Real Spin Thom isomorphisms with real and quaternionic sources

In either positive-rank branch compose the corresponding group isomorphism [F2] with [F1]. On a real bundle $E$, the volume operator on $E\otimes \mathrm{\Sigma }$ is $1\otimes \omega$, so its eigenspaces are $E\otimes {\mathrm{\Sigma }}^{\pm }$ and its clutching is $1\otimes c(v)$. Consequently the composite is exactly product by ${\tau }_V$, rather than an unspecified isomorphism. For a quaternionic bundle $E$ the same calculation is valid in the balanced tensor product since $\omega$ and $c(v)$ commute with the right scalar action on $\mathrm{\Sigma }$. This gives exactly the stated $KSp$ formula.

All constructions in step 1.1 commute with pullback of the Spin lift and of the vector bundle, proving naturality. In rank zero the repo convention is $\mathrm{Spin}(0)=\{\pm 1\}$. A given lift is a double cover and its spinor is the associated sign line $L$. Its symbol is $[L]\in K{O}^0(X)$, so the map is $x\mapsto x[L]$, an isomorphism because the transition signs square to one and $L^{\otimes 2}\cong \mathbb{R}$. For the canonical framed lift $L$ is trivial and this is the identity. This is separate from [F1] and preserves nontrivial rank-zero lifts.

For two real-branch ranks the graded tensor of the irreducible spinors is the irreducible real spinor for the orthogonal sum: its action is $c(v)\otimes 1+\mathrm{\Gamma }\otimes c(w)$; the square is $-(\parallel v{\parallel }^2+\parallel w{\parallel }^2)$ and the dimension is that of the unique simple ungraded matrix-algebra module. The product grading is the volume grading: in ranks divisible by four the product of the grouped Clifford generators gives ${\omega }_{V\oplus W}={\omega }_V\otimes {\omega }_W$. The relative product rule [F3] therefore identifies the product symbols with the sum-bundle symbol. For quaternionic factors the same computation uses the balanced tensors and conjugation from [F2]. More explicitly, for two quaternionic ranks put $M={\mathrm{\Sigma }}_V\widehat{\otimes }{\mathrm{\Sigma }}_W$ and $T={\mathrm{Hom}}_{C(V\oplus W)}({\mathrm{\Sigma }}_{V\oplus W},M)$. Its real rank is four by the simple-module dimensions. The two balancing left-$\mathbb{H}$ actions on $M$ induce a commuting left-$\mathbb{H}\otimes \mathbb{H}$ action on $T$; these are even actions, so they commute with the volume grading. Evaluation in [F2] identifies $M=T{\otimes }_{\mathbb{R}}{\mathrm{\Sigma }}_{V\oplus W}$. For right-quaternionic coefficient bundles $E,F$, the real coefficient of the sum is therefore exactly $(E{\otimes }_{\mathbb{R}}F){\otimes }_{\mathbb{H}\otimes \mathbb{H}}T$. Associativity of balanced tensors identifies its spinor symbol with the product of the original symbols. Two quaternionic units accordingly produce real rank four, not rank one. If exactly one rank is quaternionic, the tensor spinor is quaternionic irreducible by the same dimension count, and the real coefficient tensors ordinarily with the quaternionic one. Thus all branches commute with their actual tensor symbols and proper coefficient operations.