Some remarks on equivariant bundles and classifying spaces. (English) Zbl 0728.55011
Théorie de l’homotopie, Colloq. CNRS-NSF-SMF, Luminy/Fr. 1988, Astérisque 191, 239-253 (1990).
[For the entire collection see Zbl 0721.00021.]
Let \(\Pi\) be a normal subgroup of a topological group \(\Gamma\) with quotient group G and let X be a G-space. \({\mathcal B}(\Pi,\Gamma)(X)\) denote the set of equivalence classes of principal (\(\Pi\),\(\Gamma\))-bundles over X. If \(X_ G=EGX_ GX\) is the Borel construction associated to X, then \({\mathcal B}(\Pi,\Gamma)(X_ G)\) denote the set \({\mathcal B}_ G(\Pi,\Gamma)(EG\times X)\). The projection EG\(\times X\to X\) induces a natural map \(\psi\) : \({\mathcal B}_ G(\Pi,\Gamma)(X)\to {\mathcal B}(\Pi,\Gamma)(X_ G).\)
Under certain conditions for \(\Gamma\), \(\Pi\), G, the author establishes that \(\psi\) is a bijection, respectively \(\psi\) is represented by a mod p equivalence of classifying G-spaces. The results are interpretations of other theorems about equivariant classifying spaces.
The paper is very rich in ideas and comments.
Let \(\Pi\) be a normal subgroup of a topological group \(\Gamma\) with quotient group G and let X be a G-space. \({\mathcal B}(\Pi,\Gamma)(X)\) denote the set of equivalence classes of principal (\(\Pi\),\(\Gamma\))-bundles over X. If \(X_ G=EGX_ GX\) is the Borel construction associated to X, then \({\mathcal B}(\Pi,\Gamma)(X_ G)\) denote the set \({\mathcal B}_ G(\Pi,\Gamma)(EG\times X)\). The projection EG\(\times X\to X\) induces a natural map \(\psi\) : \({\mathcal B}_ G(\Pi,\Gamma)(X)\to {\mathcal B}(\Pi,\Gamma)(X_ G).\)
Under certain conditions for \(\Gamma\), \(\Pi\), G, the author establishes that \(\psi\) is a bijection, respectively \(\psi\) is represented by a mod p equivalence of classifying G-spaces. The results are interpretations of other theorems about equivariant classifying spaces.
The paper is very rich in ideas and comments.
Reviewer: I.D.Albu (Timişoara)
MSC:
| 55R91 | Equivariant fiber spaces and bundles in algebraic topology |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 55R15 | Classification of fiber spaces or bundles in algebraic topology |
