Let $(M,g)$ be a compact, smooth $n$-manifold with boundary $\partial M$ and let $f: M \to [a,b]$ be a Morse function, whose critical points are interior and which satisfies $f^{-1}(b) = \partial M$.

Via transporting points along the gradient flow lines (coming from an appropriate Riemannian metric on $M$), it is known that $M$ deformation retracts onto a finite CW-complex $S_f \subseteq M$ of dimension at most $n$, whose $k$-cells are in 1:1-correspondence with the index $k$-critical points of $f$. Call $F: M \to S_f$ the resulting retraction.

Additionally choosing an arbitrary CW-structure on $M$, we can construct a cellular approximation $F_{cell}: M \to S_f$ of $F$. Let $\tau(F_{cell}) \in Wh(\pi_1(M))$ be the Whitehead-torsion element induced $F_{cell}$.

Question: Is it known that $\tau(F_{cell}) = 0$, i.e. that $F$ is a simple homotopy equivalence ?

Note that because of Chapman's Theorem, the vanishing of the Whitehead-torison element is independent of the particular choice of CW-structure on $M$.


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