Whitehead Torsion and the S Cobordism Theorem

Choose a homotopy inverse $r:W\to M_0$ of the inclusion $\iota$ together with homotopies $H:r\iota \simeq {\mathrm{i}\mathrm{d}}_{M_0}$ and $K:\iota r\simeq {\mathrm{i}\mathrm{d}}_W$; by [F1] the inclusion is a homotopy equivalence and $r$ may be assumed cellular with cellular homotopies, so all this data is compatible with the CW structure induced by the handle decomposition.

Lift $r$ and the homotopies to the universal covers; the lifted inclusion $C_{\ast }(\tilde{\iota }):C_{\ast }({\tilde{M}}_0)\to C_{\ast }(\tilde{W})$ is a right $R$-linear chain homotopy equivalence, with the chain homotopies induced by the lifted geometric homotopies, and its algebraic mapping cone $\mathrm{Cone}(C_{\ast }(\tilde{\iota }))$ is contractible by the lifted-cone lemma of [F2].

Consider the based pair sequence $0\to C_{\ast }({\tilde{M}}_0)\stackrel{C_{\ast }(\tilde{\iota })}{\to }{C}_{\ast }(\tilde{W})\to C_{\ast }^{\mathrm{h}}(W,M_0)\to 0$ of [F4]; it is degreewise based exact and split, because the relative cells of the handle decomposition and the cells over $M_0$ together form a basis of $C_{\ast }(\tilde{W})$ in each degree.

The quotient map $\gamma :\mathrm{Cone}(C_{\ast }(\tilde{\iota }))\to C_{\ast }^{\mathrm{h}}(W,M_0)$, $\gamma (b,a):=[b]$, is a chain map for the cone differential of The mapping cone of a chain map, since $q\circ C_{\ast }(\tilde{\iota })=0$ and $q$ is a chain map; in the degreewise splitting $C_{\ast }(\tilde{W}{)}_n=C_{\ast }({\tilde{M}}_0{)}_n\oplus C_n^{\mathrm{h}}$ of [F4] its kernel consists of the pairs $(C_{\ast }(\tilde{\iota })a^{\mathrm{\prime }},a)$ and is the complex $K(a^{\mathrm{\prime }},a)=(\mathrm{\partial }{a}^{\mathrm{\prime }}+a,-\mathrm{\partial }a)$ on $C_{\ast }({\tilde{M}}_0{)}_n\oplus C_{\ast }({\tilde{M}}_0{)}_{n-1}$, which the explicit map $s(a^{\mathrm{\prime }},a)=(0,a^{\mathrm{\prime }})$ contracts, so the kernel is contractible. The displayed sequence $0\to \ker \gamma \to \mathrm{Cone}(C_{\ast }(\tilde{\iota }))\to C_{\ast }^{\mathrm{h}}(W,M_0)\to 0$ is degreewise split; choosing a graded splitting $\sigma$ and correcting it by the kernel contraction as in the proof of the based-exact-sequence clause of the AT-22 sum theorem produces a chain section ${\sigma }^{\mathrm{\prime }}$ with $d{\sigma }^{\mathrm{\prime }}={\sigma }^{\mathrm{\prime }}d$, and then $\gamma H{\sigma }^{\mathrm{\prime }}$ contracts $C_{\ast }^{\mathrm{h}}(W,M_0)$ for any contraction $H$ of the cone.

Therefore $C_{\ast }^{\mathrm{h}}(W,M_0)$ is contractible, with a right $R$-linear contraction $s$ constructed from the lifted homotopy inverse and homotopies; in particular the contraction is produced by the geometric data and not inferred from the vanishing of homology.

Apply the pairs clause of [F1] to the cellular map of pairs $f=(\iota ,{\mathrm{i}\mathrm{d}}_{M_0}):(M_0,M_0)\to (W,M_0)$: both restrictions are homotopy equivalences, the first by hypothesis and the second as an identity, so $\tau (\iota )=j_{\ast }\tau ({\mathrm{i}\mathrm{d}}_{M_0})+\tau (f_{\mathrm{r}\mathrm{e}\mathrm{l}})$ where $f_{\mathrm{r}\mathrm{e}\mathrm{l}}$ is the induced map of relative based cellular complexes.

The source relative complex of the pair $(M_0,M_0)$ is the zero complex, so $f_{\mathrm{r}\mathrm{e}\mathrm{l}}$ is the zero map from the zero complex into the based handle complex $C_{\ast }^{\mathrm{h}}(W,M_0)$; its algebraic mapping cone is therefore $C_{\ast }^{\mathrm{h}}(W,M_0)$ itself. By [F1] the cone is contractible and $\tau (f_{\mathrm{r}\mathrm{e}\mathrm{l}})$ is its contraction torsion, so the contraction torsion of $C_{\ast }^{\mathrm{h}}(W,M_0)$ is defined and, by [F2], independent of the chosen contraction.

The first summand vanishes: the identity of $M_0$ is simple, exhibited by the empty sequence of elementary operations (Simple homotopy equivalence), so its Whitehead torsion vanishes by Simple homotopy equivalences have zero torsion; since $j_{\ast }$ is a homomorphism this gives $\tau (\iota )=\tau (f_{\mathrm{r}\mathrm{e}\mathrm{l}})=\tau (C_{\ast }^{\mathrm{h}}(W,M_0))$.

For a presentation with handles only in degrees $q,q+1$, [F2] identifies the complex with $0\to C_{q+1}\stackrel{A}{\to }{C}_q\to 0$. The contraction supplied by step 2.1 satisfies $A{s}_q={\mathrm{i}\mathrm{d}}_{C_q}$ and $s_{q}A={\mathrm{i}\mathrm{d}}_{C_{q+1}}$, so $A$ is invertible and $s_q=A^{-1}$. Thus the odd-to-even map $d+s$ has matrix $A$ when $q$ is even and $A^{-1}$ when $q$ is odd, giving $\tau (C_{\ast }^{\mathrm{h}}(W,M_0))=(-1{)}^q[A]$. Step 2.2 identifies this class with $\tau (\iota )$.

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.

The class ${\tau }_H(W,M_0)$ is defined as the contraction torsion of the based handle complex of the presentation, and by [F1] it equals the AT-22 torsion of the inclusion $M_0\hookrightarrow W$ for the associated finite CW structure; consequently the auxiliary choices made in the handle-complex definition (lifts of handles, core orientations, handle order) are exactly the choices controlled by the AT-22 independence theorem and the basis-change lemma.

Independence of the chain contraction is the contraction-independence lemma for contraction torsion, and independence of the cellular representative, basepoint paths, cover identification, lifts, orientations and order is [F2] applied to the inclusion with the associated CW structures.

If $H$ and $H^{\mathrm{\prime }}$ differ by the listed elementary modifications, then by [F4] each modification preserves the class, so ${\tau }_H(W,M_0)={\tau }_{H^{\mathrm{\prime }}}(W,M_0)$; this argument covers exactly the listed moves, and the elementary modifications of the previous lemma include cancelling-pair creation and deletion, handle slides of equal-index handles, reordering and re-choices of oriented lifts.

Steps 2.1 and 3.1 give fixed-presentation auxiliary-choice independence and invariance under the listed elementary moves, and step 1.1 gives agreement with AT-22's torsion of the inclusion for the associated CW structure; nothing in the argument compares presentations that are not connected by the listed moves, so no invariance under arbitrary changes of handle presentation is asserted.

By [F1] the height function of the product is a critical-point-free Morse function and presents $M_0\times [0,1]$ relative to $M_0\times \{0\}$ with the empty handle list; the relative CW pair induced by this presentation is $(M_0,M_0)$ with no relative cells.

By [F2] the based handle complex of the empty presentation is the zero complex in every degree, because there are no handles to contribute basis vectors; its unique chain contraction is the zero map, and the parity map of the zero complex is the empty matrix, whose class is $0$ in ${\tilde{K}}_1(\mathbb{Z}[{\pi }_1(M_0)])$ and hence $0$ in the Whitehead group.

By [F3] the presentation-indexed torsion ${\tau }_{H_0}(M_0\times [0,1],M_0\times \{0\})$ is this contraction torsion, so it equals $0$ in $\mathrm{Wh}({\pi }_1(M_0))$.

Start from any given finite presentation. By the constructive rearrangement and zero-handle procedure of [F1], put it into adapted index order with no $0$-handles, using attaching isotopies, disjoint commutations and geometric $0\mathrm{/}1$ cancellations. The incoming inclusion is a homotopy equivalence. For each $1$-handle form the circle from a parallel half-core and a joining arc in the incoming boundary with its attaching balls removed. Choose the relative path class of this arc using the surjection ${\pi }_1(M_0)\to {\pi }_1(W)$ so that it cancels the loop class of the half-core and a reference joining arc; relative one-dimensional general position gives an embedded representative avoiding the other attaching balls. Shrink the existing $2$-handle attaching tubes and perturb the circle away from their core circles, using $1+1<n$, so it lies in the common part of the levels before and after the $2$-handles. Its loop is therefore null in $W$ and hence in the level after the $2$-handles, since later handles have index at least three and the reverse trace to that level has index at least three. Choose an embedded nullhomotopy disk in that level, possible because $n\ge 5$, and use its normal frame to make the circle a framed embedding trivial one level higher. It meets the selected $1$-handle belt once and avoids the other belts. [F2] trades that handle for a $3$-handle. After finitely many steps all indices are at least two.

For $r=2,\dots ,q-1$, assume all indices are at least $r$ and fix an $r$-handle $e$. Contractibility of the based handle complex and $C_{r-1}=0$ give coefficients $x_j$ with ${\sum }_j{d}_{r+1}[{\phi }_j]\cdot x_j=[e]$. Start with a trivial framed $r$-sphere $\alpha$ in the outgoing level after the $r$-handles; its class is zero. The modification construction gives a sphere $\beta$ with class $[e]$ isotopic to $\alpha$ one level higher. It supplies a framing by transporting the trivial normal frame along that higher-level isotopy; $\beta$ lies in the unchanged open part common to the two levels. Since $r\le q-1\le n-3$, the corrected homology lemma applies. For $r=2$, [F6] supplies its incoming fundamental-group injection, because the presentation still represents the same h-cobordism. It puts $\beta$ into single-point position. Carry the higher attaching data along that ambient isotopy, preserving its one-level-higher triviality and framing. The elimination lemma trades $e$ for an $(r+2)$-handle. Repeating deletes every handle of index below $q$.

Reverse the triad. An original handle of index $k$ becomes a dual handle of index $n+1-k$. Remove dual $0$- and $1$-handles by the same low-index argument, then eliminate dual indices $r=2,\dots ,n-q-1$. This upper bound is at most $n-3$, so only the already proved homology range and its $r=2$ h-cobordism complement condition are used. Trading such a dual $r$-handle creates a dual $(r+2)$-handle, which is an original handle of index $n-r-1\ge q$; the dual $1$-handle trade creates original index $n-2\ge q$. Thus this phase removes all original indices at least $q+2$ without reintroducing any original index below $q$. The remaining indices are exactly $q,q+1$, for every $2\le q\le n-2$. No arbitrary-sphere flipped endpoint is invoked.

The remaining based handle complex is concentrated in degrees $q,q+1$ and contractible. Its only differential, the intersection matrix, is therefore an isomorphism; its two free modules have equal rank and the two handle numbers agree, because tensoring this isomorphism with $\mathbb{Z}$ along the augmentation gives an isomorphism of finite free abelian groups. All changes preserve the relative diffeomorphism type and are the indicated elementary presentation modifications. This proves the full stated normal form without simple connectivity.

Put $N={\mathrm{\partial }}_1{W}_q$. Since all initial indices are at least two, connectedness of $W$ forces connectedness of its nonempty incoming collar and $W_q$. The surgery description of $N$ replaces tubes of $(q-1)$-spheres, of codimension $n-q+1\ge 3$, by $D^q\times S^{n-q}$, with connected gluing regions; thus $N$ is connected. By [F2], ${\pi }_1(N)\to {\pi }_1(W)$ is an isomorphism. Removing the higher attaching cores, of codimension $n-q\ge 2$, preserves connectedness and surjects on fundamental groups: perturb paths and loops transverse to them using [F3], with expected intersection dimensions $1+q-n<0$. Radial collar retraction replaces core complements by tube complements. Consequently every desired group label is represented by a path in $N^{\circ }$.

For handle ${\phi }_j$, take a parallel copy $t_j=S^q\times \{z\}$ of its attaching sphere, with $z$ on the boundary of the normal disk. Push it slightly into $N^{\circ }$. Its lift represents $d[{\phi }_j]$ by [F1], with orientation chosen accordingly. After the handle is attached it bounds the outgoing disk $D^{q+1}\times \{z\}$, and is therefore a trivial framed sphere in ${\mathrm{\partial }}_1{W}_{q+1}$. The disk is disjoint from $f$, which lies in the old common open region.

Fix a monomial $\epsilon \gamma$ in $x_j$. Choose a joining path from $f$ to $t_j$ in $N^{\circ }$ with the required relative homotopy class by step 1.1. Smooth it, preserve its embedded endpoint germs, approximate it by an embedded arc and perturb its interior off the two spheres by [F3]; the inequalities are $2<n$ and $1+q<n$. Thicken it to a sufficiently thin framed band as in [F3]. Lifting that band fixes the lift of its second sphere; by choosing the path class this is $T_{{\gamma }^{-1}}{\tilde{t}}_j$, namely ${\tilde{t}}_j\cdot \gamma$. The band sum thus has class $[\tilde{f}]+\epsilon d[{\phi }_j]\cdot \gamma$: collapsing the band to its core gives the pinch map whose two oriented sphere classes add, and the negative orientation gives the sign $\epsilon$. This uses right multiplication throughout.

In the higher outgoing level, the disk bounded by $t_j$ together with a thin neighborhood of the joining band lets the added sphere shrink along the band back to its end disk on $f$. This is an isotopy supported in that disk-and-band neighborhood; it preserves a supplied normal framing by transporting it along the same local motion. It is the local band-sum isotopy in Lück’s Modification Lemma, printed pp. 15–16. The resulting sphere remains in $N^{\circ }$, while its comparison isotopy takes place one level higher. When it is attaching data, [F4] transports all subsequent handles.

Expand each $x_j$ as its finite signed sum of group elements. Repeat steps 2.1–3.1 for those finitely many monomials, always choosing fresh sufficiently thin parallel copies and bands. Compatible lifts agree on the unchanged part of the sphere, so the class additions sum to the displayed right-linear formula. Concatenating the higher-level isotopies proves the isotopy and framing assertions. Countable choice is inherited through the geometric suppliers; the coefficient expansion and the number of modifications are finite.

Group the terms of the sum by label: for each $g\in \pi$ set $k_g:={\sum }_{j:g_j=g}{\epsilon }_j\in \mathbb{Z}$, a finite sum that is nonzero only for the finitely many occurring labels, so that ${\sum }_{j=1}^r{\epsilon }_j[g_j]={\sum }_{g\in \pi }{k}_g[g]$ is the expansion of the left-hand side in the basis of [F1]. By the uniqueness of that expansion, the equality ${\sum }_{j=1}^r{\epsilon }_j[g_j]=0$ holds exactly when $k_g=0$ for every $g\in \pi$, and the equality ${\sum }_{j=1}^r{\epsilon }_j[g_j]=[h]$ holds exactly when $k_h=1$ and $k_g=0$ for every $g\mathrm{\ne }h$.

Assume no two indices carry equal labels with opposite signs, so that for every occurring label $g$ all terms with $g_j=g$ share one sign and $k_g$ is $\pm$ the number of occurrences of $g$, hence a nonzero integer. If ${\sum }_{j=1}^r{\epsilon }_j[g_j]=0$, then step 1.1 forces $k_g$ to vanish for every label, contradicting the nonzero coefficient of each occurring label; therefore in the zero-sum case some pair of indices has equal labels and opposite signs.

Assume no two indices carry equal labels with opposite signs and consider the single-monomial case ${\sum }_{j=1}^r{\epsilon }_j[g_j]=[h]$ with $r\ge 2$. By step 1.1 every label $g\mathrm{\ne }h$ must have $k_g=0$, and under the assumption every occurring label has a nonzero coefficient, so no label other than $h$ occurs and all $r$ terms carry the label $h$. Then $k_h={\sum }_{j=1}^r{\epsilon }_j=r-2m$, where $m$ counts the indices with ${\epsilon }_j=-1$, and the equation $k_h=1$ with $r\ge 2$ gives an odd $r\ge 3$ and $1\le m\le r-1$; hence some index has sign $+1$ and some index has sign $-1$, both with label $h$, so an opposite-signed pair of equal labels exists in the single-monomial case as well.

Steps 2.1 and 2.2 settle the zero-sum and the single-monomial case respectively, so under the stated hypotheses and $r\ge 2$ there are always indices $j_1\mathrm{\ne }{j}_2$ with $g_{j_1}=g_{j_2}$ and ${\epsilon }_{j_1}=-{\epsilon }_{j_2}$. The argument used only the basis expansion of $\mathbb{Z}[\pi ]$, no property of $\pi$ beyond it and no choice principle.

The outgoing level $N={\mathrm{\partial }}_1{W}_q$ has the same fundamental group as $W$: its reverse trace handles have index $n+1-q\ge 4$, and the remaining forward handles have index at least $q+1\ge 3$, so van Kampen changes neither fundamental group. Isotope $f$ transverse to the finitely many belts. By [F1] its coefficients are their signed group-label sums, a single $\pm \gamma$ in the right coefficient of the distinguished handle and zero at all others. Thus the labels in ${\pi }_1(W)$ are also the actual labels in ${\pi }_1(N)$.

Unless the required single-point/disjoint configuration already holds, [F4] gives an opposite-sign equal-label pair on one belt. Choose arcs in the two spheres avoiding every other intersection. Equal labels make their Whitney circle nullhomotopic in $N$ by [F2] and step 1.1.

If $3\le q\le n-3$, both sheet dimensions are at least three, so [F3] gives the stable Whitney construction. Its disk and tube can also avoid all other attaching spheres and belts: their codimensions are at least three, and relative transversality of a disk has negative incidence dimension. For $q=2$, use [F5] and the explicit incoming injection to fill the shifted boundary circle inside the complement of all belts, then clear the $2$-sphere and any other $2$-dimensional attaching spheres. The helper supplies a clean admissibly framed disk and the corresponding two-dimensional move. This is a handle-complement argument, not an inference from simple connectivity of the level alone. Each move removes precisely the chosen pair and leaves all other intersections fixed.

Repeat finitely many times, decreasing the total intersection count by two. The surviving signed-label sums force one point at the distinguished handle and none at the others. For a diagonal family, choose each disk and tube disjoint from all other attaching spheres as in step 3.1; those spheres and the previously arranged configurations remain fixed, so the process realizes every unit diagonal entry simultaneously. Isotopy transports any given normal framing. This proves all assertions in the stated range, including the $q=2$ handle context.

By [F1], the $K_1$ class of $A$ is a finite sum of classes of units $\pm g$, with inverses again of that form. Represent that sum by a finite diagonal matrix $D$, padding $A$ and $D$ by identities to the same size. Then $(A\oplus I)D^{-1}\in \mathrm{E}(R)$. Membership in the stable union gives a further finite padding for which it is a finite product of elementary matrices. This proves the displayed factorization; $D$ has $c+b$ diagonal entries.