From: "David L. Johnson" <david.johnson@lehigh.edu>
Subject: Re: SO(3) and SO(n), and their fundamental groups
Date: Tue, 30 Jan 2001 19:03:42 -0500
Newsgroups: sci.math.research
Summary: Induced maps on homotopy groups of orthogonal groups

semorrison@hotmail.com wrote:
> 
> Hi,
> 
> The fundamental group of SO(n) for n > 2 is Z_2. Is it the case that
> the inclusion of SO(3) into SO(n) (anyway you like) induces an
> isomorphism of the fundamental groups? 

My first reaction would be to question the "any way you like" part.  I presume
you mean as a group homomorphism.  Then, you have to ask how many such
homomorphisms there are, and whether or not they are all conjugate.  Certainly
your second question:

> Similarly, the fundamental group of SO(1,n) for n >2 is Z_2. Again, we
> can include SO(3) into SO(1,n) (acting on any 3 of the last n
> coordinates say). Does this induce an isomorphism?

is more restrictive, and clearly any two such would be conjugate.  If this is
the kind of map you want to consider, then I think the answer is yes, and that
it is straightforward to prove -- since it is straightforward to prove for the
standard inclusions.  Now, there may be other homomorphisms from SO(3) to
SO(n) (I don't know, but I'm sure any expert would), and that might complicate
the issue.

-- 

David L. Johnson

   __o   | You will say Christ saith this and the apostles say this; but
 _`\(,_  | what canst  thou say?  -- George Fox.  
(_)/ (_) |
==============================================================================

From: Nikos <napostolakis@gc.cuny.edu>
Subject: RE:
Date: 31 Jan 01 14:09:52 -0500 (EST)
Newsgroups: sci.math.research

It is well known that the standard inclusion SO(n) ---> SO(n+1)
induces isomorhisms in \pi_i for i=0,...,n-2.

To see this look at the defining action of SO(n+1) in S^n. It is easy
to check that the isotropy group of a point (like (1,0,...,0) ) is
isomorphic to SO(n) , so that we get a fibration

SO(n)--->SO(n+1)--->S^n.

Now apply the homotopy exact sequence for this fibration.

In general we have similar fibrations for all classical groups and
standard inclusions beetween them. (You can check any book on fiber
bundles or discover them yourself).

Take care

   Nikos.