Primitive Alexander annihilators for compact manifolds with first homology Z

Apply [F1], and pull the given coefficient system to $K$. The induced fundamental-group map is an isomorphism and matches $\mu$, so [F2] identifies the actual $H_1$ of $K$ with that of $M$ for $\mathrm{\Lambda }$ and every $k[t,t^{-1}]$. Use [F3] on this actual finite two-complex. Its chain groups are finite free, with $C_0=\mathrm{\Lambda }$, a basis of one-cells including $e_{\mu }$, and a finite basis of two-cells. Cell lifts and orientations require only finitely many choices. Their incidence coefficients reduce under every constant-ring map, so coefficient change on these finite cellular chains is literally coefficientwise change in their Laurent matrices. In particular augmentation $t=1$ gives exactly the ordinary integral cellular complex of $K$.

If the other oriented one-cells have images $n_j\in \mathbb{Z}$, their degree-one boundaries are $t^{-n_j}-1$, while $d_1{e}_{\mu }=t^{-1}-1$. Set $z=-t{e}_{\mu }$, so $d_{1}z=t-1$. Write $q_j=(t^{-n_j}-1)\mathrm{/}(t-1)\in \mathrm{\Lambda }$, an explicit finite geometric sum also for negative exponents, and replace $e_j$ by $f_j=e_j-z{q}_j$. This is an invertible triangular basis change with $C_1=\mathrm{\Lambda }z\oplus {\mathrm{\Lambda }}^r⟨f_j⟩$ and $d_1{f}_j=0$. Multiplication by $t-1$ is injective in $k[t,t^{-1}]$ for every commutative $k$: the top nonzero coefficient of a nonzero finite Laurent list remains a nonzero top coefficient after multiplication by this monic polynomial. Thus $\ker d_1$ is exactly the $f_j$ summand, both integrally and after every constant-ring change. Since $d_1{d}_2=0$, the $z$ row of $d_2$ is zero. Let $A(t)$ be its remaining $r\times s$ matrix, giving $H_1(K;\mathrm{\Lambda })=\mathrm{coker}A(t)$ and $H_1(K;k[t,t^{-1}])=\mathrm{coker}{A}_k(t)$.

At $t=1$, $d_1=0$ and the integral first homology is therefore $\mathbb{Z}[z]\oplus \mathrm{coker}A(1)$. The marked class $z$ maps to $-[\mu ]$, a free generator of $H_1(M;\mathbb{Z})$ by hypothesis and [F1]–[F2]. Hence $\mathrm{coker}A(1)=0$ and $A(1):{\mathbb{Z}}^s\to {\mathbb{Z}}^r$ is onto. For each of the finitely many coordinate vectors choose one preimage, giving an integer $s\times r$ matrix $S$ with $A(1)S=I_r$. Put $D(t)=A(t)S$ and ${\mathrm{\Delta }}_0(t)=\det D(t)$. Then ${\mathrm{\Delta }}_0(1)=1$. The adjugate identity $D\mathrm{adj}D={\mathrm{\Delta }}_0{I}_r$, proved by finite signed minor expansion, gives ${\mathrm{\Delta }}_0{\mathrm{\Lambda }}^r\subseteq \mathrm{im}D\subseteq \mathrm{im}A$. Thus ${\mathrm{\Delta }}_0$ annihilates the presented first homology. The same identity is an identity of integer Laurent matrices, so holds after every coefficient change as well. For $r=0$, use the empty determinant one; the corresponding first homology is zero.

Normalize ${\mathrm{\Delta }}_0$ by a power of $t$ so its smallest exponent is zero, obtaining the asserted polynomial $\mathrm{\Delta }$ with nonzero constant term. This multiplies it by a Laurent unit and retains $\mathrm{\Delta }(1)=1$ and every annihilation assertion. The coefficients sum to one, so their greatest common divisor is one; reduction modulo any prime still evaluates to one at $t=1$ and is therefore nonzero. Finally finite presentation cokernels commute with the change $\mathbb{Z}\to k$ here directly: tensoring the presentation generators imposes exactly the same finite-column relations with their coefficients in $k$, so $\mathrm{coker}A{\otimes }_{\mathbb{Z}}k=\mathrm{coker}{A}_k$ by the tensor generators-and-relations construction. Via the natural identifications in steps 1.1–2.1 this is the actual coefficient map on local $H_1$, not just an abstract isomorphism of modules. All selections in step 3.1 were finite preimages of a finite standard basis, and the remaining operations were finite integer or Laurent-matrix calculations. This proves the full scope without a choice axiom or a finite CW claim about $M$.