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Rational homotopy of gauge groups. (English) Zbl 1168.55010
For \(K\rightarrow P\rightarrow^{\xi} B\) a continuous principal \(K\)-bundle (\(K \) a compact connected Lie group), \(G(\xi)\) the gauge group of \(\xi\) and \( G_{1}(\xi)\) the subgroup of the self-homeomorphisms that preserve the basepoint of \(P\), the authors obtain, from the basic results of rational homotopy theory, formulas for the rational homotopy groups of gauge groups of principal bundles in terms of the rational homotopy groups of \(K\) and the cohomology groups of \(B\): “If \(B\) has the homotopy type of a finite connected \(CW\)-complex, then for any \(q \geq 1\) we have: \( \pi_{q}(G(\xi))\otimes \mathbb{Q} \cong \sum_{r\geq0} H^{r}(B;\mathbb{Q})\otimes \pi_{r+q}(K)\) and \(\pi_{q}(G_{1}(\xi))\otimes \mathbb{Q} \cong \sum_{r\geq0} \widetilde{H^{r}} (B;\mathbb{Q})\otimes \pi_{r+q}(K)\) where \(\widetilde{H}\) denotes reduced cohomology.”

MSC:
55P99 Homotopy theory
57R91 Equivariant algebraic topology of manifolds
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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