Presentation chains compute twisted first homology and peripheral cycles

Construct the actual presentation map. Begin with the generator circles and attach the finitely many relator disks. The circle wedge has the free fundamental group in [F1]. For one disk attachment, the old complex together with the outer region $r>1\mathrm{/}3$ in the attached disk is an open set deforming to the old complex; the inner region $r<2\mathrm{/}3$ is an open contractible disk. Their overlap is an annulus, and its generator maps to the attaching word in the old complex. These are open sets also for a noninjective attaching map, as can be checked on the old complex and the disk before taking the adjunction quotient. Van Kampen therefore quotients the old group by the normal closure of that word. Finite iteration proves ${\pi }_1(P,p)=⟨x_j\mid r_i⟩$ with the stated generators. Send the generator circles to the chosen loops in $Y$. Each relator is trivial in ${\pi }_1(Y,y)$ by the given presentation isomorphism, so its based nullhomotopy descends from the cone on the circle to a filling disk. Choose these finitely many fillings and glue along their attaching circles; the quotient topology gives a continuous map $f:P\to Y$. It induces exactly the prescribed isomorphism. No approximation or CW structure on the target is needed.

Verify the actual augmented cover complexes. Put $R=\mathbb{Z}[\mathrm{\Gamma }]$ and identify the two deck groups by $f_{\ast }$. Choose pointed universal covers by [F2]. Use cellular integral chains $C_{\ast }(\tilde{P})$ for the presentation cover and singular integral chains $D_{\ast }=S_{\ast }(\tilde{Y};\mathbb{Z})$ for the target cover. Both covers are path-connected and simply connected. Their singular integral $H_0$ is $\mathbb{Z}$: differences of points bound paths and the sum-of-coefficients augmentation induces this isomorphism. Their singular integral $H_1$ is zero by Hurewicz [F3]. For $\tilde{P}$, ordinary cellular comparison in [F3] gives the same two groups. Thus both augmented complexes are exact in degrees1/0 and at the common trivial right module $\mathbb{Z}$. The right action is $c\cdot g=T_{g^{-1}}c$, and commutes with boundaries. For the cellular domain a lift of each of its finitely many cells is a free right $R$ basis. For the singular target, every singular simplex downstairs has a lift after a lift of its first vertex is prescribed, since its domain is a simply connected simplex; two lifts differ by exactly one deck transformation. The deck action on any lifted simplex is free, since a deck transformation fixing that simplex fixes one of its image points. Consequently one lifted simplex for each downstairs simplex is a free right $R$ basis of $D_k$. AC supplies these simultaneous choices, and projectivity follows from [F3]. This proves the needed exactness and projectivity without a CW target, a UCT calculation or asphericity.

Identify the target singular tensor complex without a CW hypothesis. Write $p_Y:\tilde{Y}\to Y$ and fix the lifted basepoint $\tilde{y}$. For a lifted singular simplex $\tilde{\sigma }$, choose a path $\tilde{\beta }$ from $\tilde{y}$ to its first vertex and put $\beta =p_Y\tilde{\beta }$. Define $\mathrm{\Phi }(\tilde{\sigma }\otimes a)=(T_{\beta }a)(p_Y\tilde{\sigma })$ in the intrinsic singular complex of [F5]. Two choices of $\tilde{\beta }$ are homotopic relative endpoints in the simply connected cover, so their projected transports agree. Under the no-reversal deck convention, a path to $\tilde{\sigma }\cdot g=T_{g^{-1}}\tilde{\sigma }$ represents $g^{-1}$ followed by $\beta$. Its transport is $T_{\beta }{T}_{g^{-1}}$, and $T_{g^{-1}}a=ga$ by the supplied coefficient convention. Hence $\mathrm{\Phi }((\tilde{\sigma }\cdot g)\otimes a)=\mathrm{\Phi }(\tilde{\sigma }\otimes ga)$, exactly the balanced relation. For the zeroth face, the path to its first vertex is $\tilde{\beta }$ followed by the lifted edge from vertex0 to vertex1. Thus its coefficient is $T_{\sigma [0,1]}{T}_{\beta }a$, precisely the exceptional face coefficient in [F5]. Each other face retains the first vertex and coefficient; the alternating signs are the ordinary simplex signs. So $\mathrm{\Phi }$ commutes with every boundary. Conversely, for an intrinsic generator $m\sigma$, lift $\sigma$ and choose $\tilde{\beta }$ as above, and send it to $\tilde{\sigma }\otimes T_{\beta }^{-1}m$. Replacing the lift by $\tilde{\sigma }\cdot g$ replaces this coefficient by $g^{-1}{T}_{\beta }^{-1}m$, and the balanced relation gives the same tensor. Path independence follows as before. Each construction is additive on the direct sum, and their two composites are the identity on every simplex summand. They are therefore inverse chain isomorphisms. They also commute with based maps and coefficient morphisms: the image of $\tilde{\beta }$ is a path to the first vertex of the image simplex, and the defining coefficient morphism commutes with its transport. This proves the actual target identification for the stated locally path-connected, semilocally simply connected $Y$, with no CW or local contractibility assumption on $Y$.

Compute each word by the actual lifted path. The lifted edge $e_j$ has endpoint $T_{x_j}{e}_0$, so its boundary is $e_0\cdot x_j^{-1}-e_0$. A positive occurrence after prefix $q$ traverses $T_q{e}_j=e_j\cdot q^{-1}$. A negative occurrence after prefix $q$ traverses the reverse of $T_{q{x}_j^{-1}}{e}_j$, giving exactly the negative contribution in the Statement. The endpoint differences telescope along these consecutive lifted edges, giving $\mathrm{\partial }{c}_w=T_w{e}_0-e_0=e_0\cdot w^{-1}-e_0$. In tensor coordinates its positive contribution is $e_j\otimes q^{-1}u$ and its negative contribution is $-e_j\otimes (q{x}_j^{-1}{)}^{-1}u$. These are exactly the initial local coefficients on the successive oriented edges, because forward transport along a path of class $q$ is the action of $q^{-1}$ in [F5]. If $wu=u$, also $w^{-1}u=u$, and $\mathrm{\partial }(c_w\otimes u)=0$. The boundary of each lifted relator disk is the same finite expression $c_{r_i}$, with its chosen orientation; its boundary is zero since $r_i=1$ in $\mathrm{\Gamma }$. Thus the presentation complex really gives finite coefficient matrices, whose entries are the displayed finite word sums, including their inverse prefixes. No untyped identification of forward conjugation with a right module action is made.

Build a comparison which is the actual map in degree one. The based lift $\tilde{f}:\tilde{P}\to \tilde{Y}$ exists by [F2]; uniqueness, or lifting a loop $g$, gives $\tilde{f}{T}_g=T_{f_{\ast }(g)}\tilde{f}$. Send the lifted presentation vertex to its actual singular image point, and each lifted generator edge to the singular path obtained by composing its parametrization with $\tilde{f}$; extend right $R$-linearly to maps $b_0,b_1$. They commute with the degree-one boundary, because a path boundary is its final point minus its initial point. For each of the finitely many relator cells, $b_1\mathrm{\partial }{e}_{r_i}$ is therefore a singular1cycle upstairs. The actual $H_1(\tilde{Y};\mathbb{Z})=0$ in step2.1 supplies a finite singular2chain bounding it. Choose one for each relator and extend right-linearly to $b_2$. This is an actual augmentation-preserving comparison through degree2. Apply [F4] over $R^{\mathrm{o}\mathrm{p}}$: a right $R$-module $C$ becomes a left module by $r^{\mathrm{o}\mathrm{p}}c=cr$, while $A$ becomes a right module by $a\cdot r^{\mathrm{o}\mathrm{p}}=ra$. The flip $a\otimes c\mapsto c\otimes a$ identifies $A{\otimes }_{R^{\mathrm{o}\mathrm{p}}}C$ with $C{\otimes }_{R}A$, since $(ra)\otimes c=a\otimes cr$ becomes $c\otimes ra=cr\otimes a$. Step2.1 supplies all hypotheses of [F4], so $b\otimes 1$ gives an isomorphism on first homology. Use step3.1 for the target singular tensor complex, and [F5] for cellular comparison on the CW domain $P$. Every presentation first-homology class is represented by a local edge cycle on $P^1$, and cellular comparison identifies that edge cycle with its actual singular cycle. The map $b_1$ sends it to its actual singular image under $f$, by construction. Hence this isomorphism is precisely $f_{\ast }$; the freely chosen $b_2$ changes no first-homology image. Neither a target CW structure nor a presentation spine is used.

Identify the actual peripheral inclusion and its algebraic kernel. Under cellular comparison [F5], the subdivided loop carrying initial coefficient $u$ is the sum of its successive oriented edge chains with transported coefficients, which is exactly step 3.2; subdividing or concatenating the finite edges does not change its homology class, by the prism relation in [F5]. Applying $f$ sends it to the loop class represented by $w$ in $Y$. Any other based loop representing that group element is based-homotopic to it; the homotopy transport at the fixed basepoint is the identity, so [F5] identifies their coefficient homology classes. Two choices of relator fillings give maps agreeing on generator edges and therefore give the same images of these cycles, independently of their behavior on $2$-cells. Apply this to the two loops of $j$. The isomorphism of step4.1 takes the two finite word cycles to $j_{\ast }[u\otimes \lambda ]$ and $j_{\ast }[u\otimes \mu ]$, in the stated order. In field coordinates they are the columns $U$; being cycles is $CU=0$, and $CB=0$ is the cellular boundary identity. Their kernel is precisely the preimage of $\mathrm{im}B$ under $U$. Finally appending two columns to $B$ adds exactly the dimension of their span modulo $\mathrm{im}B$, so their actual image has dimension one exactly when $\mathrm{rank}(B\mid U)=\mathrm{rank}B+1$. This uses the actual comparison, not the assumption that the presentation complex computes exterior $H_2$.

Adjoint and knot applications, with the conventions retained. For $A={\mathfrak{s}\mathfrak{l}}_2(F)$ and its left coefficient action $gX=\rho (g)X\rho (g{)}^{-1}$, a generator action and its inverse have polynomial entries in the $S{L}_2$ entries: matrix inversion there is the polynomial adjugate. Every finite word sum in step 3.2 and each cycle column is consequently polynomial in those entries and in $u$. On the peripheral torus, forward transports are $\mathrm{Ad}\rho (\lambda {)}^{-1}$ and $\mathrm{Ad}\rho (\mu {)}^{-1}$ in [F5]. Apply [F6] to these two inverse matrices. Their meridian trace is still $t$, and their invariant vector is $\rho (\mu {)}^{-1}-tI\mathrm{/}2=-u$ by the direct $S{L}_2$ identity $\rho (\mu )+\rho (\mu {)}^{-1}=tI$. Multiplying both peripheral basis vectors by $-1$ preserves their being a basis and does not change the common kernel line or its ordered coordinates. Thus the actual two torus classes with coefficient $u$ have exactly the asserted basis property, including characteristic different from2. An actual compact smooth knot exterior is a particular $Y$: its manifold charts give local path-connectedness and small simply connected coordinate neighborhoods, including half-space charts at its boundary, while its connectedness supplies path-connectedness. Its actual based torus inclusion is a particular $j$. No CW structure on the exterior, deformation retract to a presentation complex, generic representation family, generic rank assertion or valuation/tree/surface conclusion has been presumed. All coefficient, lift and cellular choices are either supplied, finite, or covered by the declared full AC.

By [F7], the compact exterior has finite CW type. A connected finite CW complex has finitely presented fundamental group: choose a maximal tree in its finite $1$-skeleton, take the remaining finitely many $1$-cells as generators, and use the finitely many attaching maps of its $2$-cells as relators; cells of dimension at least $3$ do not change the fundamental group. Thus the exterior supplies the finite presentation required in the Statement, and the comparison applies to its actual based peripheral torus map.