Re: base points in moduli spaces

Jing M Lim wrote:
I don't really know anything about moduli spaces, but I heard the
following vague statement and wondered what it means and why it is
true:

If you have a family of spaces (say topological spaces) depending on
some parameter t, it is not always possible to pick a base point in
each space varying with t.

Can anyone explain what this means?

I don't know whether this addresses the matter you are
interested in, but if the parameter t belongs to a
topological space T, one can view the situation as
being described by a mapping

p: X ---> T

where the mapping takes X_t to t in T. One desirable
feature might be that the base point *_t in X_t be
represented by a continuous (wrt t) selection in X.

The selection s: t |---> *_t would then be a right inverse
to the map p; that is, p(s(t)) = t for all t in T.

If the X is a principal G bundle over T, with G a
topological group (including as a special case a
discrete group), then the existence of such a selection
would be a section of that bundle, which in turn
implies that the space X is homeomorphic to the
cartesian product T x G.

A simple example of this situation can be had by
considering the mapping S^1 --> S^1 by z |--> t = z^2,
where I've identified S^1 with the set of complex
numbers of unit magnitude. The spaces X_t in this
case would be pairs of antipodal points in S^1, and
one can readily see that there is no continuous
selection of one of these points that holds over the
whole range of parameters t.

Of course, this may be nothing at all related to your
actual question; just an example of where one is not
free to make a continuous global selection based on
values of the parameter t.

Dale


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