# Twisted Tensor Products Related to the Cohomology of the Classifying Spaces of Loop Groups

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*Katsuhiko Kuribayashi; Mamoru Mimura; Tetsu Nishimoto*

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

# Table of Contents

## Twisted Tensor Products Related to the Cohomology of the Classifying Spaces of Loop Groups

- Contents 6370
- 1. Introduction 18 free
- 2. The mod 2 cohomology of BLSO(n) 916 free
- 3. The mod 2 cohomology of BLG for G = Spin(n) (7 ≤ n ≤9) 1017
- 4. The mod 2 cohomology of BLG for G = G[sub(2)], F[sub(4)] 1320
- 5. A multiplication on a twisted tensor product 1522
- 6. The twisted tensor product associated with H*{Spin(N);Z/2) 3542
- 7. A manner for calculating the homology of a DGA 3946
- 8. The Hochschild spectral sequence 4552
- 9. Proof of Theorem 1.6 5057
- 10. Computation of a cotorsion product of if H*(Spin(10); Z/2) and the Hochschild homology of H*(BSpin(10); Z/2) 6370
- 11. Proof of Theorem 1.7 7077
- 12. Proofs of Proposition 1.9 and Theorem 1.10 7380
- 13. Appendix 7784
- Bibliography 8390