Date: Thu, 9 Mar 1995 13:56:46 -0500 (EST)
To: Rod Gomez
From: Dave Rusin
Subject: Re: When can a map be lifted from S^2 to S^3?
In article <3jf6uv$305@lastactionhero.rs.itd.umich.edu> you write:
>Let M, and N be manifolds such that N is a quotient space of M i.e. N = M/~.
>for some relation ~. Let F: M ---> N denote the quotient map.
>
>It is well-known that S^3 has a fibration with fiber S^1 (the Hopf fibration)
>and base space S^2; i.e. there exists a quotient map p: S^3 ----> S^2 with the
>pre-image of any point of S^2 being a circle S^1 in S^3.
>
>Now suppose h: S^2 -->N is a smooth map. Under what conditions can h be
>lifted to a smooth map h': S^3 ---> M?
What do the equivalence classes in M look like? If they are discrete, there
is an h'. If they are connected and simply connected,
then since the S^1's have to map to equivalence classes, there is a lifting
h' iff the original map h: S^2 --> N lifts to M. (Hmm: at least this
is OK in the topological category, I'll have to think about smoothness).
dave