Module derivations and cohomological splitting of adjoint bundles

Akira Kono, and Katsuhiko Kuribayashi. "Module derivations and cohomological splitting of adjoint bundles." Fundamenta Mathematicae 180.3 (2003): 199-221. <http://eudml.org/doc/283297>.

@article{AkiraKono2003,
abstract = {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.},
author = {Akira Kono, Katsuhiko Kuribayashi},
journal = {Fundamenta Mathematicae},
keywords = {mod- cohomological splitting; -cohomological splitting; Hochschild homology; module derivation; the bar type Eilenberg–Moore spectral sequence},
language = {eng},
number = {3},
pages = {199-221},
title = {Module derivations and cohomological splitting of adjoint bundles},
url = {http://eudml.org/doc/283297},
volume = {180},
year = {2003},
}

TY - JOUR
AU - Akira Kono
AU - Katsuhiko Kuribayashi
TI - Module derivations and cohomological splitting of adjoint bundles
JO - Fundamenta Mathematicae
PY - 2003
VL - 180
IS - 3
SP - 199
EP - 221
AB - 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.
LA - eng
KW - mod- cohomological splitting; -cohomological splitting; Hochschild homology; module derivation; the bar type Eilenberg–Moore spectral sequence
UR - http://eudml.org/doc/283297
ER -