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Show that $S^n$ can't be decomposed diffeomorphic in a product of manifolds $X \times Y$ with $dim(X), dim(Y) >0$

I try to prove that using tools of differential topology (basically the first 2 chapters of Guillemin Pollack) i think to these way is the hard way and I not sure to it's posible, I read in another post the proof but using algebraic topology...

First I try to show that $X$ or $Y$ necessarily they should be one point manifolds then i try to find one contradiction if I suppose that $dim(X), dim(Y) >0$ but i have not been able to come up with something

Any hint or help I will be very grateful

Nick
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    @Surb: A sphere is never homeomorphic to a product, I do not know where did you get the idea that $S^n\cong S^1\times S^{n-1}$. – Moishe Kohan Nov 21 '22 at 12:14
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    As a hint: Use the unoriented intersection number. Namely, show that for any two compact submanifolds $M, N$ of $S^n$ of dimension $<n$, $I_2(M,N)=0$. – Moishe Kohan Nov 21 '22 at 12:23
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    @MoisheKohan Oh ok, I have already achieved the statement you give me, so since we are assuming that $S^n \cong X \times Y$ then naturally in $S^n$ we will have two diffeomorphic copies of $X$ and $Y$ respectively let's call them $X'$ and $Y'$ then by the fact already proven we will have that $I_2 (X', Y')=0$ is these correct? – Nick Nov 21 '22 at 17:57
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    @MoisheKohan i not sure how get a contradiction .I hope not to cause any inconvenience and I apologize if this is the case. – Nick Nov 21 '22 at 18:08

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Since it is likely a homework problem, I will only leave you with hints (some of these, you probably already proved in your class):

  1. Suppose that $Z$ is a smooth manifold diffeomorphic to the product of two compact manifolds, $X=M\times N$. Pick a point $(p,q)\in Z$ and consider the submanifolds $M'=M\times \{q\}, N'=\{p\}\times N$ in $Z$. Prove that the submanifolds $M', N'\subset Z$ intersect transversely and at a single point. From this, compute the unoriented intersection number $I_2(M', N')$.

  2. Prove that every smooth map $M\to S^n$ is smoothly homotopic to a constant map provided that $\dim(M)<n$.

  3. Using (2), given two compact submanifolds $M', N'$ of $S^n$ of dimensions strictly less than $n$, compute $I_2(M', N')$.

  4. Finish the proof that $S^n$ is not diffeomorphic to the product of two manifolds of positive dimension by verifying that you get different intersection numbers in (1) and in (3).

Moishe Kohan
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  • i really appreciate your help and that you took the trouble to write these suggestions, the problem appeared in an algebraic topology class and the professor proved the problem using tools from that area but he mentioned that it could also be solved with intersection mod 2 so I tried it. – Nick Nov 22 '22 at 06:12