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A comparison principle and extension of equivariant maps. (English) Zbl 0829.55002

This work unifies and extends many generalizations of Borsuk’s theorem. The authors prove an equivariant version of the Kuratowski-Dugundji Theorem. In order to do this, they show the existence of a (quasi)- fundamental domain for any free action of a compact Lie group on a metric space \(X\). Then for \(G\) finite and \(\Psi, \varphi : M \to S\) a pair of equivariant maps where \(M\) is a manifold of the same dimension \(n\) as the sphere \(S\), they show that \(\deg \Psi - \deg \varphi \equiv 0 \pmod {\text{GCD} \{|G/H_i |\}}\) under certain hypotheses. Finally they show that \(n - k\) is a lower bound for the genus of \(S\smallsetminus B\) where \(S\smallsetminus B\) is a free \(G\)-subspace and \(B\) the image of a smooth mapping of a \(k\)-dimensional compact manifold.

MSC:

55M20 Fixed points and coincidences in algebraic topology
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
57S17 Finite transformation groups
55P91 Equivariant homotopy theory in algebraic topology
57S15 Compact Lie groups of differentiable transformations
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References:

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