Date: Fri, 5 Mar 1999 13:51:00 -0500 (EST)
From: "Daniel H. Gottlieb" <gottlieb@math.purdue.edu>
Subject:  Re: Hovey response


Here is an idea for problems for Mark Hovey's list, especially ones that
combine Algebraic and geometric topology. There was a problem list attached
to the conference on "Algebraic and Geometric Topology" held at Stanford
in the summer of 1976. The problems appear in

Algebraic and geometric topology. Part 2. Proceedings of the Symposium in Pure
Mathematics of the American Mathematical Society (Twenty-fourth Summer Research
Institute), held at
Stanford University, Stanford, Calif., August 2--21, 1976. Edited by R. James Mi
lgram. Proceedings of
Symposia in Pure Mathematics, XXXII. American Mathematical Society, Providence,
R.I., 1978


The two problems I submitted are still open. They are problems 6 and 7 on
page 252 and 253. They
are still unsolved. ( I guess nobody got past problem 5, which I just noticed
is the Sullivan Conjecture).

Problem number 7 is a conjecture: Any Hurewicz fibration with base space
a sphere and fibre a connected finite complex whose Euler Characteristic is
not zero, must have the projection map induce an onto homomorphism in
integral homology.

The only advance I have been able to make over the discussion appearing with
the problem is that if the conjecture is true for smooth fibre bundles with
fibre a smooth closed manifold, then it is true in general. So the proof
(or counter example) may very well come from differential topology.

Problem 6 is strictly homotopy theory, but it is very easy to state. I haven't
thought about it since the seventies, so it may very well be amenable to
some of the powerful techniques since developed. The reference in the
discussion is updated to

Gottlieb, Daniel Henry; Fibering suspensions. Houston J. Math. 4 (1978),
no. 1, 49--65.

Conjecture: A compact suspension whose Euler characteristic is not zero and
whose total reduced integral homology group is not finite cannot be the
total space of a Hurewicz fibration with compact and noncontractible fibre
and base.

Dan Gottlieb
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Date: Fri, 5 Mar 1999 15:31:00 -0600 (CST)
From: Brayton Gray <brayton@math.nwu.edu>
Subject: Re: comment and question

Well, we all have our favorite problems. But I think that there is a
collection of well recognized problems: the Barratt conjecture, the Moore
conjecture, questions about the v_1 exponent, and many others. I would
like to assemble a moderately sized list to delivered to Mark Hovey for
inclusion on his page. Please send me your candidates. I am happy to have
coworkers in this.

Brayton Gray
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