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Notice added Draw attention by Matthew Kvalheim
Bounty Started worth 300 reputation by Matthew Kvalheim
Removed “compact” from the title since I do not want to assume that the codomain is necessarily compact
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Matthew Kvalheim
Matthew Kvalheim
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Maps with small fibers between compact manifolds of equal dimension

Added more bold to indicate revisions from previous (linked) question
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Matthew Kvalheim
Matthew Kvalheim
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The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold.

Are there any known examples of a compact Riemannian manifold $M$ with (possibly empty) boundary and a smooth manifold $N$ such that $\dim M = \dim N$ and no topological embeddings $M\hookrightarrow N$ exist, but for each $\varepsilon > 0$ there is a continuous map $f:M\to N$ whose fibers $f^{-1}(n)$ all have diameter $< \varepsilon$?

For example, take $N=\mathbb{R}^2$ and $M$ to be a 2-torus with an open diskan open disk removed. Does there exist a sequence $f_k: M\to \mathbb{R}^2$ of continuous maps such that $$\sup_{x\in \mathbb{R}^2} \text{diam} f_k^{-1}(x) \to 0$$ as $k\to \infty$ with respect to some fixed metric on $M$?

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold.

Are there any known examples of a compact Riemannian manifold $M$ with (possibly empty) boundary and a smooth manifold $N$ such that $\dim M = \dim N$ and no topological embeddings $M\hookrightarrow N$ exist, but for each $\varepsilon > 0$ there is a continuous map $f:M\to N$ whose fibers $f^{-1}(n)$ all have diameter $< \varepsilon$?

For example, take $N=\mathbb{R}^2$ and $M$ to be a 2-torus with an open disk removed. Does there exist a sequence $f_k: M\to \mathbb{R}^2$ of continuous maps such that $$\sup_{x\in \mathbb{R}^2} \text{diam} f_k^{-1}(x) \to 0$$ as $k\to \infty$ with respect to some fixed metric on $M$?

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold.

Are there any known examples of a compact Riemannian manifold $M$ with (possibly empty) boundary and a smooth manifold $N$ such that $\dim M = \dim N$ and no topological embeddings $M\hookrightarrow N$ exist, but for each $\varepsilon > 0$ there is a continuous map $f:M\to N$ whose fibers $f^{-1}(n)$ all have diameter $< \varepsilon$?

For example, take $N=\mathbb{R}^2$ and $M$ to be a 2-torus with an open disk removed. Does there exist a sequence $f_k: M\to \mathbb{R}^2$ of continuous maps such that $$\sup_{x\in \mathbb{R}^2} \text{diam} f_k^{-1}(x) \to 0$$ as $k\to \infty$ with respect to some fixed metric on $M$?

Source Link
Matthew Kvalheim
Matthew Kvalheim
  • 501
  • 1
  • 4
  • 14

Maps with small fibers between compact manifolds of equal dimension

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold.

Are there any known examples of a compact Riemannian manifold $M$ with (possibly empty) boundary and a smooth manifold $N$ such that $\dim M = \dim N$ and no topological embeddings $M\hookrightarrow N$ exist, but for each $\varepsilon > 0$ there is a continuous map $f:M\to N$ whose fibers $f^{-1}(n)$ all have diameter $< \varepsilon$?

For example, take $N=\mathbb{R}^2$ and $M$ to be a 2-torus with an open disk removed. Does there exist a sequence $f_k: M\to \mathbb{R}^2$ of continuous maps such that $$\sup_{x\in \mathbb{R}^2} \text{diam} f_k^{-1}(x) \to 0$$ as $k\to \infty$ with respect to some fixed metric on $M$?