We show vanishing of some cup products on the rational cohomology of the mapping space from the k-sphere to a Poincaré duality space. When k is odd, it can be proved by using the brane coproduct, which is a generalization of the loop coproduct. In this talk, we focus on the case that k is even. In that case, the key of the proof is a convenient algebraic model of the mapping space, which is constructed by using the notion of regular sequences (from commutative ring theory) and its relationship to Poincaré duality.

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