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4

The space of immersions has $2^{2g}$ components. Let's go back to the proof of the fact for $g=0$: Smale-Hirsch immersion theory. The form of the result I want is from here. Theorem: The space of immersions $\Sigma_g \to \Bbb R^3$ is homotopy equivalent to the space of bundle injections $T\Sigma_g \to \bf{3}$, the trivial bundle of rank 3 over $\Sigma_g$. ...


2

We define the distance between two sets, $A,B$, in a metric space $(X,d)$ to be $$d(A,B)=\inf_{(a,b)\in A\times B}d(a,b)$$ It's simple to see that with this definition, the distance between the circles is zero.


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This question seems to be getting at the fact that there are different possible notions of “distance” and it's not always obvious which one to pick. Stella's answer suggests one common definition (minimum pairwise distance of the points). But even though there are other useful approaches, pretty much anything we call “distance” includes as an axiom that the ...


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This is an answer which I am not comfortable assessing the correctness of as I cannot find all the relevant sections of Whitney's paper and I don't understand Massey's. This apparently a somewhat well-known question of Hassler Whitney. It seems the Euler number can be nonzero. I think the Euler number (with twisted coefficients) here ...


1

"I mean for anyone who trusts the eye theorem is obvious." How can you trust your eye when you are unable to fully visualize general continuous curve, e.g. a fractal? If a point $z_0$ lies on a fractal that is a Jordan curve, then there is no way to tell in what direction the "inside" is. For this reason we must "be blind" and resort in our proof only to ...


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Start with the standard Morse embedding of the torus into $\Bbb R^3$ (that is, a donut standing up on its side). Rotate this in the plane "perpendicular to the torus" until you're laying flat on the table. An interim stage looks like this picture, and for every time $t \in [0,1)$, the height function is Morse. One of the critical points (when we started, the ...


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Since this question has been hanging around for a few years, I think I should post Prof. Mazur's official answer below: Hello Bombyx mori, Take a look at J.-P. Serre, Exemples de variétés projectives conjuguées non homéomorphes, C. R. Acad. Sci. Paris 258 (1964), 4194-4196. which actually gives two conjugate algebraic varieties with different ...



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