Subject:
Re: three postings
From:
Tom Goodwillie <tomg@math.brown.edu>
Date:
Thu, 24 Mar 2005 15:43:57 -0500
To:
Don Davis <dmd1@lehigh.edu>

>
>
> Thanks
> I as trying to be provocative (cf. Larry Summers)
> expecting an answer in terms of conditions on k in terms of n_i, m
>
>    Jim Stasheff        jds@math.upenn.edu
>  >>  >>
>
>>>   >> 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
>>>   >> ??
>
>  >>  >>


As people have noted, Jim, the best you can say is that it's true if k + n_i < m for all i.

But this reminds me of another more intricate question, which might conceivably be what you were really after. Anyway, I'll describe it since it is dear to my heart:

Suppose for example that N is the disjoint union of N_1 and N_2, and that two embeddings f_1 : P ---> M and f_2 : P ---> M are disjoint from N_1 and N_2 respectively, and that f_1 is isotopic to f_2. Then in fact these are isotopic to an embedding f : P ---> M disjoint from both N_1 and N_2 if

k < 2m - n_1 -n_2 - 2 .

In fact, you can isotop the whole isotopy away from the union of N_1 and N_2 without ever violating the disjointness of f_i from N_i. The better, stronger, statement is that in the diagram of spaces of embeddings

Emb(P,M-N_1-N_2) --> Emb(P,M-N_1)
               |                              |
              V                             V
    Emb(P,M-N_2)    -->       Emb(M)

the canonical map from the upper left to the homotopy pullback of the others is

(2m - n_1 -n_2 - 3)-connected .

This is known (if John Klein and I ever finish our paper about it) in almost all cases (like maybe all except when m=3). The general statement, involving say r disjoint N_i's, involves an r-dimensional cubical diagram of embedding spaces. This sort of multirelative connectivity info is the key to applying "calculus of functors" methods to embeddings.

Tom Goodwillie

P.S.  The fact that John and I have not finished our paper is all my fault.