Let \(G\) be a compact, simply connected, simple Lie group. By applying the notion of a twisted tensor product in the senses of Brown as well as of Hess, we construct an economical injective resolution to compute, as an algebra, the cotorsion product which is the \(E_2\)-term of the cobar type Eilenberg-Moore spectral sequence converging to the cohomology of classifying space of the loop group \(LG\). As an application, the cohomology \(H^*(BLSpin(10); \mathbb{Z}/2)\) is explicitly determined as an \(H^*(BSpin(10); \mathbb{Z}/2)\)-module by using effectively the cobar type spectral sequence and the Hochschild spectral sequence, and further, by analyzing the TV-model for \(BSpin(10)\).

Table of Contents

• Introduction
• The mod 2 cohomology of \(BLSO(n)\)
• The mod 2 cohomology of \(BLG\) for \(G=Spin(n)\ (7\leq n\leq 9)\)
• The mod 2 cohomology of \(BLG\) for \(G=G_2,F_4\)
• A multiplication on a twisted tensor product
• The twisted tensor product associated with \(H^*(Spin(N);\mathbb{Z}/2)\)
• A manner for calculating the homology of a DGA
• The Hochschild spectral sequence
• Proof of Theorem 1.6
• Computation of a cotorsion product of \(H^*(Spin(10);\mathbb{Z}/2)\) and the Hochschild homology of \(H^*(BSpin(10);\mathbb{Z}/2)\)
• Proof of Theorem 1.7
• Proofs of Proposition 1.9 and Theorem 1.10
• Appendix
• Bibliography