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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
##### Keywords:
cohomology; classifying space; exceptional Lie group
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##### References:
 [1] J. F. Adams, Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. · Zbl 0866.22008 [2] M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23-64. · Zbl 0107.02303 [3] A. Borel, Sur l’homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math. 76 (1954), 273-342. · Zbl 0056.16401 [4] N. Bourbaki, Lie groups and Lie algebras. Chapters 4-6, translated from the 1968 French original by Andrew Pressley, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. · Zbl 0983.17001 [5] A. Kono and M. Mimura, Cohomology $$\text{mod}\,2$$ of the classifying space of the compact connected Lie group of type $$E_6$$, J. Pure Appl. Algebra 6 (1975), 61-81. · Zbl 0297.55013 [6] W. S. Massey and F. P. Peterson, The cohomology structure of certain fibre spaces. I, Topology 4 (1965), 47-65. · Zbl 0132.19103 [7] M. Mimura and T. Nishimoto, On the Stiefel-Whitney classes of the representations associated with $$\text{Spin}(15)$$, in Proceedings of the School and Conference in Algebraic Topology, 141-176, Geom. Topol. Monogr., 11, Geom. Topol. Publ., Coventry, 2007. · Zbl 1145.55015 [8] M. Mimura and H. Toda, Topology of Lie groups. I, II, translated from the 1978 Japanese ed. by the authors, Translations of Mathematical Monographs, 91, American Mathematical Society, Providence, RI, 1991. [9] D. Quillen, The $$\text{mod}\,2$$ cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197-212. · Zbl 0225.55015 [10] H. Toda, Cohomology of the classifying space of exceptional Lie groups, in Manifolds-Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), 265-271, Univ. Tokyo Press, Tokyo, 1975. [11] A. Vavpetič and A. Viruel, On the homotopy type of the classifying space of the exceptional Lie group $$F_4$$, Manuscripta Math. 107 (2002), no. 4, 521-540. · Zbl 1007.55011
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