The Ghost length, levels and shriek maps on classifying spaces

TL;DR

This paper investigates the cochain level of the diagonal map on classifying spaces of Lie groups, using ghost maps and shriek maps, and explores their triviality in derived categories affecting loop products.

Contribution

It provides a lower bound for the cochain level of the diagonal map and analyzes the triviality of shriek maps in the context of classifying spaces.

Findings

Lower bound for the cochain level of the diagonal map.

Conditions for triviality of shriek maps in derived categories.

Implications for the loop product structure.

Abstract

We give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by using the ghostness of a shriek map. Moreover, in a derived category, we discuss the triviality of the shriek map which induces the loop product on the classifying space of a Lie group.