From: "W. Dale Hall" <wd_hall@pacbell.net>
Subject: Re: Fundamental group of a space
Date: Wed, 21 Feb 2001 02:08:13 -0800
Newsgroups: sci.math
Summary: Constructing KG_1 (space with fundamental group equal to G)

Hans Shtrulevacht wrote:
> 
> If G is any group it is possible to find a topological space such that
> its fundamental group isomorphic to G ?
> 
> Thanks

Given a presentation of G in terms of generators and relations, one can
produce a CW complex with a 1-cell for each generator, and a 2-cell for
each relation. The result has the appropriate fundamental group, at
least for finitely-presented groups G. Whether this process runs into
difficulty for non-finite presentations, I don't recall. I'm sure the
resulting space can be somewhat complicated (imagining the additive
group of the reals, for instance).

Any more, and I'll have to start looking stuff up, so I'll let it go at
that.

Dale.
==============================================================================

From: "Charles Matthews" <charles@sabaki.demon.co.uk>
Subject: Re: Fundamental group of a space
Date: Wed, 21 Feb 2001 10:19:57 -0000
Newsgroups: sci.math

Hans Shtrulevacht wrote

>If G is any group it is possible to find a topological space such that
>its fundamental group isomorphic to G ?

Yes - one can find such a space X which is also connected, and has
contractible universal cover.  Such spaces are called K(G,1) in homotopy
theory, and should be discussed in standard algebraic topology books.

In fact I think it should be easy to construct a (possibly infinite)
2-complex to do this, taking 1-cells for each g in G and 2-cells for each
relation gg'=g".  This looks like the low-dimensional part of the complexes
used in group cohomology.

Charles

==============================================================================

From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Fundamental group of a space
Date: 22 Feb 2001 16:32:21 GMT
Newsgroups: sci.math

In article <prh26y7df43t@forum.mathforum.com>,
Hans Shtrulevacht <hansshtru@netvis.com> wrote:
>If G is any group it is possible to find a topological space such that
>its fundamental group isomorphic to G ?

Yes, and you can even have all the higher homotopy groups trivial, and
you can even construct such a space in your very own home! I can 
finesse a few details by assuming  G  finite, but that's not really
necessary:

Give  G  the discrete metric, and consider the space F = L^1( [0,1], G ), 
the space of all integrable functions from the unit interval into  G. 
If you like, you may view the elements of  G  as being the unit coordinate
vectors in a  |G|-dimensional space, so this  F  is a subspace of
L^1( [0,1], R^n ), which is a little more familiar to analysts. But
the only reason to mention analysis is so that I can disregard changes
to functions on sets of measure zero.

Within  F  we have the subspace  E  of piece-wise constant functions,
e.g.  f(t) = { g0, if 0 < t < t1;  g1, if t1 < t < t2;  g2, if t2 < t < 1 }.
This  E  is almost the space you want. Here are some things to prove:

1. E is a metric space with  d(f,f') = sqrt(2)*m( { t ; f(t) <> f'(t) } ).

2. E is contractible  (Hint:  H(f,u)(t) = { f(t), if t < u ; 1_G, if t > u } )

3. E is the disjoint union of the subspaces  E_k  (k >= 0)  consisting of
   the functions with exactly  k  'breakpoints'. (Above I gave an example
   of an element of  E_2.)

4. E_k  is the union of |G| (|G|-1)^k  components each homeomorphic to the
   interior of the  k-simplex.  (Hint: e.g. in E_2 fix  g0, g1, g2 and
   send  f ->  (t1, t2-t1, 1-t2). )  We may label the cells in an
   obvious way with labels of the form  [ g0, g1, g2, ..., g_k ].

5. The topological boundary of   [ g0, g1, g2, ..., g_k ]  is the union of
   the cells   [ g0, g1, ..., g_{i-1}, g{i+1}, ..., g_k ]  with 
   appropriate conventions when  g_{i-1} = g_{i+1}

6. G  acts freely on  E , as a group of isometries in fact, and furthermore
   permutes the cells according to the rule
           g [ g0, ..., g_k ] = [ g g0, g g1, ..., g g_k ].
   (Hint for freeness:  for  g <> 1_G, the distance  d( f, g f ) is  1.)

So the space  B = E/G  of orbits under the action of  G  is a space
which not only has  G  as its fundamental group, but has  E  as its
universal cover, making  B = K(G,1); it follows that this  B  is unique
up to homotopy. This construction give  B  an obvious decomposition
into cells (of all dimensions). For example, you might want to show

7. When  |G|=2,  E_k  is the  k-sphere, decomposed into a union of 
   exactly 2 cells of each dimension, so that  E  itself is  S^\infty
   and  B = RP^\infty  (the union of the spaces  RP^k,  each being a
   single  k-cell  larger than its subspace  RP^{k-1}).

dave