MathOverflow Asked by Fredy on December 21, 2021
The Morse perfection of a closed differentiable manifold $Sigma^n$ is defined to be the largest integer $k$, such that there exists a smooth mapping
$$p:S^ktimesSigma^nrightarrowmathbb{R}$$
where $S^k$ is the standard sphere, such that
(i). For any $xin S^k$, $p|_{(x,Sigma)}$ restricts to a Morse function with two critical points over $Sigma$.
(ii). $p|_{(x,Sigma)}=-p|_{(-x,Sigma)}$.
Clearly, a manifold with positive Morse perfection is a topological sphere. Using Borsuk-Ulam theorem we know that the Morse perfection of a homotopy sphere is no greater than its dimension.
My question is, what happens if a homotopy sphere has Morse perfection equal to its dimension, is it diffeomorphic to the standard sphere?
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