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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeNov 6th 2021

Created:

Definition

Denote by $Emb_n$ the site of $n$-dimensional smooth manifolds and open embeddings.

An (∞,1)-sheaf $F\colon Emb_n^op\to Top$ of topological spaces is microflexible if for any closed inclusion $K\to K'$ of compact spaces, the induced map $F(K')\to F(K)$ is a Serre microfibration.

An (∞,1)-sheaf $F\colon Emb_n^op\to Top$ of topological spaces is flexible if for any closed inclusion $K\to K'$ of compact spaces, the induced map $F(K')\to F(K)$ is a Serre fibration.

Gromov’s theorem

Given an open manifold $M$, the inclusion of microflexible sheaves into flexible sheaves on the site $Emb_n/M$ is an equivalence of (∞,1)-categories.

Related concepts

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 6th 2021

IS there a canonical reference to point to? I have added pointer to:

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 6th 2021

also:

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeNov 6th 2021

The canonical reference is Section 2.2.1 of

(Why is the section title “Literature” instead of “References”? Is there a difference?)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 6th 2021

Thanks.

Recently I noticed that some people read “References” as meaning “The above text relies on these” instead of the more encompassing “Here is various relevant and useful literature on the topic”. That made me think that the last section of an $n$Lab entry should maybe better be titled “Literature” than “References”. But maybe it’s not so important.

• CommentRowNumber6.
• CommentAuthorTim_Porter
• CommentTimeNov 6th 2021

I have changed References to Reference and Literature in the entry on microbundles. Tht would cover both of the points made by Urs.

I recall lectures at the Liverpool singularity conference on Gromov’s thesis from 1970 (?) and reference to “microgibki” things. Is this the same concept? I could not find a reference and have mislaid my copy of his thesis. The lectures were by André Haefliger I think and are in the Springer Lecture notes volume of the conference. I do not have access to that volume.

• CommentRowNumber7.
• CommentAuthorTim_Porter
• CommentTimeNov 6th 2021
• (edited Nov 6th 2021)

I note that Larry Siebenmann used the term ’micro-gibki’ in his article on Topological Manifolds in the ICM 1970 proceedings. That article may be relevant somewhere on this entry or on the related ones.

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeNov 6th 2021

Микрогибкий translates as microflexible from the Russian, so it is the same notion.

• CommentRowNumber9.
• CommentAuthorTim_Porter
• CommentTimeNov 6th 2021

That explains it.

• CommentRowNumber10.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 27th 2022

"the induced map F(K is a Serre fibration."


The source is correct. It appears that the apostrophe character is creating problems.

1. I have now fixed this issue, thanks again for raising it.

• CommentRowNumber12.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 2nd 2022

The original reference is

• M. L. Gromov, STABLE MAPPINGS OF FOLIATIONS INTO MANIFOLDS, Mathematics of the USSR-Izvestiya 3:4 (1969), 671-694. doi.
• CommentRowNumber13.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 2nd 2022

In Revision 5, I added a reference to the original paper by Gromov.

I then edited the article again in order to fix a problem with superscripts.

What happened then is that in Revision 6 all changes made in Revision 5 were discarded!

2. Thank you for raising this, Dmitri. I made some tweaks to the cache which might have caused this, but have not been able to work on anything for several days for personal reasons, and have now resigned from my involvement in the nLab for reasons explained in the migration thread.