From: Douglas Zare <zare@math.columbia.edu>
Subject: Re: topology question
Date: Thu, 24 May 2001 20:59:12 -0400
Newsgroups: sci.math
Summary: House with two rooms (contractible but non-collapsible complexes)

John Rickard wrote:

> Consider the usual picture of a Klein bottle immersed in R^3, and
> remove the disk cut off by the self-intersection, giving a space like
> a glass "Klein bottle" (such as one finds at www.kleinbottle.com).
>
> A slice through the plane of symmetry looks something like this:
>
>     __   __
>    /  \  | \
>   |    \  \A\
>   |     \  \-+----_
>   |      \         \
>   |       \----+-_  \
>   |            |  \  \
>   |            |   \  \
>    \            \ B |  )
>     \            \_/  /
>      \               /
>       \______ _____ /
>
> Now attach two disks within the plane of symmetry, attached along the
> loops surrounding A and B.  The resulting complex is contractible!
> [...]
> I'm pretty sure that I've seen this construction or something similar
> before, but I don't know where.

I think this is somehow meaningfully equivalent to a variation on the "house
with two rooms" in Hatcher's notes,
http://www.math.cornell.edu/~hatcher/AT/AT2001.pdf (page 12 or page 4 of
chapter 0). Instead of the one pictured there, I think of two boxes on top of
each other with a chimney from the lower one through the upper and a pipe from
the upper through the lower. Both chimney and pipe are then attached to the
wall of the room they pass through with a disk. In that example, and in your
Klein bottle construction, the triple locus can be three circles, and the two
which bound disks intersect the third in one point. I don't see whether they
are homeomorphic, though that should be an easy check of the pieces and gluing
map, but even if they are, it still seems like the wrong notion of equivalence.

Douglas Zare
==============================================================================

From: Douglas Zare <zare@math.columbia.edu>
Subject: Re: topology question
Date: Fri, 25 May 2001 08:48:45 -0400
Newsgroups: sci.math

John Rickard wrote:

> It seems this is due to R. H. Bing; Lee Rudolph also drew my attention
> to it by email.  There is another picture at
>
>   http://www.qci.jst.go.jp/~hachi/math/library/bing_eng.html
>
> which seems to be your variant rather than Hatcher's.  I do remember
> seeing it before, but I don't know whether my Klein bottle example was
> my own perversion of my indistinct memories of the principle of the
> house with two rooms, or something that I'd also seen somewhere
> before.

I'd like to understand the principle of the house with two rooms better.
The Klein bottle construction is very nice; is there a sense in which it
generates all other contractible but non-collapsible complexes embedded in
R^3? I can't remember where I first encountered the chimney version,
though it may have been something Greg Kuperberg told me about the "animal
problem:" Consider the unions of unit lattice cubes in R^d. Generate an
equivalence relation by saying that two are equivalent if they are
homeomorphic and differ by the addition or removal of a single cube. Are
all unions of cubes homeomorphic to a ball equivalent?

For R^2 this is surprisingly nontrivial, but elementary.
For R^3, the house with two rooms shows that one cannot always descend to
a single cube. This is open. (Should I say that credit is due to Krystyna
Kuperberg and leave it at that? It's counterintuitive that this is a hard
problem.)
For R^6, Siddhartha Gadgil and I showed that finding a path between
homeomorphic unions of cubes is nonalgorithmically hard (perhaps
impossible?) for thickened wedges of 2-spheres, and for balls depending on
some conjectures in algebra. One can even throw in the operation of
subdivision with the same result.

Douglas Zare