A topologically induced 2-in/2-out operation on loop cohomology

TL;DR

This paper demonstrates a novel application of the Transfer Algorithm to construct a space with a nontrivial 2-in/2-out operation on its loop cohomology, revealing new algebraic structures beyond classical methods.

Contribution

It introduces a new method for transferring A_ Infinity-algebra structures using the Transfer Algorithm, enabling the construction of nontrivial operations on loop cohomology.

Findings

Constructed a space with nontrivial  operation on loop cohomology.

Applied Transfer Algorithm to transfer complex algebraic structures.

Revealed algebraic operations not accessible via classical perturbation methods.

Abstract

We apply the Transfer Algorithm introduced in arXiv:1106.5090 to transfer an A_\infty-algebra structure that cannot be computed using the classical Basic Perturbation Lemma. We construct a space X whose (base pointed) loop cohomology H = H^*(\Omega X; Z_2) comes equipped with a nontrivial operation \omega : H x H --> H x H.