Subject: Re: Goodwillie posting From: "Prof. A. R. Shastri" Date: Sat, 26 Mar 2005 11:22:35 +0530 (IST) To: Don Davis I find Tom Goodwille's repsonse interesting. Once again, I believe the result he described must be true if k> Subject: Re: three postings >> From: Tom Goodwillie >> Date: Thu, 24 Mar 2005 15:43:57 -0500 >> > >> > 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. >>