A contractible relative group-ring complex with a pi-one isomorphism detects a homotopy equivalence

Since the inclusion induces an isomorphism ${\pi }_1(A)\to {\pi }_1(X)$ and $A$ is connected, the covering $p^{-1}(A)\to A$ induced by the universal cover $p:\tilde{X}\to X$ is connected, because the image of ${\pi }_1(A)$ is the whole deck group, lifting loops in $A$ joins every pair of points in a fibre and makes $p^{-1}(A)$ connected. An upstairs loop projects to a loop in $A$ trivial in $X$; injectivity of ${\pi }_1(A)\to {\pi }_1(X)$ makes that projected loop null in $A$, and its nullhomotopy lifts by [F3] to contract the upstairs loop. Thus $\tilde{A}=p^{-1}(A)$ is simply connected; hence $\tilde{A}$ is the universal cover of $A$ and both $\tilde{A}$ and $\tilde{X}$ are simply connected CW complexes.

Because the based relative complex is contractible, its homology vanishes, so by [F1] the integral relative homology $H_i(\tilde{X},\tilde{A};\mathbb{Z})$ vanishes for all $i\ge 0$. The pair $(\tilde{X},\tilde{A})$ is simply connected: both terms are simply connected and path connected by step 1.1, so the relative group ${\pi }_1(\tilde{X},\tilde{A})$ sits between two trivial groups in the long exact sequence and is trivial.

Show by induction on $n\ge 2$ that ${\pi }_n(\tilde{X},\tilde{A})=0$. For $n=2$ the pair is $1$-connected by step 2.1, the space $\tilde{A}$ is nonempty, path connected and simply connected, and $H_1(\tilde{X},\tilde{A};\mathbb{Z})=H_2(\tilde{X},\tilde{A};\mathbb{Z})=0$ by step 2.1, so relative Hurewicz gives ${\pi }_2(\tilde{X},\tilde{A})\cong H_2(\tilde{X},\tilde{A})=0$. For the induction step, if ${\pi }_j(\tilde{X},\tilde{A})=0$ for $2\le j<n$ then the pair is $(n-1)$-connected and the hypotheses of [F2] hold, so ${\pi }_n(\tilde{X},\tilde{A})\cong H_n(\tilde{X},\tilde{A})=0$ by step 2.1.

By the long exact sequence of relative homotopy groups and step 3.1 the inclusion $\tilde{A}\hookrightarrow \tilde{X}$ induces isomorphisms on all homotopy groups, and it is a bijection on path components because both spaces are connected; hence it is a weak homotopy equivalence between CW complexes and therefore a homotopy equivalence by [F3] with the assumed AC.

Descend to the pair $(X,A)$. Covering maps induce isomorphisms on ${\pi }_i$ for $i\ge 2$: every based $S^i$ map lifts uniquely because $S^i$ is simply connected, and its based homotopies lift from the chosen initial lift. A nullhomotopy disk lifts as well; its boundary lift is the original sphere lift by uniqueness. This proves both surjectivity and injectivity. Therefore for every $i\ge 2$ the composite ${\pi }_i(A)\to {\pi }_i(\tilde{A})\to {\pi }_i(\tilde{X})\to {\pi }_i(X)$, in which the outer maps are the covering isomorphisms and the middle map is induced by the homotopy equivalence of step 4.1, is the isomorphism induced by the inclusion $A\hookrightarrow X$; and on ${\pi }_1$ the inclusion is an isomorphism by hypothesis. Therefore $A\hookrightarrow X$ is a weak homotopy equivalence between CW complexes and hence a homotopy equivalence by [F3].

For the cobordism consequence, use the chosen finite CW pair $(X^{\mathrm{\prime }},K)\simeq (W,M_0)$ of [F4]. Contractibility gives $H_1(X^{\mathrm{\prime }},K)=H_0(X^{\mathrm{\prime }},K)=0$, so the homology sequence makes $H_0(K)\to H_0(X^{\mathrm{\prime }})$ an isomorphism; since $W$ is nonempty and connected, so is $M_0$. The fundamental-group hypothesis and relative contraction transfer to $(X^{\mathrm{\prime }},K)$, and step 5.1 shows that $K\hookrightarrow X^{\mathrm{\prime }}$ is a homotopy equivalence. The equivalence of pairs therefore gives the same conclusion for $M_0\hookrightarrow W$.

Suppose in addition that the presentation has relative cells only in degrees $2$ and $3$, with $n\ge 5$ and $\dim W=n+1\ge 6$. Then the dual presentation relative to $M_1$ has handles only in degrees $n-2$ and $n-1$, both at least $3$, and attaching a handle of index $i\ge 3$ to an $n$-manifold preserves the fundamental group: the attaching region $S^{i-1}\times D^{n+1-i}$ is path connected with fundamental group ${\pi }_1(S^{i-1})$, which is trivial for $i\ge 3$, and the handle $D^i\times D^{n+1-i}$ is contractible, so the Seifert--van Kampen pushout over the connected attaching region adds no generator and no relation (Seifert–van Kampen identifies the fundamental group with a group pushout); hence ${\pi }_1(M_1)\to {\pi }_1(W)$ is an isomorphism. For the dual complex, exchange core and cocore in every handle. A lifted attaching/belt intersection with label $g$ becomes the reversed incidence with label $g^{-1}$; translating its ambient orientation contributes $w(g)$, where $w:{\pi }_1(W)\to \{\pm 1\}$ is the orientation character. Thus, up to degree signs and oriented-lift basis units, the new differential is the adjoint transpose $A^{\ast }=(\overline{a_{ji}})$ for $\bar{g}=w(g)g^{-1}$. This is the handle-local calculation of Ranicki’s handle-duality proposition, printed pp. 177–178. The identity $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$ shows $(A^{-1}{)}^{\ast }$ is a two-sided inverse of $A^{\ast }$, and a two-term invertible differential has contraction its inverse. Hence the dual complex is contractible; so step 5.1 applies to the pair $(W,M_1)$ and shows that $M_1\hookrightarrow W$ is a homotopy equivalence as well; with step 6.1 both boundary inclusions are homotopy equivalences and $(W;M_0,M_1)$ is an h-cobordism.