Handle slides and cancelling-pair creations preserve Whitehead torsion

Consider one elementary modification of $H$ of the listed types. For a cancelling-pair insertion or deletion, [F1] shows that the relative based complex of the presentation changes by an elementary contractible two-term complex, i.e. by an elementary expansion or contraction whose torsion class is $0$; by the based sequence clause of the AT-22 sum theorem the torsion of the complex is unchanged.

For a slide over handle $h_i$, attach $h_i$ first. Its parallel attaching sphere bounds a core-parallel disk in the new outgoing region, so the framed band sum defining the slid attachment is isotopic to the old attachment in this new boundary: shrink the parallel sphere across that disk and back along the band. The argument includes the attaching $0$-sphere endpoint interpretation for $1$-handles. The isotopy comparison in [F4] preserves the total manifold and carries later data. Lifting the same band gives the core-basis calculation of [F2], which shows that the presented manifold and the CW model are unchanged relative to $M_0$ up to diffeomorphism and homotopy equivalence, while the handle chains change by an elementary basis change; the corresponding change of the based complex is the changes $P^{-1}AQ$ with elementary matrices $P,Q$, whose classes in $K_1$ and hence in $\mathrm{Wh}$ are $0$, so the contraction torsion is unchanged.

For a reordering of equal-index handles, a change of core orientation or a change of oriented lift, [F3] identifies the change of the displayed basis as a permutation, a replacement of a basis vector by its negative, or a replacement by a unit $\pm g$; each of these basis changes has zero class in the Whitehead group, so again the contraction torsion is unchanged. For an isotopy of full attaching embeddings, transport every later attachment by the isotopy comparison of [F4]. The resulting filtration comparison takes each oriented lifted handle core to its corresponding core, hence induces the identity in these relative handle bases and commutes with the cellular boundaries. The based complexes therefore have equal torsion. Commuting two disjoint attaching regions leaves their glued manifold and core cells unchanged, so it gives the same cellular complex in its degree-ordered handle bases.

By [F4] the presentation-indexed torsion depends only on the contraction torsion of the based handle complex, so each single elementary modification leaves ${\tau }_H(W,M_0)$ unchanged in $\mathrm{Wh}({\pi }_1(M_0))$; composing the finitely many modifications relating $H$ to $H^{\mathrm{\prime }}$ gives ${\tau }_H(W,M_0)={\tau }_{H^{\mathrm{\prime }}}(W,M_0)$. The statement concerns exactly the listed moves, and no claim is made about presentations not connected by them.