Spin-c lifts with fixed determinant are stabilized Spin lifts

Let $e_j^{+}$ and $e_j^{-}$ denote positive- and negative-square generators. In the complexification of the negative Clifford algebra the elements $i{e}_j^{-}$ square to $+1$ and have the positive Clifford anticommutation relations. By [F1], $e_j^{+}\mapsto i{e}_j^{-}$ extends to an algebra map; on ordered even monomials of length $2r$ it is multiplication by $i^{2r}=(-1{)}^r$ followed by the corresponding negative monomial. Therefore its restriction is a real algebra isomorphism on the even parts, with inverse given by the same basis signs. A positive even product of $2r$ unit vectors maps to $(-1{)}^r$ times the corresponding negative even product, which belongs to the negative norm-one Spin group. The inverse has the same property, so the map identifies the groups. Conjugation on the generators intertwines the orthogonal actions: the identity for $i{e}_j^{-}$ cancels the common scalar $i$. Scalars, in particular $\pm 1$, are fixed. Finite-dimensional linearity makes the group isomorphism a homeomorphism. Thus principal lifts and their central twisting actions transport between these two conventions over the identity on $SO(n)$.

In the last negative-square plane put $B=e_{n+1}{e}_{n+2}$; then $B^2=-1$ and $z=a+ib\mapsto a+bB$ identifies its Spin group with $U(1)$: every such element factors as $(a{e}_{n+1}+b{e}_{n+2})(-e_{n+1})$, and its Clifford norm is $a^2+b^2=1$. Conjugation by $\cos \theta +B\sin \theta$ rotates the last plane through $2\theta$, so its map to $SO(2)=U(1)$ is $z\mapsto z^2$. Even monomials from the first summand commute with $B$, since moving them through the two last generators gives an even number of sign changes. Thus $(s,z)\mapsto s(a+bB)$ is a continuous group homomorphism into $\mathrm{Spin}(n+2)$. By [F1], the first even algebra and the last even algebra meet exactly in real scalars. By [F2], their Spin groups therefore intersect in $\{\pm 1\}$; the homomorphism kernel is precisely $\{(1,1),(-1,-1)\}$. Its projection is the block pair $({\rho }_n(s),z^2)$. Every block pair has such a lift by [F2] and the circle square map. Any other lift of that pair differs by $\pm 1$, already in the image, so the image is the whole preimage subgroup. The induced continuous bijection from the compact quotient Spin-c to a Hausdorff subgroup is a homeomorphism. For $n=1$ the same calculation uses $\mathrm{Spin}(1)=\{\pm 1\}$ and $SO(1)=1$, with no universal-cover assertion in that rank.

The character $\chi ([s,z])=z^2$ is well-defined because replacing $(s,z)$ by $(-s,-z)$ leaves $z^2$ unchanged. Given $P_c$ and a unitary identification of its determinant with $L$, its frame maps give an equivariant map to $P_{SO}(V){\times }_M{P}_{U(1)}(L)$ with the homomorphism of step 2.1. Extend its structure group along the resulting inclusion into Spin$(n+2)$. The product frame bundle is the block-adapted reduction of $P_{SO}(V\oplus L_{\mathbb{R}})$, and its extension to $SO(n+2)$ is the full frame bundle. Consequently the extended principal Spin bundle carries the required double-cover frame map and is a Spin lift of $V\oplus L_{\mathbb{R}}$. These assertions are literal in every bundle chart: a local point $(x,h)$ maps to its local block frame times $\rho (h)$, and extension sends $(x,h)$ to the same point in the larger group chart.

Conversely take a Spin lift $P$ of $V\oplus L_{\mathbb{R}}$ and restrict it to the inverse image of the block-adapted frame reduction. Local lifts of block frame sections exist by the double-cover charts in [F2]; these exhibit the inverse image as a principal bundle for the subgroup in step 2.1. Its projection to the first block gives a Spin-c lift of $V$. Projection to the second block identifies the associated $\chi$ line with the specified line $L$, including its unitary frame map. Restriction after extension and extension after restriction are inverse: the local group-chart maps are respectively $h\mapsto h$ and $[p,g]\mapsto pg$, and these glue on overlaps. They commute with maps of lifts and pullback. Thus the two constructions induce inverse bijections on the stated isomorphism classes, with the determinant identification part of the data. The empty base has one empty object in each class.