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Define $\mathbb{R}P^k$ as the quotient of $S^k$ by the antipodal map, with smooth structure defined so that the projection $p: S^k \to \mathbb{R}P^k$ is a local diffeomorphism. Suppose that $k$ is odd and $Z_1, Z_2 \subset \mathbb{R}P^k$ are compact submanifolds of positive dimension for which the oriented intersection number $I(Z_1, Z_2)$ is defined. Prove $I(Z_1, Z_2) = 0$. (Hint: What conditions are guaranteed by well-definedness of $I(Z_1, Z_2)$?) Is the corresponding statement true for the mod-2 intersection number $I_2$?

Because $k$ is odd and $Z_1, Z_2$ are of complementary dimension, $I(Z_1, Z_2) = I(Z_2, Z_1)$. But then I found trouble to proceed. Could someone point out what is the missing piece? Or if I am on the right track at all?

WishingFish
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1 Answers1

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Hints: First of all, note that for any $\ell<k$, $\mathbb RP^\ell$ and $\mathbb RP^{k-\ell}$ (generically) intersect in a single point. But the hypotheses of the problem are very specific: If $k$ is odd, $\mathbb RP^k$ is orientable, but to make sense of $I(Z_1,Z_2)$, we need both $Z_1$ and $Z_2$ to be oriented. One more hint: What is the intersection number of complementary dimension (and positive-dimensional) oriented submanifolds of $S^k$?

Ted Shifrin
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  • Let $l = 2, k = 4$, so $\mathbb RP^2$ and $\mathbb RP^2$ intersect in a single point? – WishingFish Jul 24 '13 at 00:57
  • Well, yes, if you take two different ones. Remember that you get $\mathbb RP^2\subset\mathbb RP^4$ by choosing a $3$-dimensional subspace of $\mathbb R^5$ (or a great $S^2\subset S^4$). So take two generic such. – Ted Shifrin Jul 24 '13 at 01:01
  • Still not quite get it - should I go with the fact that dimensional complementarity gives isomorphism, which is orientation preserving...? – WishingFish Jul 24 '13 at 01:27
  • I don't understand. Are we on the second part or still on your previous question? And what do you mean by "... Gives isomorphism"? – Ted Shifrin Jul 24 '13 at 01:30
  • I am still trying to show $I(Z_1, Z_2) = 0$. – WishingFish Jul 24 '13 at 01:31
  • Oh. So did you start by figuring out what I asked about submanifolds of the sphere? And then you need to relate that to the original question. – Ted Shifrin Jul 24 '13 at 01:37
  • No, still don't have a clue. I have been thinking about direct sum, but don't think it will work out. – WishingFish Jul 24 '13 at 02:38
  • First prove the intersection number in the sphere is $0$. Then use the fact that the antipodal map will be orientation-preserving on $p^{-1}(Z_i)$. – Ted Shifrin Jul 24 '13 at 02:57
  • I have trouble proving the intersection number in the sphere is $0$. I am also not convinced that $Z_1, Z_2$ intersect in a single point. As submanifolds of $\mathbb{R}P^2, Z_1 ,Z_2$ can be a loop, hence they can intersect twice. – WishingFish Jul 24 '13 at 21:47
  • I didn't say they intersected in a single point. That was a specific example -- $\mathbb RP^m$'s sitting inside. Reread carefully. The sphere question is absolutely essential and basic. What do you know about the sphere topologically that makes it unique to you? – Ted Shifrin Jul 24 '13 at 21:53
  • Symmetric? I have the idea to think of $\mathbb{R}P^m$ as $S^m$, whose orientation cancels out by antipodal points, ending Lefschetz number 0. But I find trouble showing $\mathbb{R}P^m$ has the same orientation as $S^m$. – WishingFish Jul 24 '13 at 21:56
  • I don't know what course you're taking, but I honestly don't think you have the appropriate background. You need to go talk to your professor and work with him/her. – Ted Shifrin Jul 24 '13 at 22:11
  • Hi Ted, it is summer... – WishingFish Jul 24 '13 at 23:25
  • This is a non-standard problem, so I have to assume you got it from a particular class, not a generic textbook. I don't know why differential topology has become so popular this summer! It requires some serious sophistication, particularly this problem. At any rate, I'm off for 2 weeks, so perhaps someone more generous will help :P – Ted Shifrin Jul 25 '13 at 00:42
  • Hi Ted! Thanks for your kind comment! I will be missing your kind help! Have fun for your vacation, hope I will struggle a better understanding (I doubt, but I will try) when you come back! – WishingFish Jul 25 '13 at 01:04
  • I don't have a solid foundation on real analysis, but I am putting a significant amount of time reviewing it now. But what do you mean by background exactly? – WishingFish Jul 25 '13 at 01:09