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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm180-3-1,
author = {Akira Kono and Katsuhiko Kuribayashi},
title = {Module derivations and cohomological splitting of adjoint bundles},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {199-221},
zbl = {1070.55012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm180-3-1}
}
Akira Kono; Katsuhiko Kuribayashi. Module derivations and cohomological splitting of adjoint bundles. Fundamenta Mathematicae, Tome 177 (2003) pp. 199-221. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm180-3-1/