Two responses to yesterday's question...........DMD
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Subject: Re: three postings
From: "Prof. A. R. Shastri" <ars@math.iitb.ac.in>
Date: Wed, 23 Mar 2005 10:03:20 +0530 (IST)

 I think Jim forgot to tell us that the (expletive withheld)
k (dimension of P) should satisfy the condition k+n_i<m, without which the 
statement is false. With the above assumption, it follows by 
relative version of transversality. 

Anant R. Shastri 
ars@math.iitb.ac.in

>> Subject: query
>> From: James Stasheff <jds@math.upenn.edu>
>> Date: Tue, 22 Mar 2005 08:44:37 -0500 (EST)
>> 
>> Query:
>> Is the following ( or something similar) a therem inthe literature?
>> note I am old enough to write M - N for the complement of N \subset M
>> instead of the (expletive delted) M\N
>> 
>> Let M be a smooth manifold of dim m
>> N a closed submanifold with components N_i of dims n_i
>> If P is a compact smooth manifold of dim k with non-trivial boundary
>> and f : P --> M with \partial P --> M-N
>> then f can be deformed off N keeping \partial P fixed
>> ??
>> 
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Subject: RE: three postings
From: "Lynn Dover" <ldover@math.ualberta.ca>
Date: Wed, 23 Mar 2005 10:34:31 -0700

Jim:

Unless I have mis-understood your question, your statement is false.
Consider, for example,

M = standard 2-sphere
N = the circle at the equator
P = the unit disk
f maps P in the expected way onto the top 2/3 of the sphere (hence covering
N with the boundary of P ending up "below" N)

There is no way to deform f off of N while fixing the boundary of f(P).

Hope this helps.

Lynn Dover
Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, Canada