Newsgroups: sci.math.research
From: gk00@quads.uchicago.edu (Greg Kuperberg)
Subject: Re: Fundamental group of the Hawaiian earring?
Date: Sun, 5 Mar 1995 07:42:03 GMT
In article <9503031517.AA03802@math46.sma.usna.navy.MIL>,
Mark D. Meyerson wrote:
>Can anyone describe or point me to a reference that describes the
>fundamental group of the "Hawaiian earring"? It's the subspace of
>E^2 formed by a countable union of circles C_n of diameter 1/n
>(n a natural number) tangent to (and above) the x-axis at the origin.
I communicated only a brief and unhelpful comment as moderator
in response to this question (namely, try Munkres, "Topology"), but I
had more thoughts on the matter since then. The fundamental
group is a subgroup of an inverse limit of free groups with n
generators with n --> infinity, which you get by considering
the earring as an intersection of spaces consisting of n circles
and a filled-in disk of radius 1/(n+1). But it is not the entire
inverse limit, because the inverse limit contains elements such
as
[L_1,L_2][L_1,L_3][L_1,L_4],[L_1,L_5]...
where [a,b] is the commutator and L_n is a loop going around the
n'th circle. Rather, it is that subgroup of the inverse limit
in which you only see each L_n finitely many times.
This is in some weak sense an explicit description, because inverse
limits are standard fare in descriptions of topological groups. On the
other hand, it isn't much better than saying that "the fundamental
group of the earring is what it is". My conclusion is that one may
have to be content with this type of non-answer.
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Newsgroups: sci.math.research
From: Geoffrey Mess
Subject: Re: Fundamental group of the Hawaiian earring?
Date: Mon, 6 Mar 1995 00:38:55 GMT
In article gk00@quads.uchicago.edu
(Greg Kuperberg) writes:
> In article <9503031517.AA03802@math46.sma.usna.navy.MIL>,
> Mark D. Meyerson wrote:
> >Can anyone describe or point me to a reference that describes the
> >fundamental group of the "Hawaiian earring"? It's the subspace of
> >E^2 formed by a countable union of circles C_n of diameter 1/n
> >(n a natural number) tangent to (and above) the x-axis at the origin.
> The fundamental
> group is a subgroup of an inverse limit of free groups with n
> generators with n --> infinity, which you get by considering
> the earring as an intersection of spaces consisting of n circles
> and a filled-in disk of radius 1/(n+1). But it is not the entire
> inverse limit, because the inverse limit contains elements such
There is a canonical homomorphism from the fundamental group of the
Hawaiian earring to the inverse limit described above, but it has
nontrivial kernel. An example of an element in the kernel
is the commutator [L_1,[L_2,[L_3,...[L_n,...]...]]].
(Thanks to Greg Kuperberg for suggesting this description.)
Infinite words don't always make sense, though this particular one
does, so here's a more explicit description.
At the first step in the construction, a map from [0, 1] to the earring
has been completely specified on two of the four intervals [0, 1/4],...
[3/4, 1], which are mapped to C_1. After n steps, the map is still
unspecified on 2^n subintervals each of length 1/4^n, and where the map
has been defined it maps into the union of C_1, ... C_n. In the end one
gets a continuous map from [0. 1] to the earring which represents an
element in the kernel. (Perhaps this is one of the rare occasions
where Brouwer's theorem that all functions are continuous could be used
to simplify an exposition. I doubt it really helps, because one would
have to show that the construction was constructive, and it's probably
easier to see the continuity directly.)
>it isn't much better than saying that "the fundamental
>group of the earring is what it is". My conclusion is that one may
>have to be content with this type of non-answer.
Agreed. It's known that the fundamental group isn't free and even the
abelianization of the fundamental group isn't free. (Sam White,
Matthew Wiener and I discussed this in the distant past. I don't know
when it was first proved.) Is there anything in particular that you
want to know about the fundamental group? I suppose it could be shown
that every finitely generated subgroup of the fundamental group is
free, though I don't know if this has been done.
If you look in Steenrod's Reviews, subject van Kampen's theorem, you'll
find a reference to an article which discusses a one point union of two
cones over the earring. This space has no nontrivial covering space,
though it is locally connected and has uncountable fundamental group.
--
Geoffrey Mess
Department of Mathematics, UCLA. geoff@math.ucla.edu
==============================================================================
Newsgroups: sci.math.research
From: ruberman@maths.ox.ac.uk (Prof Daniel Ruberman)
Subject: Re: Fundamental group of the Hawaiian earring?
Date: Mon, 6 Mar 1995 11:41:13 GMT
Check out J. Morgan and I. Morrison, "A van Kampen theorem for weak joins,"
Proc Lond Math. Soc 53 (1986) 562-576, for a very complete discussion.
Daniel Ruberman
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Newsgroups: sci.math.research
From: geoff@math.ucla.edu (Geoffrey Mess)
Subject: Re: Fundamental group of the Hawaiian earring?
Date: Mon, 6 Mar 1995 23:12:48 GMT
Woe is me. The infinite commutator I said was in the kernel is in fact
null homotopic, (by an explicit homotopy) and the map to the inverse
limit is injective according to the Morgan-Morrison paper.
--
Geoffrey Mess
Department of Mathematics, UCLA. geoff@math.ucla.edu
==============================================================================
Newsgroups: sci.math.research
From: greg@dent.uchicago.edu (Greg Kuperberg)
Subject: Re: Fundamental group of the Hawaiian earring?
Date: Tue, 7 Mar 1995 00:16:19 GMT
In article <3jg4tg$fu5@saba.info.ucla.edu>,
Geoffrey Mess wrote:
>Woe is me. The infinite commutator I said was in the kernel is in fact
>null homotopic, (by an explicit homotopy) and the map to the inverse
>limit is injective according to the Morgan-Morrison paper.
Well, you were morally correct in the sense that it didn't occur to me
that such a map might not be injective. If you consider the map from
pi_1 of an inverse limit to the inverse limit of pi_1, it clearly might
not be injective. The example you mentioned of the one-point union of
two cones over a Hawaiian earring (as long as you don't connect them at
their tips) is an example of this, because it's cell-like (an inverse
limit of balls), but pi_1 is large.
I guess the outside of the Alexander horned sphere is also an
example.
On the other hand, for the Hawaiian earring itself, if you let H_n be
an approximation to the Hawaiian earring H with n circles and a disk,
then the diameter of H_n - H goes to zero. Presumably Morgan and
Morrison use this condition (but maybe not this condition alone) to
show that if L is a loop in H that bounds a disk D_n in each H_n, then
after repairing the disks there is a convergent subsequence. I
would be interested to see a more general condition that implies that
pi_1 of an inverse limit of polyhedra injects into the inverse limit of
pi_1.
==============================================================================
Newsgroups: sci.math.research
From: dsmit@cs.few.eur.nl (Bart de Smit)
Subject: Re: Fundamental group of the Hawaiian earring?
Date: Fri, 10 Mar 1995 14:34:51 GMT
greg@dent.uchicago.edu (Greg Kuperberg) writes:
>In article <3jg4tg$fu5@saba.info.ucla.edu>,
>Geoffrey Mess wrote:
>> ... the map to the inverse
>>limit is injective according to the Morgan-Morrison paper.
>Well, you were morally correct in the sense that it didn't occur to me
>that such a map might not be injective.
I wrote up a short proof for injectivity to the projective limit
which is very different from Morgan-Morrison's. Reference:
"The fundamental group of the Hawaiian earring is not free",
Internat. J. Algebra Comput. Vol. 2, No. 1 (1992), 33-37.
As the title suggests, the paper also contains an argument showing that
the fundamental group is not free. It uses the description that Greg
mentioned before:
> .... The fundamental
> group is a subgroup of an inverse limit of free groups with n
> generators with n --> infinity, which you get by considering
> the earring as an intersection of spaces consisting of n circles
> and a filled-in disk of radius 1/(n+1). But it is not the entire
> inverse limit ... Rather, it is that subgroup of the inverse limit
> in which you only see each L_n finitely many times.
The fact that this subgroup is not free was actually proved first
in a purely group theoretic paper of G. Higman (J London Math Soc (2)
27 (1952), 73-81).
Bart de Smit (dsmit@wis.few.eur.nl)