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On the homotopy fixed point sets of circle actions on product spaces. (English) Zbl 1509.55008

Summary: For arbitrary \(S^1\)-actions on \(S^m_\mathbb{Q}, S^n_\mathbb{Q}\), and \(S^m_\mathbb{Q} \times S^n_\mathbb{Q}\), we show the conditions for the tenability of the homotopy equivalence \((S^m_\mathbb{Q})^{hS^1} \times (S^n_\mathbb{Q})^{hS^1} \simeq (S^m_\mathbb{Q} \times S^n_\mathbb{Q})^{hS^1}\). Here, \(X^{hS^1}\) denotes the homotopy fixed point set of an \(S^1\)-action on an space \(X\).

MSC:

55P62 Rational homotopy theory
55P91 Equivariant homotopy theory in algebraic topology
57S10 Compact groups of homeomorphisms
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