The low-degree coefficient Bockstein on a compact manifold without choice

Fix the paths from [F1]. For a degree-one cocycle $\alpha$ in a local system $\mathcal{E}$, the two-simplex identity of [F2] gives $\alpha (uv)=\alpha (u)+T_u^{-1}\alpha (v)$ for concatenated paths, with the second term transported to the initial fiber. A triangle realizing the concatenation proves this formula, and a triangulated homotopy square proves invariance for endpoint-fixed path homotopies; common internal edges cancel with their inverse transports. A constant two-simplex shows that a cocycle evaluates to zero on a constant path, and the concatenation formula then handles reversed paths. Define a zero-cochain $v_x=T_{{\gamma }_x}\alpha ({\gamma }_x)$ and $v_o=0$. The normalized cocycle ${\alpha }^0=\alpha -\delta v$ vanishes on every ${\gamma }_x$. If $g_{\sigma }=[{\gamma }_x\sigma {\bar{\gamma }}_y]$ for a path $\sigma :x\to y$, the concatenation formula consequently gives ${\alpha }^0(\sigma )=T_{{\gamma }_x}d(g_{\sigma })$, where $d(g)={\alpha }^0(g)$ on based loops. Here $d(gh)=d(g)+\rho (g)d(h)$ with $\rho (g)=T_g^{-1}$ in the cohomological convention. This is a crossed homomorphism; conversely that formula assigns a normalized cocycle to every such $d$, since the three edges of a singular two-simplex satisfy the same loop relation.

Apply step 1.1 to a $\mathcal{C}$-cocycle. Choose one lift $b_s\in {\mathcal{B}}_o$ of $d(s)$ for each of the finitely many generators $s$ of ${\pi }_1(M,o)$ from [F1]. For its inverse use $b_{s^{-1}}=-\rho (s^{-1})b_s$. Order the finite alphabet and use shortlex order on its finite words. Every group element $g$ has a least representing word $w(g)$; the identity has the empty word. On a word $t_1\cdots t_k$ form the finite sum $D(w)={\sum }_j\rho (t_1\cdots t_{j-1})b_{t_j}$ in ${\mathcal{B}}_o$, and put $Q(g)=D(w(g))$. Its image in ${\mathcal{C}}_o$ is $d(g)$ by the crossed-homomorphism law. Thus $b(\sigma )=T_{{\gamma }_x}^{\mathcal{B}}Q(g_{\sigma })$ gives an actual $\mathcal{B}$-valued one-cochain lifting ${\alpha }^0$ on every singular path, without selecting a fraction representative for each path. Its coboundary has zero $\mathcal{C}$-image, so unique inverse images under the injective fiber maps $i$ define a two-cochain $a$ in $\mathcal{A}$ with $i(a)=\delta b$. It is a cocycle since $i$ is injective and ${\delta }^{2}b=0$. Define $\beta [\alpha ]=[a]$.

Any other lift of the same normalized one-cochain differs from $b$ by the image of a unique $\mathcal{A}$-valued one-cochain, so its coboundary gives the same class $[a]$. This includes changes in the finite generators, their lifts and the shortlex order. If $\alpha$ is changed by $\delta c$ for a $\mathcal{C}$ zero-cochain, its normalization changes by $\delta \widehat{c}$, where ${\widehat{c}}_x=T_{{\gamma }_x}{c}_o$: indeed the change in the normalizing zero-cochain is $c_x-T_{{\gamma }_x}{c}_o$. A single lift of $c_o$ in ${\mathcal{B}}_o$, transported along all ${\gamma }_x$, lifts $\widehat{c}$. Its coboundary adjusts $b$ without changing $\delta b$. Thus $\beta$ depends only on the cohomology class. Sums of lifts are lifts of sums, and their coboundaries add, proving additivity.

The map also does not depend on the finite-atlas path assignment. Compare a second family ${\eta }_x$ with ${\gamma }_x$, and first use the $\gamma$-normalized cocycle. Its evaluation on ${\eta }_x$ is $d(k_x)$, where $k_x=[{\eta }_x{\bar{\gamma }}_x]$. Thus its $\eta$-normalizing zero-cochain is $T_{{\eta }_x}^{\mathcal{C}}d(k_x)$. The explicit $\mathcal{B}$ zero-cochain $T_{{\eta }_x}^{\mathcal{B}}Q(k_x)$ lifts it, so subtracting its coboundary from $b$ produces a lift of the $\eta$ normalization with the same two-coboundary. Step 3.1 removes the initial change of representative. Coefficient morphisms commute with transport and the formulas, and images of chosen finite lifts are valid lifts for the image cocycle; independence of lifts therefore proves naturality. A commuting scalar action is treated by taking scalar multiples of lifts and using step 3.1.

If $[\alpha ]$ comes from a $\mathcal{B}$ one-cocycle, normalize that cocycle using its own path evaluations. It lifts the normalized image cocycle and has zero coboundary, so its Bockstein is zero. Conversely if $\beta [\alpha ]=0$, the lift $b$ from step 2.1 has $\delta b=i(\delta a_1)$ for some $\mathcal{A}$ one-cochain $a_1$. Then $b-i(a_1)$ is a $\mathcal{B}$ one-cocycle whose image is ${\alpha }^0$, so $[\alpha ]$ is in the image of $q_{\ast }$. This proves exactness at $H^1(M;\mathcal{C})$.

Every Bockstein two-cocycle becomes a coboundary in $\mathcal{B}$ by construction. Conversely let an $\mathcal{A}$ two-cocycle $a$ become a coboundary $i(a)=\delta b$ for one $\mathcal{B}$ one-cochain $b$. Its image $\alpha =q(b)$ is a $\mathcal{C}$ one-cocycle. The $\mathcal{B}$ zero-cochain $v_x=T_{{\gamma }_x}^{\mathcal{B}}b({\gamma }_x)$, with $v_o=0$, lifts the normalizing zero-cochain for $\alpha$. Hence $b-\delta v$ lifts ${\alpha }^0$ and still has coboundary $i(a)$. By step 3.1 it computes the same Bockstein, so $\beta [\alpha ]=[a]$. This proves exactness at $H^2(M;\mathcal{A})$ and the asserted isomorphism when the two surrounding groups vanish. All lifts used either finitely many generator values, one base-fiber value, or a previously supplied whole cochain. No surjectivity of an infinite product of coefficient quotients was assumed.