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Topological applications of graded Frobenius \(n\)-homomorphisms. II. (English. Russian original) Zbl 1273.13002
Trans. Mosc. Math. Soc. 2012, 167-182 (2012); translation from Tr. Mosk. Mat. O.-va 73, No. 2, 207-228 (2012).
In the main result of the paper, the authors prove that if \(X\) is a connected Hausdorff space homotopy equivalent to a CW-complex such that \(\mathrm{dim}H^q (X;\mathbb{Q}) < \infty\) for all \(q> 0\) then the algebra \(H^\ast (X;\mathbb{Q})\) has a structure of graded \(n\)-Hopf prealgebras provided that \(X\) has a structure of \(nH\)-space, respectively \(H^\ast (X;\mathbb{Q})\) has a structure of a graded \(n\)-bialgebra if \(X\) has a structure of a \(nH\)-monoid (Theorem 1). Moreover, if \(X\) admits a structure of a \(2H\)-space, then its fundamental group \(\pi_1(X)\) does not belong to the class \(\mathcal{C}\) (Theorem 2).
MSC:
13A02 Graded rings
16T05 Hopf algebras and their applications
55P45 \(H\)-spaces and duals
57N65 Algebraic topology of manifolds
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