin of your observation that the spheres are a natural set of building blocks. If you insist on actual product decompositions (say) you'll find there are obstructions to decomposing the spaces. At the very least you have the problem that the cohomology rings of the spaces (with coefficients in finite fields, say) would have to respect that decomposition, both as rings and as modules over the appropriate Steenrod algebra. Let me observe, finally, that there is another kind of decomposition which grows out of the comments in the previous paragraph. One can ask for decompositions of the space into wedge products (=one-point unions =sums in the category of pointed topological spaces), either as is or stably (meaning, after iterated suspensions, or in the category of spectra). In all these viewpoints the actions of the Steenrod algebras put an upper bound on the amount of decomposition possible, since you are asking for a decomposition of a module into a direct sum of smaller modules. I don't know how much people have done for compact Lie groups G but there has been quite a bit of work done on their classifying spaces BG. In particular, Priddy and Martino have many nice results about finite groups G and Feschbach has extended some of this to compact groups. dave