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Une interprétation géométrique des nombres de Radon-Hurwitz. (French) Zbl 0159.53503

The author proves that the Radon-Hurwitz numbers \(q(n,F)\) over the fields \(F=\mathbb R\), \(\mathbb C\) or \(\mathbb H\) have the following common geometric significance, stated in two equivalent theorems.
Theorem I. The geometric dimension of the tangent bundle \(\tau(P^{n-1}(F))\) to the \((n-1)\)-\(F\)-dimensional projective space over \(F\) is \(d(F)\cdot n-q(n,F)\), where \(d(F) =1, 2, 4\) resp. when \(F=\mathbb R\), \(\mathbb C\) or \(\mathbb H\).
Theorem II. If \(F_0^*\) is the multiplicative group of elements of \(F\) of length 1, then the tangent bundle to \(S^{n-1}(F)\) (the unit sphere \(S^{n\cdot d(F)-1}\) in \(F^n\cong \mathbb R^{n\cdot d(F)})\) admits a trivial \(F_0^*\)-invariant factor of dimension \(q(n,F)-1\), and has no such of higher dimension.
The result is known when \(F= \mathbb R\) from the work of Adams and the author is extending to the cases \(F=\mathbb C\) and \(\mathbb H\). The only nontrivial case is when \(n\) is even. Using the fact that \(q=q(n,F)\) is the greatest integer such that \(C_q(F)\), the Clifford algebra over \(F\), has an irreducible module \(M^0\otimes M^1\) with \(M^0\simeq M^1 \simeq F^k\) and \(k\mid n\), the author shows how to construct from such a module \(q-1\) \(F_0^*\)-independent fields on \(S^{n-1}(F)\), in cases \(F = \mathbb C, \mathbb H\), thus proving the existence of the factor of theorem II, or that \(d(F)\cdot n-q(n,F)\) in theorem I is a bound on the geometric dimension. He finishes the proof by using \(K\)-theory to prove the geometric dimension cannot be less than \(d(F)\cdot n - q(n,F)\). Namely, he shows that a necessary condition that a \(2n-1\) dimensional vector bundle \(\mu\) over a finite CW complex \(X\) with \(\mathrm{Spin}(2n-1)\) structure have geometric dimension \(< 2n - 2r\) is that there exist \(\beta\in K(X)\) such that \(2^{r+1}\beta=\Delta(n)(\mu)\), where \(\Delta(n)\) is the representation \(\mathrm{Spin}(2n-1)\rightarrow U(2^{n-1})\). Then the author finishes by calculating that \(\Delta(n)(\tau\otimes 1)=\pm \tfrac12 (2+\alpha)^n\cdot (1+\alpha)^{-n/2}\) when \(F=\mathbb C\) and \(\Delta(n)(\tau\otimes 3)=\pm \tfrac12 (4+\alpha)^n\) when \(F=\mathbb H\) where \(\alpha\) generates the polynomial ring \(K(\pm \tfrac12(2+\alpha)^n)\), and \(\tau\) is the real tangent bundle to \(\pm \tfrac12(2+\alpha)^n\).
Reviewer: St. Weingram

MSC:

55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
55Q40 Homotopy groups of spheres
55R25 Sphere bundles and vector bundles in algebraic topology
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