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Let's consider closed simply connected manifold $M^n$ and a $a\in H^k(M)$ and $a*\in H^{n-k}(M)$ is the dual to $a$. Is it true that dual to $a$ is realisable as a immersed sphere or $ a*=bc $ for some $b,c\in H^*(M)$ ? Edit: it is more natural to ask about possibility to decompose dual to $a$ as a product, see example in the answer below. Edit2: let's assume that $M$ is not decompasable, so there is no $X,Y$ such that $M = X\times Y$ |
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Let's consider closed simply connected manifold $M$ M^n$ and a $a\in H^k(M)$ . and $a*\in H^{n-k}(M)$ is the dual to $a$. Is it true that dual to $a$ is realisable as a immersed sphere or $ a=bc a*=bc $ for some $b,c\in H^*(M)$ ? Edit: it is more natural to ask about possibility to decompose dual to $a$ as a product, see example in the answer below. |
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Realisability cohomological class as product or as immersed sphereLet's consider closed simply connected manifold $M$ and a $a\in H^k(M)$. Is it true that dual to $a$ is realisable as a immersed sphere or $ a=bc $ for some $b,c\in H^*(M)$ ?
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