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Misha
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The maps $\psi_k$ are all isomorphisms; this is a simple application of the transfer ("averaging") construction. See Theorem 2.4, Chapter IIIII, of Bredon's book "Introduction to compact transformation groups".

The maps $\psi_k$ are all isomorphisms; this is a simple application of the transfer ("averaging") construction. See Theorem 2.4, Chapter II, of Bredon's book "Introduction to compact transformation groups".

The maps $\psi_k$ are all isomorphisms; this is a simple application of the transfer ("averaging") construction. See Theorem 2.4, Chapter III, of Bredon's book "Introduction to compact transformation groups".

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Misha
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The maps $\psi_k$ are all isomorphisms; this is a simple application of the transfer ("averaging") construction. See Theorem 2.4, Chapter II, of Bredon's book "Introduction to compact transformation groups".