Real Clifford–Thom isomorphism

First construct compatible metrics and local regular embeddings without assuming Clifford-local triviality of the module bundle. On a local orthonormal frame of $V$, the signed Clifford monomials form a finite group $G=\{\pm e_I\}$, closed under multiplication by the Clifford relations. Average any ordinary bundle metric over its action. Each frame generator is then orthogonal and has square $-I$, hence is skew-adjoint. For arbitrary $v$, the Clifford relations give $c(v{)}^{\ast }c(v)=\mathrm{\parallel }v{\mathrm{\parallel }}^{2}I$, so this metric compatibility is independent of the frame. Patch these local metrics with a finite partition of unity from [F6]; positive definite compatible metrics form a convex set, so the resulting global metric is compatible. Apply the same construction to regular module bundles. For the fiber embedding, [F5] gives $C(V_x)\cong M_r(D)$, $D=\mathbb{R}$ or $\mathbb{H}$. Its matrix units decompose any module into copies of the column module: $L={⨁}_i{E}_{ii}L$, and $E_{i1}$ identifies each $E_{11}L$ with $E_{ii}L$. A finite $D$-basis of $E_{11}L$ identifies $L$ with a finite sum of column modules, while the regular module contains $r$ such columns. Thus enough regular copies contain $L$. Fix such an intertwining injection at one point. In ordinary bundle charts extend its matrix constantly to a local real-linear map $a_x$; this uses no module-bundle trivialization. The averaged map ${\bar{a}}_x=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{g\in G}{\rho }_{\mathrm{r}\mathrm{e}\mathrm{g},x}(g{)}^{-1}{a}_x{\rho }_{L,x}(g)$ is continuous and intertwines $G$, by reindexing $g\mapsto gh$, and hence intertwines the Clifford algebra generated by $G$. At the chosen point it equals the original injection. Injectivity is open for finite matrices, so it remains an embedding on a neighborhood. Cover the compact base by finitely many such neighborhoods and choose a subordinate partition $({\rho }_i)$ with [F6]. The weighted direct sum $a={⨁}_i\sqrt{{\rho }_i}{\bar{a}}_i$ extends by zero off each chart, because each support lies inside its chart, and is injective at every point. It is a global module map into finitely many regular copies. Since source and target have compatible metrics, $a^{\ast }a$ commutes with all Clifford generators; hence $a(a^{\ast }a{)}^{-1\mathrm{/}2}$ is an isometric module embedding, and its orthogonal complement is a module. Consequently equality of module Grothendieck classes is equivalently isomorphism after equal regular summands: complete a common arbitrary stabilizer to a regular module. The ordinary version uses [F3].

Define $T(e,\lambda )={\pi }^{\ast }e+[H(\lambda )]$ by additivity of clutching. For surjectivity, [F1] expresses any actual bundle after trivial and regular-Hopf shifts as a pulled-back bundle plus a Hopf bundle; move the shifts to the other side, then pass to differences. For injectivity, equality of two such actual sums in $K{O}^0(\widehat{V})$ gives an isomorphism after a common arbitrary bundle. The total sphere-bundle space is compact Hausdorff, so [F4] supplies a complement there; adding it makes the stabilizer trivial, and [F2] applies. Both coordinates agree stably. For arbitrary formal coordinates first move their negative parts to opposite sides and repeat this argument. Hence $T$ is an isomorphism of groups, with zero kernel.

Under $T$, restriction to the north section is $(e,\lambda )\mapsto e+[{\lambda }^{+}]$. Thus its kernel is the graph $\{(-[{\lambda }^{+}],\lambda )\}$. The map $\chi (z)=z-{\pi }^{\ast }{s}_{+}^{\ast }z$ is a projection onto this section kernel, with kernel ${\pi }^{\ast }K{O}^0(X)$. Consequently $\chi T$ has kernel $K{O}^0(X)\oplus \{0\}$ and identifies the remaining summand with the section kernel. This proves the asserted cancellation and kernel computations before using relative KO.

We prove the relative section-kernel assertion from the actual triple model [F4]. Let $Z=\widehat{V}$ and $A=s_{+}(X)$; the retraction is $r=s_{+}\pi :Z\to A$. Forgetting a relative triple $(E,F,g)$ gives $[E]-[F]$, whose restriction to $A$ vanishes. Conversely, if $[E{\mathrm{\mid }}_A]=[F{\mathrm{\mid }}_A]$, the common-trivial-stabilization clause of [F4] supplies an isomorphism after adding the same trivial bundle to both, hence a triple lifting the original ordinary class. Thus the image is the restriction kernel. For injectivity, suppose a triple forgets to zero in $K{O}^0(Z)$. By [F4], stabilize both entries trivially so there is a global isomorphism between them. Transport along it to represent the triple as $(E,E,h)$ with an automorphism $h$ on $E{\mathrm{\mid }}_A$. Add the elementary triple of a global complement of $E$, also supplied by [F4], and choose a trivialization of the resulting common bundle. The stabilized section automorphism is now a matrix map $b:A\to {\mathrm{G}\mathrm{L}}_N(\mathbb{R})$. The map $b\circ r:Z\to {\mathrm{G}\mathrm{L}}_N(\mathbb{R})$ extends it globally, so this triple is elementary and its relative class is zero. Hence forgetting is injective and identifies $K{O}^0(Z,A)$ with $\ker (s_{+}^{\ast })$. The quotient-space clause of [F4] then gives $\ker (s_{+}^{\ast })\cong K{O}^0(\widehat{V},s_{+}X)={\widetilde{KO}}^0(\widehat{V}\mathrm{/}{s}_{+}X)\mathrm{.}$ The disk-half description identifies the quotient with $\mathrm{Th}(V)$. Therefore the remaining summand of step 3.1 maps by exactly $[\mathrm{\Lambda }]\mapsto [H(\mathrm{\Lambda })]-{\pi }^{\ast }[{\mathrm{\Lambda }}^{+}]$, proving the asserted isomorphism. With the standard relative convention $[E,F,\sigma ]$ restricting to $[E]-[F]$ at the disk zero section, the volume-positive symbol is its negative: the uncollapsed south restriction of this reduced Hopf class is $[{\mathrm{\Lambda }}^{-}]-[{\mathrm{\Lambda }}^{+}]$. Hence ${\mathrm{\Phi }}_V=-{\mathrm{\Psi }}_V$; the sign must be retained when choosing a multiplicative Bott class. At rank zero, $\mathrm{Th}(0_X)=X_{+}$ and the basepoint split gives the separate identity. No raw graded-module group is identified with $K_{C(V)}(X)$.