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I am trying to understand lens spaces. While understanding the basic definition is rather easy, everything that follows appears to be quite hard. I wanted to understand the classification up to homotopy equivalence, which is outlined as two exercises in Hatcher's book on algebraic topology, page 310 in Chapter 3.E and page 391 in Chapter 4.2.

There I am given a very specfic map between spheres $f\colon S^{2n-1}\to S^{2n-1}$ defined by $$f(r_1 e^{i\theta_1},\dots,r_n e^{i\theta_n})=(r_1 e^{ik_1\theta_1},\dots,r_n e^{ik_n\theta_n}).$$ For this map I need to compute the mapping degree, which is equal to the product $k_1 \cdot\dots \cdot k_n$. For these computations I have never actually seen a general way to do them. I know the rules for mapping degrees, but do not see how they apply, since I have only ever proven things about degrees, but never computed them. How do I tackle such a problem? How do I even begin?

Cheers

Sellerie
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    Here's the general procedure using differential topology, each step can be completed using at most Calculus. The first step to computing the mapping degree is to choose a regular value $p$ in the image of $f$ (in this case it turns out every point is regular). Then for each $x\in f^{-1} p$ you have to compute $sgn(x)$, the sign of the determinant of the Jacobean of $f$ at $x$. Then the mapping degree of $f$ is $\sum_{x\in f^{-1}(p)} sgn(x)$. – William Jun 20 '19 at 16:11
  • Have you learned degree theory from differential topology? That should be very useful. – Lee Mosher Jun 20 '19 at 16:13
  • I have not, but this is quite interesting. Where can I find a proof that these notions of mapping degree are actually equivalent? – Sellerie Jun 20 '19 at 16:15
  • What is the notion of mapping degree that you're supposed to be using? Computing the effect on $H_{2n-1}$? – William Jun 20 '19 at 16:29
  • Yes. At least that's the only notion I have ever been taught before this. Also, it is the only notion introduced in Hatcher's book. – Sellerie Jun 20 '19 at 16:31
  • I think there are two earlier exercises which solve your problem: problems 8 and 9 on page 258. Problem 8 gives you a way of computing the degree of a map between smooth manifolds, and problem 9 is a corollary for when you consider covering spaces. It looks like the meat of the argument is in Exercise 8, I'm not sure off the top of my head how to solve it so I'll have to think about it. – William Jun 20 '19 at 16:45
  • Exercise 8 by itself is not quite enough though. One also needs to augment Exercise 8 with a differential topology exercise. – Lee Mosher Jun 20 '19 at 16:57
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    I'll say that I've noticed this "gap" in the textbook literature of degree theory for some time. I don't know where to find a full textbook proof of the equality between the differential topology and the algebraic topology notions of degree. When teaching such a course one can easily fill this gap, but it is annoying. – Lee Mosher Jun 20 '19 at 17:01
  • How would you then compute the degree of the given map? Are the techniques outlined here the easiest techniques to do so, in your opinion? – Sellerie Jun 20 '19 at 17:03
  • The easiest way to compute is to use the formula given in the opening comment of @William. As he says, you just need calculus. – Lee Mosher Jun 20 '19 at 20:39

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I think I found a nice answer to my question which doesn't involve any calculus whatsoever, but I would like confirmation that this actually works. The argument goes as follows:

The only degree computation I have ever actually seen was the degree of the map $S^1\to S^1,\,z\mapsto z^k$ with degree $k$. Now looking at the original map, I can guarantee, by permuting the entries, that above $f$ can be written as a composition of functions which only act on one entry of $S^{2n-1}$. The degree of $f$ can then be calculated as the product of the degrees of the single entry action. Thus I only need to consider the case $k_1=k$ and $k_i=1$ for $i\neq 1$. So far I am very certain that my argument works. The next step though might be a problem:

Isn't the above map with only one $k_i\neq 1$ exactly the $2n-1$-fold suspension of the exponentiation on $S^1$? Can anyone verify this claim?

Sellerie
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