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Here's an example I learned from John Franks. This is a nice example because it's used to produce an example of Smale's sphere eversion problem. It also generalizes to include (for example) Will Sawin's comment above.

Consider the Boy's surface. This is an immersion of $\mathbb{R}P^2$ into $\mathbb{R}^3$. If you look at its normal bundle, there's no sense of +1 or -1 (because it's non-orientable) but you can look at its associated unit sphere bundle. This is the orientation double cover, immersed cover--a.k.a. the sphere--immersed into $\mathbb{R}^3$. By scaling the fibers of the normal bundle from 1 to 0, you see the "covering homotopy" of $S^2$ onto the Boy's surface($\mathbb{R} P^2$), as you ask for.

So that's the thing you're looking for, but let's go further--instead of just scaling from 1 to 0, scale from 1 to -1. This is a homotopy, through immersions, of $S^2$ to itself, and it leads to one way in which you can evert the sphere (i.e., turn it inside out). I think this strategy originally came from Shapiro, though any historical corrections are welcome!

More generally, if you immerse any 2-dimensional object (i.e., an un/orientable surface with or without boundary) you can perform the same trick, examining the unit-length elements of its normal bundle. In Will Sawin's example, take an embedded Mobius band and examine its unit normal bundle--this gives the non-standard embedding of the cylinder you're asking for, and scaling the normal bundle to the zero section gives you the "covering homotopy" you seek.

2 Indicated that the Boy's surface example generalizes; added 93 characters in body

Here's an example I learned from John Franks. This is a nice example because it's used to produce an example of Smale's sphere eversion problem. It also generalizes to include (for example) Will Sawin's comment above.

Consider the Boy's surface. This is an immersion of $\mathbb{R}P^2$ into $\mathbb{R}^3$. If you look at its normal bundle, there's no sense of +1 or -1 (because it's non-orientable) but you can look at its associated unit sphere bundle. This is the orientation double cover, immersed into $\mathbb{R}^3$. By scaling the fibers of the normal bundle from 1 to 0, you see the "covering homotopy" of $S^2$ onto the Boy's surface ($\mathbb{R} P^2$), as you ask for.

So that's the thing you're looking for, but let's go further--instead of just scaling from 1 to 0, scale from 1 to -1. This is a homotopy, through immersions, of $S^2$ to itself, and it leads to one way in which you can evert the sphere (i.e., turn it inside out). I think this strategy originally came from Shapiroand Phillips, though any historical corrections are welcome!

More generally, if you immerse any 2-dimensional object (i.e., an un/orientable surface with or without boundary) you can perform the same trick, examining the unit-length elements of its normal bundle. In Will Sawin's example, take an embedded Mobius band and examine its unit normal bundle--this gives the non-standard embedding of the cylinder you're asking for, and scaling the normal bundle to the zero section gives you the "covering homotopy" you seek.

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Here's an example I learned from John Franks. This is a nice example because it's used to produce an example of Smale's sphere eversion problem.

Consider the Boy's surface. This is an immersion of $\mathbb{R}P^2$ into $\mathbb{R}^3$. If you look at its normal bundle, there's no sense of +1 or -1 (because it's non-orientable) but you can look at its associated unit sphere bundle. This is the orientation double cover, immersed into $\mathbb{R}^3$. By scaling the fibers of the normal bundle from 1 to 0, you see the "covering homotopy" of $S^2$ onto the Boy's surface ($\mathbb{R} P^2$), as you ask for.

So that's the thing you're looking for, but let's go further--instead of just scaling from 1 to 0, scale from 1 to -1. This is a homotopy, through immersions, of $S^2$ to itself, and it leads to one way in which you can evert the sphere (i.e., turn it inside out). I think this strategy originally came from Shapiro and Phillips, though any historical corrections are welcome!