Module derivations and cohomological splitting of adjoint bundles

Tom 180 / 2003

Akira Kono, Katsuhiko Kuribayashi Fundamenta Mathematicae 180 (2003), 199-221 MSC: 55T20, 57T35, 55S05. DOI: 10.4064/fm180-3-1


Let $G$ be a finite loop space such that the mod $p$ cohomology of the classifying space $BG$ is a polynomial algebra. We consider when the adjoint bundle associated with a $G$-bundle over $M$ splits on mod $p$ cohomology as an algebra. In the case $p = 2$, an obstruction for the adjoint bundle to admit such a splitting is found in the Hochschild homology concerning the mod $2$ cohomologies of $BG$ and $M$ via a module derivation. Moreover the derivation tells us that the splitting is not compatible with the Steenrod operations in general. As a consequence, we can show that the isomorphism class of an $SU(n)$-adjoint bundle over a $4$-dimensional CW complex coincides with the homotopy equivalence class of the bundle.


  • Akira KonoDepartment of Mathematics
    Faculty of Science
    Kyoto University
    Kyoto 606, Japan
  • Katsuhiko KuribayashiDepartment of Applied Mathematics
    Faculty of Science
    Okayama University of Science
    Okayama 700-0005, Japan

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