Subject: Question
Date: Sun, 11 Feb 2001 19:33:27 -0500 (EST)
From: Adam Sikora <sikora@CRM.UMontreal.CA>
To: Don Davis <dmd1@lehigh.edu>

Hello,

could you post my question on your list?

Thank you.

-- Adam Sikora

I am interested in pairs of manifolds M, N
(both closed, orientable, of the same dimension)
which have the following property:
 Any map M->N of degree m is (freely) homotopic to a
"branched covering" M->N of degree m.
For example, M=S^n and N=S^n have this property,
but it is not true that any map between closed surfaces is homotopic
to a branched covering.

Is there anything known about such pairs? For example, a simple
characterization of such pairs? (dimension 3 is of particular interest
for me).

We can consider various categories of mflds (eg. simplicial mflds) and
some variations of the definition of the branched cover.
Roughly speaking, I would call a map f:M->N to be
a branched cover of deg m if it has a restriction M-M_s -> N-N_s which
is a cover of degree m, and the set of singular points, M_s, is a
CW-complex of dimension < n-1.
(Maybe, the term "CW-complex" should be replaced by something else).

-- Adam Sikora