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On the mod 2 cohomology of the classifying space of the exceptional Lie group \(E_6\). (English) Zbl 1451.55004
A. Kono and M. Mimura [J. Pure Appl. Algebra 6, 61–81 (1975; Zbl 0297.55013)] showed that the mod \(2\) cohomology ring of the classifying space \(BE_6\) of the exceptional Lie group \(E_6\) is \[ H^* (BE_6) = \mathbb{Z}/2 [y_4, y_6, y_7, y_{10}, y_{18}, y_{34}, y_{32}, y_{48}]/I \] such that deg\((y_i) = i\) and \(I\) is the ideal generated by \[ y_7 y_{10}, y_7 y_{18}, y_7 y_{34}, \tau_{68}, \] where \[ \tau_{68} = y^2_{34} + y^2_{18}y_{32} + y^2_{10} y_{48} + \text{ higher terms}. \] Let \(\rho^*_6 : H^* (BSU(27)) \rightarrow H^* (BE_6)\) be the homomorphism in cohomology induced by the representation \(\rho_6 : E_6 \rightarrow SU(27)\) via classifying spaces. Then the \(i\)-th Chern class \(c_i(\rho_6)\) of \(\rho_6\) is defined by \(\rho^*_6 (c_i)\), where \(c_i\) is the \(i\)-th universal Chern class in \(H^* (BSU(27))\).
In this paper, the authors define the generators \(y_{32}, y_{48}\) by \[ y_{32} = \rho^*_6 ( c_{16} + c_4 c_{12} + c^2_4 c_8 ) \] and \[ y_{48} = \rho^*_6 ( c_{24} + c_{10} c_{14} + c_6 c_{18} + c^2_6 c_{12} + c_4 c_6 c_{14} ). \] By applying a squaring operation to the Chern class, the authors give a more precise description of \(\tau_{68}\) of \(H^* (BE_6)\) as an algebra over the mod \(2\) Steenrod algebra as follows: \[ \tau_{68} = y^2_{34} + y^2_{18}y_{32} + y^2_{10} y_{48} + y_6 y_{10} y_{18} y_{34} + y_4 y_{10} y^3_{18} + y_4 y^3_{10} y_{34}. \]

MSC:
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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