The Chern character on the odd spinor groups
This paper is a continuation of one by the author [Osaka J. Math. 22, 463-488 (1985; Zbl 0579.57020)]. Here the author calculates explicitly the Chern character, ch: \(K^*(Spin(2\ell +1\)))\(\to H^*(Spin(2\ell +1); {\mathbb{Q}})\), for \(\ell \geq 5\). Since \(K^*(Spin(2\ell +1))\) is generated multiplicatively by the image of the map \(\beta\) : R(Spin(2\(\ell +1))\to K^*(Spin(2\ell +1))\) introduced by \textit{L. Hodgkin} [Topology 6, 1-36 (1967; Zbl 0186.571)], one needs to compute ch \(\beta\) (\(\lambda)\), for \(\lambda\) in the representation ring \(R(Spin(2\ell +1))={\mathbb{Z}}[\lambda_ 1',...,\lambda '_{\ell -1}\), \(\Delta_{2\ell +1}].\) The author shows how to compute ch \(\beta\) (\(\lambda\) \({}_ k')\), ch \(\beta\) (\(\Delta\) \({}_{2\ell +1})\) once ch \(\beta\) (\(\lambda\) \({}_ 1')\) is known. Therefore, the core result of the paper, theorem 1, gives an explicit formula for ch \(\beta\) (\(\lambda\) \({}_ 1')\). Finally, an application is given to computing the transgression in the Serre spectral sequence of the fibration \(Spin(2\ell +1)\to Spin(2\ell +1)/T\to BT\), for T a maximal torus of \(Spin(2\ell +1)\).
fibration defined by factoring out the maximal torus
scientific article; zbMATH DE number 4009358