Uniqueness of Solutions to the Relative Lifting Problem
Jeffrey Rolland
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.
(For enquiring minds who want to know, X = domain = S^1xS^7 and Y = range =
BG+ for G = ZxP and Z (semi-direct product) P, Z = integers, P = binary
icosahedral group aka SL(2,5) aka the Poincare group.)
I start with a map on the 2-skeleta (induced by a homomorphism of the
fundamental groups). Then I wish to solve the Relative Lifting Problem, as
outlined in _Algebraic Topology_ by Hatcher P. 415-418. to extend this map
on 2-skeleta to a map on all of the domain.
Proposition 4.72 gives a condition for existence, which I am able to solve
for the two particular CW complexes in question. Proposition 4.72 says an
extension exists if and only if the map from X^(n+1) \cup C(X^n) to a
K(\pi_j(Y),j+1) induces the 0 class in H^(j+1)(X^(n+1) \cup C(X^n);
\pi_j(Y)) = H^(j+1)(X^(n+1), X^n; \pi_j(Y)) for 1 <= j <= n (where C(X^n)
is the cone on the n-skeleton of X).
The problem is uniqueness.
I know the solution to a similiar problem, the Group Extension Problem.
This seeks to find, for proscribed groups K and Q, all groups G which
satisfy 1 -> K -> G -> Q -> 1, up to an equivalence called congruence
(slightly stronger than isomorphism). For a given "outer action", that is,
for a given homomorphism from Q into Out(K), an extension corresponding to
this outer action exists iff the induced class in H^3(Q;Z(K)) is the 0
class (where Z(K) is the center of the kernel group). Moreover, for an
outer action which induces the 0 class, all solution corresponding to this
outer action are in bijective correspondence with H^2(Q;Z(K)).
What I would like is a similiar kind of uniqueness result (or
classification result) for the Relative Lifting Problem for CW Complexes,
in the case that a relative lift exists.
If anyone can post some kind of uniqueness (or classification) result or a
pointer to one in the literature, I would be greatly indebted.
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,
--
Jeffrey Rolland
<wilds...@hotmail.com>
Maarten Bergvelt
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
<wilds...@hotmail.com>
In article <fso859$hmm$1...@news.ks.uiuc.edu>,
Jeffrey Rolland <wilds...@hotmail.com> wrote:
> Hello, all!
>
> I am stuck trying to do a relative lifting problem.
>
> The basic premise is, I have two 8-dimensional (well, the domain is
> 8-dim'l and the range is infinite dimensional, but all maps are to be
> cellular, so the range may as well be 8-dim'l) (both have finitely many
> cells in each dimension) CW complexes, and I would like to find all
> homotopy classes of maps between them.
<snip>
Jeffrey Rolland
I'm not sure how Maarten Bergvelt got credit for my last post, but here
is yet another follow-up from *me*.
{Moderator's remark: something went wrong in approving the previous message.
I apologize for the confusion this may have caused.]
I have been trying to classify homotopy classes of maps between the
closed surfaces, which, as I indicated in my original post, was one of
the ways I wanted to build intuition.
Prof. Geoghegan pointed out in his response that for a range manifold R
that is aspherical (as all except that sphere and the real projective
plane are), [M,R] is in bijective correspondence with Hom(\pi_1(M),\pi_1
(R)).
Vagn Lundesgaard Hansen has written an aricle "On the Space of Maps of a
Closed Surface into the 2-Sphere" in which he not only classifies all
maps into the sphere, but he gives the fundamental groups of path-
components of maps (up to a solution of the Group Extension Problem).
As Hansen notes at the very end of his paper, this leaves just maps into
the real projective plane.
So, does anyone know of any work that has been done to classify maps of
a closed surface into the real projective plane (or will I get a short
paper published out of doing so)?
Thank you in advance for any assistance you can provide.
Sincerely,
--
Jeffrey Rolland
<wilds...@hotmail.com>
be...@math.uiuc.edu (Maarten Bergvelt) wrote in
news:futm1i$bju$1...@news.ks.uiuc.edu:
Zbigniew Karno
> Hello, once again, all!
>
<snip>
>
> 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
>
Certain homotopy classification of all maps of a closed surface
into the real projective plane was done in the following papers by
Paul Olum:
Mappings of manifold and the notion of degree, Ann. Math. 58
(1953), 458-480 (in particular Section 9).
Invariants for effective homotopy classification and extension of
mappings, Mem. Amer. Math Soc. 37 (1961)
Cocycle formulaas for homotopy classification; maps into
projective and lens spaces, Trans. Amer. Math Soc. 103(1962),
30-44.
Also in the following:
Allan L. Edmonds, Deformation of maps to branched coverings in
dimension two, Annals of Math. 110 (1979) 113-125.
Richard Skora, The degree of map between surfaces, Math. Annalen
276 (1987) 415-423.
Maybe these references will be helpful.
--
Z. Karno