Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lannes' $ T$ functor on summands of $ H\sp *(B({\bf Z}/p)\sp s)$

Authors: John C. Harris and R. James Shank
Journal: Trans. Amer. Math. Soc. 333 (1992), 579-606
MSC: Primary 55S10
MathSciNet review: 1118825
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ H$ be the $ \bmod$-$ p$ cohomology of the classifying space $ B({\mathbf{Z}}/p)$ thought of as an object in the category, $ \mathcal{U}$, of unstable modules over the Steenrod algebra. Lannes constructed a functor $ T:\mathcal{U} \to \mathcal{U}$ which is left adjoint to the functor $ A \mapsto A \otimes H$. In this paper we evaluate $ T$ on the indecomposable $ \mathcal{U}$-summands of $ {H^{ \otimes s}}$, the tensor product of $ s$ copies of $ H$. Our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring $ {{\mathbf{F}}_p}[{{\mathbf{M}}_{s,}}_s({\mathbf{Z}}/p)]$. The main application is to determine the homotopy type of the space of maps from $ B({\mathbf{Z}}/p)]$ to $ X$ when $ X$ is a wedge summand of the space $ \Sigma (B{({\mathbf{Z}}/p)^s})$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55S10

Retrieve articles in all journals with MSC: 55S10

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society