We study the analogue of tautological rings of fibre bundles in the context of fibrations with Poincar\' e fibre, i.e. the ring obtained by fibre integrating powers of the fibrewise Euler class. We discuss how to compute the Euler ring with tools from rational homotopy theory and completely determine the tautological ring for even spheres, complex projective spaces and some products of odd spheres.
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Nils Prigge (2026) studied this question.
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