Real Bott periodicity from the rank-eight Clifford–Thom map

Give the trivial oriented bundle ${\mathbb{R}}_X^8$ its framed Spin lift and let $\mathrm{\Sigma }$ be its real irreducible Clifford module, graded by the volume involution. For a real bundle $E\to X$, its spinor module has volume eigenspaces $E\otimes {\mathrm{\Sigma }}^{\pm }$ and symbol $1_E\otimes c(v)$. Thus the actual isomorphism [F1] is precisely $B_X:x\mapsto x⊠\beta$ into ${\widetilde{KO}}^0(X_{+}\wedge S^8)$ by [F2]. This fixes the generator sign before any coefficient computation. Pullback commutes with the displayed eigenspaces and symbol, so $B_X$ is natural, and group completion gives the assertion for virtual bundles.

For based finite CW $Y$, basepoint evaluation splits $K{O}^0(Y)$ as ${\widetilde{KO}}^0(Y)\oplus K{O}^0(\mathrm{p}\mathrm{t})$. Target restriction to the basepoint sphere $S^8$ is also split, by $Y\to \mathrm{p}\mathrm{t}$. Naturality in step 1.1 gives a commuting square of these split projections with vertical maps $B_Y$ and $B_{\mathrm{p}\mathrm{t}}$, both isomorphisms. The induced map on kernels is an isomorphism: injectivity is inherited, and a lift of an element of the target kernel has zero basepoint restriction because $B_{\mathrm{p}\mathrm{t}}$ is injective. By the split quotient identification [F2], this is ${\widetilde{KO}}^0(Y)\cong {\widetilde{KO}}^0(Y\wedge S^8)$, with map again multiplication by $\beta$.

Apply step 2.1 to $Y=X\mathrm{/}A$, using $X_{+}$ when $A$ is empty. It gives $K{O}^0(X,A)\cong K{O}^{-8}(X,A)$. Applying it to $S^k\wedge Y$ for $k\ge 0$ gives $K{O}^{-k}(X,A)\cong K{O}^{-k-8}(X,A)$. Smash associativity and the suspension product convention of [F2] identify every one of these maps with multiplication by the same $\beta$. All spaces used remain finite CW; no arbitrary-base continuity assertion is needed.

For any integer $j$, define its periodic continuation by the groups $K{O}^{j-8t}(X,A)$ for sufficiently large integers $t$ with $j-8t\le 0$, identifying successive groups by the isomorphisms just proved. Formally this is the direct limit under multiplication by $\beta$ as $t$ increases. Since every transition is an isomorphism, each term maps isomorphically to the limit; all choices are canonically identified by finite composites of these maps. In nonpositive degrees this recovers the original groups. The same natural maps define every pair map and connecting map in positive degrees. Shifting $t$ proves the asserted periodicity for all $j$, rather than presupposing positive-degree groups or their periodicity.

Exterior products of the relative symbols are associative by [F2]. Therefore $(xy)\beta =x(y\beta )$ and $(x\beta )(y\beta )=(xy){\beta }^2$ after the canonical rearrangement of sphere factors. The interchange sign involving the Bott sphere is $(-1{)}^{8k}=1$ in every suspended degree $k$. Consequently products pass to the continuation of step 4.1 independent of representatives, and $\beta$ becomes invertible there: its inverse is the unique coefficient sent to the unit by its isomorphism. This proves natural multiplicative periodicity on finite pairs and all degrees. The only choice assumption is that already used in the real Thom and relative bundle constructions [F1]–[F2].