Module derivations and cohomological splitting of adjoint bundles

Akira Kono¹ ; Katsuhiko Kuribayashi²

¹Department of Mathematics Faculty of Science Kyoto University Kyoto 606, Japan
²Department of Applied Mathematics Faculty of Science Okayama University of Science Okayama 700-0005, Japan

Fundamenta Mathematicae, Tome 180 (2003) no. 3, pp. 199-221

Résumé

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.

Détail

DOI : 10.4064/fm180-3-1

Keywords: finite loop space mod cohomology classifying space polynomial algebra consider adjoint bundle associated g bundle splits mod cohomology algebra obstruction adjoint bundle admit splitting found hochschild homology concerning mod cohomologies via module derivation moreover derivation tells splitting compatible steenrod operations general consequence isomorphism class adjoint bundle dimensional complex coincides homotopy equivalence class bundle