14
$\begingroup$

Why every smooth orientable 4-dimensional manifold admits an immersion into $\mathbb{R}^{6}$?

This is a one-line question as I see the statement in a comment by Michael Hopkins(update: this is wrong, it is by Peter Kronheimer) . I thought about it for a long time but I do not know how to prove it or to approach it. Characteristic classes provide a way of showing "if this...", but does not help to show the existence of such an immersion. (comment: this is stupid line of thinking because obviously I did not make use of all characteristic tools available to me, as evident in reading Kirby's book)

update: The correct statement is provided in Kirby's book, page 44 Lemma 1, which states such immersion exists iff there exists a characteristic class $x\in H^{2}(M^{4};\mathbb{Z})$ such that $x_{(2)}=-w_{2}$ and $x^{2}=-p_{1}$.

  • 4
    This is probably overkill but there's a result of Cohen saying that every $n$-manifold immerses into a $S^{2n-\alpha(n)}$ where $\alpha(n)$ is the number of $1$'s in the binary expansion of $n$. Now $\alpha(4) = 1$, so every $4$-manifold immerses into $S^7$ and since such an immersion won't be onto, you can project stereographically to $R^6$. Here's [Cohen's paper](http://www.jstor.org/stable/1971304).2011-11-14
  • 2
    @t.b.- wouldn't that be $\mathbb{R}^7$?2011-11-14
  • 0
    @t.b.: I am aware that this might have generalization like every $2n$ manifold have an immersion into $R^{3n}$, etc. But it feels a mile away as I am only attacking on a small problem. I will try to read Cohen's paper in my sparetime. As the previous commentor noted, stereographical projection only offer one from $\mathbb{S}^{n}$ to $\mathbb{R}^{n}$. In your case it should be $\mathbb{R}^{7}$.2011-11-14
  • 1
    @SteveD: sure, you're absolutely right, this was just plain silliness on my part, I leave the comment despite its stupidity. Thanks for the correction :)2011-11-14
  • 0
    @t.b. Making mistakes is common in mathematics. Thank you for providing the link for the paper - I really like it. I double checked from (http://www.math.harvard.edu/archive/272b_spring_05/assignments/Homework_2.pdf), it is $\mathbb{R}^{6}$. So maybe we are just missing something subtle in here.2011-11-15
  • 0
    Thank you very much for this. I'm sure that Ryan Budney or some of the other topologists will have something more pertinent and substantial to say about this, I'm way out of my comfort zone here.2011-11-15
  • 8
    t.b. uh oh, now I have to think about this!2011-11-15
  • 0
    @ChangweiZhou: I don't think the statement you linked to is meant to be part of that problem! Merely an aside, not something you have to prove.2011-11-15
  • 0
    @Steve D: Thank you for the comment. As you may noticed I am not studying in Harvard. I did these exercises in my spare time as I want to consolidate my algebraic topology background, which is quite poor. I ask this in here because I thought it would be impolite to ask Prof. Hopkins on such a "trivial" question. I think Problem 2 can be tackled by standard techniques he described in Problem 1.2011-11-15
  • 0
    @t.b. Again, thank you for your comment - after all the discussions it turns out $\mathbb{R}^{7}$ is the best we can have. So you are correct.2011-11-15

3 Answers 3

1

I emailed Michael Hopkins and he said this problem set is not his, but composed by Peter Kronheimer. So the content of my question is quite inappropriate and I apologize for the misnomber in here. I will leave the question unchanged (for otherwise the answers may be incomprehensible). I will update this "answer" once I worked out Kirby's proof.

9

I suspect you either mis-read the Hopkins statement, or Hopkins made an error. Likely the statement he intended is that every orientable 4-manifold is cobordant to one that immerses in $\mathbb R^6$ -- this is in Kirby's book on 4-manifolds and is a commonly used step in the proof of Rochlin's theorem.

But not every orientable 4-manifold immerses in $\mathbb R^6$, as $w_4$ is an obstruction. This is a theorem of K. Sakuma's. Sakuma Reference

So for example, $\mathbb CP^2$ does not immerse in $\mathbb R^6$ according to Sakuma, since it has odd Euler characteristic.

  • 0
    I think your suspicion is justified. I did not mis-read Hopkins' statement; it is in here (http://www.math.harvard.edu/archive/272b_spring_05/assignments/Homework_2.pdf). Your explanation is very clear. So thank you a lot for clearing my thoughts. Kirby's book give an if and only if condition for such an immersion to exist, and obviously $w_{1}$ is not enough. I shall update my post.2011-11-15
  • 4
    A great thing about being a prof at MIT is you can feel comfortable giving students homework problems where they're asked to prove false statements. It keeps those students on their toes, and (as a prof) you get to find out which ones are sharp! I don't have the Kirby book at home -- could you include Kirby's if and only if statement?2011-11-15
  • 0
    I updated the question contents. I am a student in Bard, not in MIT, so I feel sympathetic for Hopkins's real students. I know I am not brave enough to challenge a master by finding a counter-example myself. I need some time to digest Kirby's proofs. You may find his book in here: (http://free-books.dontexist.com/search?req=The+Topology+of+4-Manifolds&nametype=orig&column%5B%5D=title&column%5B%5D=author&column%5B%5D=series&column%5B%5D=publisher&column%5B%5D=year)2011-11-15
  • 0
    I suspect this is simply an error on Hopkins's part since it is not an exercise, but rather a "cultural aside" after an exercise.2011-11-15
  • 1
    @Adam Smith: it may actually be an error of someone else, as the files showed the signature of Peter Kronheimer, who is another professor in Harvard. So Hopkins may just make use of his HW sheets. I would not pursue this line of thinking as it is obviously unhelpful to help achieve a better understanding of this problem. I will try to write an essay on Kirby's proof and post it in here.2011-11-15
  • 2
    Nice. I'm happy to see that you were up to the challenge I carelessly imposed on you :)2011-11-15
6

This is the main topic of Chapter VI (entitled "Immersing $4$-manifolds in $\mathbb{R}^6$") of Kirby's book "The Topology of 4-Manifolds".

  • 0
    This comment is very helpful. I found Kirby's book and his discussion is clear and understandable. Thanks a lot for the information, I really do not know much about 4-dimensional manifolds.2011-11-15
  • 1
    Kirby's book is a fantastic source for this kind of stuff. I also recommend Scorpan's book "The Wild World of 4-Manifolds".2011-11-15