Timeline for Do all topological manifolds admit locally flat embeddings into R^n?
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6 events
| when toggle format | what | by | license | comment | |
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| Jan 31, 2017 at 16:48 | comment | added | Igor Belegradek | Yes, this works too. | |
| Jan 31, 2017 at 14:50 | comment | added | Dmitri Pavlov | @IgorBelegradek: Many thanks for the clarification. Isn't such a closed embedding into R^n itself a proper map? (Proper means preimages of compact subsets are compact, in R^n compact subsets are precisely bounded closed subsets, and intersecting a bounded closed subset witha closed subset again gives a closed subset.) This seems to make the other two steps (distance function and multiplication by some R^N) redundant. | |
| Jan 30, 2017 at 16:25 | comment | added | Igor Belegradek | Disclaimer: I did not read Dancis paper carefully which is why I am not posting this as an answer. Regarding "distance to a point": any topological manifold is a finite dimensional locally compact metric ANR, and hence it embeds into some $\mathbb R^n$ as a closed subset (google the above phrase to locate a reference). The distance to a point in $\mathbb R^n$ defines a proper continuous function on any closed subset. | |
| Jan 30, 2017 at 16:11 | comment | added | Dmitri Pavlov | @IgorBelegradek: Indeed, the cited paper by Dancis does not make any assumptions about the manifold. What does “distance to a point” mean in the context of arbitrary topological manifolds? | |
| Jan 29, 2017 at 23:56 | comment | added | Igor Belegradek | Remark 5.21.(2) in sciencedirect.com/science/article/pii/S0166864100000134 states "Results of Dancis show that a proper continuous map of a q-manifold to an n-manifold can be approximated, taking boundaries into account, by locally flat embeddings if $q$ is much less than $n$". The paper linked above is "Topologically, quasiconformally or Lipschitz locally flat approximation of embeddings" by Jouni Luukkainen. Thus start from a proper function on your manifold (eg, distance to a point), multiply the codomain by some $R^N$ and approximate. | |
| Jan 29, 2017 at 21:14 | history | asked | Dmitri Pavlov | CC BY-SA 3.0 |