Spin structures exist exactly when w2 vanishes and then form an H1 torsor

Each nonempty finite intersection of the good cover is contractible by [F6], hence path-connected and locally path-connected. For every nonempty double intersection choose a basepoint and a point over its transition value under the cover [F1]. Since the intersection is simply connected, [F10] lifts $g_{ij}:U_i\cap U_j\to SO(n)$ to a continuous ${\tilde{g}}_{ij}$. Full AC supplies these simultaneous choices. Choose ${\tilde{g}}_{ji}={\tilde{g}}_{ij}^{-1}$ and ${\tilde{g}}_{ii}=1$. Since $\rho ({\tilde{g}}_{ij}{\tilde{g}}_{jk}{\tilde{g}}_{ki})=1$, the triple product is a locally constant central sign ${ϵ}_{ijk}\in \{\pm 1\}$.

Associativity and centrality give ${ϵ}_{ijk}{ϵ}_{ikl}={ϵ}_{jkl}{ϵ}_{ijl}$ on every nonempty quadruple intersection: substitute the four chosen lifts, cancel each shared lift with its inverse, and commute only the resulting central signs. Thus $ϵ$ is a degree-two cocycle on the cover nerve. If the lifts are changed by signs $a_{ij}=a_{ji}^{-1}$, the triple product changes by the Čech coboundary $\delta a$; hence its cohomology class $\eta (E)$ is independent of these choices. Under pullback, transitions and chosen lifts compose with the base map, so the sign cocycle pulls back; on a common refinement, restriction gives the simplicial cochain pullback. By [F7], the nerve comparison is natural, so $\eta (E)$ is a well-defined element of $H^2(M;{\mathbb{F}}_2)$, natural under pullback and refinement.

If $\eta (E)=0$, choose signs $a_{ij}$ with $ϵ=\delta a$ and replace ${\tilde{g}}_{ij}$ by $a_{ij}{\tilde{g}}_{ij}$, using additive ${\mathbb{F}}_2$ notation for the signs. The new triple products are $1$, so these transitions glue a principal Spin bundle and the maps $\rho ({\tilde{g}}_{ij})=g_{ij}$ glue its equivariant map to $P_{SO}(E)$. Conversely, transitions of any Spin lift have triple product $1$, so relative to the chosen lifts their signs make $ϵ$ a coboundary. Thus a lift exists exactly when $\eta (E)=0$.

For $n\ge 3$, [F5] gives the CW model $BSO(n)={\mathrm{Gr}}_n^{+}({\mathbb{R}}^{\mathrm{\infty }})$ and its contractible stable Stiefel total space. Its tautological frame map is locally trivial by the graph charts, and is a Serre fibration: pullbacks over the compact parameter spaces $D^k\times [0,1]$ are numerable by [F5], hence Hurewicz fibrations. The long exact sequence, contractibility of the total space, connectedness of $SO(n)$, and [F2] give ${\pi }_1(BSO(n))=0$ and ${\pi }_2(BSO(n))\cong {\pi }_1(SO(n))\cong \mathbb{Z}\mathrm{/}2$. The base is path-connected because the stable Stiefel space is nonempty, contractible, and maps onto it. Hurewicz and UCT [F5] give $H^2(BSO(n);{\mathbb{F}}_2)\cong \mathrm{Hom}(\mathbb{Z}\mathrm{/}2,{\mathbb{F}}_2)\cong {\mathbb{F}}_2$. We construct its universal lifting obstruction on cells. The universal frame bundle is trivial over the 1-skeleton: it is trivial over each edge by the classifying property in [F5], and edge sections can be adjusted at their endpoints using path-connectedness of $SO(n)$. Choose such a frame section on the 1-skeleton. For each oriented 2-cell, choose a frame over its characteristic disk and compare its boundary with the chosen 1-skeleton section; the comparison is a loop in $SO(n)$. Assign the cell value $0$ if this loop lifts to a closed loop in Spin and $1$ otherwise. Changing the disk frame multiplies the comparison by a loop that extends over the disk, so the value is unchanged. To check independence of the 1-skeleton section, compare two such sections by a map into $SO(n)$, choose a lift at every vertex, and lift its path along each oriented edge from the chosen initial lift. Record whether each terminal lift agrees with the chosen lift at the terminal vertex. On the boundary of each 2-cell, these edge signs change its obstruction value by their cellular incidence sum, exactly the coboundary of this 1-cochain. An isomorphism of principal $SO(n)$-bundles transports frame sections and comparison loops, so it preserves this obstruction class. For a 3-cell, the pulled-back frame bundle on its characteristic disk is trivial and has a Spin lift; hence its boundary obstruction on the attaching 2-sphere is zero. The cellular incidence formula identifies that value with the coboundary of the 2-cochain on the 3-cell. Thus the values form a cocycle and determine a choice-independent class ${\eta }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}\in H_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^2(BSO(n);{\mathbb{F}}_2)$, which [F11] identifies with singular $H^2(BSO(n);{\mathbb{F}}_2)$. Pulling back the universal frame bundle pulls back this cellwise obstruction class: on each 2-cell, the comparison loop is the pullback of the corresponding boundary-frame comparison, and the cellular incidence maps give the induced cohomology pullback. Let $e:\mathrm{\mid }N\mathrm{\mid }\to M$ be the good-cover homotopy equivalence and let $f:M\to BSO(n)$ classify $P_{SO}(E)$. Replace $f\circ e$ by a cellular map using [F12]; its pulled-back frame bundle is classified by the nerve transition cocycle. On each 2-simplex, the comparison loop is the triple product of transition lifts in step 1.1, so its cell value is ${ϵ}_{ijk}$. Hence the nerve comparison [F7] identifies $(f\circ e{)}^{\ast }{\eta }_{\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}}$ with $\eta (E)$.

If a lift exists, any other lifted cocycle differs from it by a sign-valued degree-one cochain $b_{ij}$. Both are cocycles, so $\delta b=0$. An isomorphism over the fixed $SO$ bundle is given locally by signs $b_i$; it changes the difference cocycle by $\delta (b_i)$. Therefore isomorphism classes of lifts are a free transitive $H^1(M;{\mathbb{F}}_2)$-set. The action is twisting by the associated principal $\{\pm 1\}$-bundle; freeness follows because a twisting cocycle that is a coboundary is removed by that local sign isomorphism, and transitivity follows from the cocycle difference.

For $n\ge 3$, the clutching loop that rotates an oriented two-plane through $2\pi$ and is the identity on its orthogonal complement represents the generator of ${\pi }_{1}SO(n)$, hence corresponds under the fibration boundary isomorphism in step 3.2 to a generator of ${\pi }_{2}BSO(n)$. Its Spin lift is $t\mapsto \cos (\pi t)+e_1{e}_2\sin (\pi t)$, which starts at $1$ and ends at $-1$, so the universal obstruction evaluates to $1$. The associated oriented plane bundle over $S^2$ is the underlying real Hopf line, whose $c_1$ is a generator; [F4] makes its $w_2$ the nonzero reduction. Adding a trivial $(n-2)$-plane preserves $w_2$ by [F3]. Hence $w_2$ and $\eta$ are both the unique nonzero element of $H^2(BSO(n);{\mathbb{F}}_2)$ computed in step 3.2, and are equal.

For $n=2$, $SO(2)\cong S^1$ and direct Clifford multiplication gives $\mathrm{Spin}(2)\cong S^1$ with $z\mapsto z^2$. Thus $BSO(2)\simeq {\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}$, $H^2(BSO(2);{\mathbb{F}}_2)={\mathbb{F}}_2$, and the degree-one clutching loop again fails to lift closed. Its universal oriented plane is the realification of the Hopf line; [F4] gives $w_2={\rho }_2{c}_1$, which evaluates to $1$ on that generator. The two classes are again equal. For $n=1$, the oriented orthonormal frame bundle has structure group $SO(1)=1$ and $w_2(E)=0$ by rank; its lifts are principal $\mathrm{Spin}(1)=\{\pm 1\}$ bundles, whose sign transitions modulo sign-valued vertex changes are exactly $H^1(M;{\mathbb{F}}_2)$.

By [F8], $P_{SO}(E)$ is numerable; [F9] verifies the paracompact Hausdorff CGWH base hypotheses of the oriented-bundle classifier in [F5], which supplies an oriented bundle isomorphism from $E$ to the pullback of the tautological bundle. The positive square-root metric correction in [F5] identifies their $SO(n)$ frame bundles, giving a classifying map $f:M\to BSO(n)$ for the actual $P_{SO}(E)$. Naturality of the local lifting obstruction and of $w_2$, together with the nerve comparison [F7], pulls the universal equality $\eta =w_2$ back to $\eta (E)=w_2(E)$. Step 3.1 proves existence iff this class vanishes, and step 4.1 proves the torsor statement. The empty-base case has one empty lift and $H^1=0$; rank zero lies outside this item's stated rank range; no claim that the repo's Spin(0) convention lacks a sign-bundle torsor is made. The only nonempty choices are the local lifts in step 1.1 and the cochain primitive in step 3.1, both covered by full AC [A1].