Timeline for Homotopy groups of spaces of embeddings
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| May 28, 2014 at 15:29 | comment | added | Tom Goodwillie | "Embeddings from the point of view of immersion theory I", I meant. | |
| May 28, 2014 at 12:50 | comment | added | Tom Goodwillie | The description of the fiber of the tower is in Weiss's paper "Embeddings from the point of view of immersion theory". The "finite complex" that is the base of the fibration is the space of unoriented configurations of $k$ points in $M$, and the "subcomplex" is "infinity". Each fiber of the fibration is the "total" or "iterated" homotopy fiber of a $k$-dimensional cubical diagram consisting of the spaces $Emb(S,N)$ for all subsets $S\subset\lbrace 1,\dots ,k\rbrace$. | |
| May 28, 2014 at 12:46 | comment | added | Tom Goodwillie | Each homotopy fiber of $T_k\to T_{k-1}$ has the homotopy type of the space of sections of a fibration over a finite complex with fixed behavior on a subcomplex. The fiber of the fibration will be simply connected under these hypotheses, and will have finitely generated homology groups if $N$ does. | |
| May 28, 2014 at 12:11 | comment | added | Igor Belegradek | Thank you! I wish this pretty statement would appear in the literature. I gather the base of the tower equal to the space of monomorphisms $TM\to TN$, which is a nilpotent space. It seems your are saying that the homotopy fiber of ${\mathcal T}_k\to {\mathcal T}_{k-1}$ is also nilpotent. Why? | |
| May 28, 2014 at 12:06 | comment | added | John Klein | Yes, I noticed the updates. | |
| May 28, 2014 at 11:34 | comment | added | Tom Goodwillie | You're welcome. By the way, I'm working hard on final (?) revisions of our paper. | |
| May 28, 2014 at 4:47 | history | edited | Tom Goodwillie | CC BY-SA 3.0 |
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| May 28, 2014 at 3:58 | history | answered | Tom Goodwillie | CC BY-SA 3.0 |