The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups 𝑃𝑈(𝑝) and exceptional Lie groups

For a fixed prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we compute the Brown-Peterson cohomologies of classifying spaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P upper U left-parenthesis p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">PU(p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B upper P Superscript asterisk Baseline left-parenthesis upper B upper P upper U left-parenthesis p right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mi>P</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">BP^*(BPU(p))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis n right-parenthesis Superscript asterisk Baseline left-parenthesis upper B upper P upper U left-parenthesis p right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>B</mml:mi> <mml:mi>P</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K(n)^*(BPU(p))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are even dimensionally generated.