Re: Uniqueness of Solutions to the Relative Lifting Problem

Hello, once again, all!

I'm not sure how Maarten Bergvelt got credit for my last post, but here
is yet another follow-up from *me*.

{Moderator's remark: something went wrong in approving the previous message.
I apologize for the confusion this may have caused.]

I have been trying to classify homotopy classes of maps between the
closed surfaces, which, as I indicated in my original post, was one of
the ways I wanted to build intuition.

Prof. Geoghegan pointed out in his response that for a range manifold R
that is aspherical (as all except that sphere and the real projective
plane are), [M,R] is in bijective correspondence with Hom(\pi_1(M),\pi_1
(R)).

Vagn Lundesgaard Hansen has written an aricle "On the Space of Maps of a
Closed Surface into the 2-Sphere" in which he not only classifies all
maps into the sphere, but he gives the fundamental groups of path-
components of maps (up to a solution of the Group Extension Problem).

As Hansen notes at the very end of his paper, this leaves just maps into
the real projective plane.

So, does anyone know of any work that has been done to classify maps of
a closed surface into the real projective plane (or will I get a short
paper published out of doing so)?

Thank you in advance for any assistance you can provide.

Sincerely,
--
Jeffrey Rolland
<wildstar200@xxxxxxxxxxx>

bergv@xxxxxxxxxxxxx (Maarten Bergvelt) wrote in
news:futm1i$bju$1@xxxxxxxxxxxxxxxx:

Hello, all!

I have been helped by Ross Geoghegan of Binghamton University, who
suggested the obstruction theory chapter of _Homology Theory_ by
Hilton & Wylie (Peter Hilton also being of Binghamton University).

_Homology Theory_ in turn led me to "Obstructions to Extensions and
Homotopies" by Paul Olum. I cannot recommend this paper strongly
enough for the beginner. It's a little light on examples, but the
exposition is unparalleled. If you want a nickel's worth of free
advice, stay away from Hatcher's exposition. His excursion into
Postinikov towers is misleading for the beginner.

Hope this helps. I just wanted to record this for posterity.

Sincerely,
--
Jeffrey Rolland
<wildstar200@xxxxxxxxxxx>

In article <fso859$hmm$1@xxxxxxxxxxxxxxxx>,
Jeffrey Rolland <wildstar200@xxxxxxxxxxx> wrote:

Hello, all!

I am stuck trying to do a relative lifting problem.

The basic premise is, I have two 8-dimensional (well, the domain is
8-dim'l and the range is infinite dimensional, but all maps are to be
cellular, so the range may as well be 8-dim'l) (both have finitely
many cells in each dimension) CW complexes, and I would like to find
all homotopy classes of maps between them.

<snip>

Additionally, if someone knows the set of homotopy classes of maps
between closed, orientable surfaces (the sphere to the torus, the
torus to the torus, the torus to the the double-torus, the
double-torus to the sphere, etc.), and how to derive these sets from
data like the standard CW decompositions of the closed, orientable
surfaces and perhaps their homology and/or homotopy groups, that
would be tremendously helpful.

(Of couse, the set of homotopy classes of maps of the sphere to the
sphere I know :))

Thank you in advance for any assistance you can provide.

Sincerely,



.

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