# Is the deformation along flow lines a simple homotopy equivalence?

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 $$M$$.