Document Zbl 0994.55009 - zbMATH Open

The paper starts by regarding an element \(x\in X^H\) as the \(G\)-map \(G/H\rightarrow X\) such that \(eH\mapsto x\). Under this interpretation, for a \(G\)-space \(X\), the fundamental equivariant groupoid \(\Pi_GX\) is defined as being the category whose objects are the \(G\)-maps \(x:G/H\mapsto X\) and whose morphisms \(x\mapsto y\) for \(y:G/K\mapsto X\) are the pairs \((\omega,\alpha)\) where \(\alpha:G/H\mapsto G/K\) is a \(G\)-map and \(\omega\) is an equivalence class of paths \(x\mapsto y\circ\alpha\) in \(X^H\) relative, as usual, to their endpoints. Thus one has that a \(G\)-map \(f:X\mapsto Y\) induces a covariant functor \(f_*:\Pi_GX\mapsto \Pi_GY\) and that a \(G\)-homotopy \(f\cong g\) induces a natural isomorphism \(h_*:f_*\mapsto g_*\). Then, imitating the categorical construction of the ordinary orientation theory, the authors define a \(G\)-bundle as a real \(G\)-vector bundle with orthogonal structure group. Afterwards, the category \(\overline{\mathcal V}_G\) is defined as having objects that are \(G\)-vector bundles and morphisms that are equivalence classes of \(G\)-bundle maps. Here two maps are equivalent if they are \(G\)-bundle homotopic with the homotopy inducing the constant homotopy on base spaces. Now \({\mathcal V}_G(n)\) is defined to be the full subcategory of \(\overline{\mathcal V}_G\) whose objects are the \(n\)-plane bundles of the form \(G\times_H\mathbb R^n\mapsto G/H\) where \(H\) acts on \(\mathbb R^n\) through some representation \(\lambda:H\mapsto O(n)\). After defining the universal orientable representation \({\mathcal SV}_G(n)\) a functor \(S:{\mathcal SV}_G(n)\mapsto{\mathcal V}_G(n)\) is determined such that \(p:E\mapsto B\) is orientable if and only if \(p^*:\Pi_GB\mapsto{\mathcal V}_G(n)\) factors through \({\mathcal SV}_G(n)\). The main result in this paper is the following
Theorem. A \(G\)-vector bundle \(p:E\rightarrow B\) of dimension \(n\) is orientable iff \(p^*:\Pi_GB\rightarrow{\mathcal V}_G(n)\) can be lifted to a functor \(F:\Pi_GB\rightarrow{\mathcal SV}_G(n)\) together with a natural isomorphism \(\phi:S\circ F\rightarrow p^*\). A choice of such a lift \((F,\phi)\) is an orientation of \(p\).
It is proved here that for \(G\) is a finite group of odd order, then a \(G\)-vector bundle is \(G\)-orientable if and only if it is orientable. It is also shown that an orientation of a \(G\)-bundle \(p:E\rightarrow B\) induces orientations of the \(H\)-fixed point bundle over \(B^H\) and of its complementary bundle over \(B^H\) for all subgroups \(H<G\). After reproducing an account of universal spaces due to S. Waner, this paper ends with providing a good account of the classification of oriented \(G\)-bundles, giving some examples of spherical \(G\)-fibrations and PL \(G\)-bundles.

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