Known uniqueness results say that an embedded submanifold has a unique smooth structure making it an embedded submanifold with the subspace topology, and immersed submanifolds have a unique smooth structure making them immersed if we have a prior fixed topology.

This seems to imply that there is an instance of some immersed submanifold $S\subseteq M$ with a given topology and smooth structure, where it's possible to make $S$ an immersed submanifold with different smooth structure if we're also allowed to endow it with a different topology.

Is there a known example of such a submanifold? My definition is the stricter one that a smooth manifold is the image of an injective immersion. If no example exists, I wouldn't mind relaxing the definition to the image of just an immersion.

  • $\begingroup$ I've long since forgotten some of the terms here, but my wild guess would be that you only need to consider exotic $\mathbb{R}^4$. There are a continuum of homeomorphic, non-diffeomorphic such structures. Or exotic spheres. $\endgroup$ – zibadawa timmy Aug 13 '14 at 4:37
  • $\begingroup$ It would help if you were to clarify what do you mean by an "immersed submanifold". It could be: 1) The image of an immersion, 2) a topological submanifold which is the image of an immersion, 3) a smooth submanifold which is the image of an immersion (and maybe something entirely different). Also: What do you mean by "different topology"? Not the subspace topology induced from $M$? $\endgroup$ – Moishe Kohan Aug 18 '14 at 16:32

The key is to find a set that is the image of two different injective immersions that induce different topologies on it. Here's a hint:

enter image description here $\qquad$ enter image description here

EDIT: For an example where the two submanifolds are not homeomorphic, try this: enter image description here

In one case, the submanifold has two connected components, neither of which is compact. In the other, there are three components, one of which is compact.

  • $\begingroup$ In this case, however, the two resulting immersed subamnifolds are related by a homeomorphism. Is it possible to construct an example where a set $S$ can be made into an immersed submanifold in two different ways so that the resulting manifolds are not homeomorphic? $\endgroup$ – GFR Oct 13 '16 at 22:01
  • 1
    $\begingroup$ @GFR: Sure. See my addition above. $\endgroup$ – Jack Lee Oct 13 '16 at 22:30

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