Coproducts in brane topology

Abstract

We extend the loop product and the loop coproduct to the mapping space from the k-dimensional sphere, or more generally from any k-manifold, to a k-connected space with finite dimensional rational homotopy group, k ≥ 1. The key to extending the loop coproduct is the fact that the embedding M → M S k−1 is of "finite codimension" in a sense of Gorenstein spaces. Moreover, we prove the associativity, commutativity, and Frobenius compatibility of them.

A reduction of the string bracket to the loop product

“…So is the product ⊙ on H * (LM ) if M is the classifying space BG of a connected Lie group G (see [8] and [25,Theorem B.1]). Moreover, so are both of • and ⊙ if M is a Gorenstein space with dim( n π n (M ) ⊗ Q) < ∞ (see [37,Theorem 1.1] and [47,Theorem 1.5]).…”