Classifying space for U(n): Difference between revisions

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==K-theory of BU(''n'')==

Let us consider topological complex K-theory as the cohomology theory represented by the spectrum <math>KU</math>. In this case, <math>KU^*(BU(n))\cong \mathbb{Z}[t,t^{-1}][[c_1,...,c_n]]</math><ref> Adams 1974, p. 49</ref>, and <math> KU_*(BU(n))</math> is the free <math>\mathbb{Z}[t,t^{-1}]</math> module on <math>\beta_0</math> and <math>\beta_{i_1}\ldots\beta_{i_r}</math> for <math>n\geq i_j > 0</math> and <math>r\leq n</math>.<ref> Adams 1974, p. 47</ref> In this description, the product structure on <math> KU_*(BU(n)) </math> comes from the H-space structure of <math>BU</math> given by Whitney sum of vector bundles. This product is called the [[Pontryagin product]].

{{warning| The following seems to be a computation of <math>KU_*BU(n)</math>, where <math>KU_*BU(n)</math> gets a ring structure from the tensor product H-space structure on <math>BU</math>. The statement needs clarification.}}

{{warning|The description of the K-theory of BU(n) below is incomplete, and unclear. The author of the secion below should clarify what they are computing. It seems more like a computation of <math>KU_*KU</math>.}}

The [[topological K-theory]] is known explicitly in terms of [[numerical polynomial|numerical]] [[symmetric polynomial]]s.