# New construction of the brane coproduct and vanishing of cup products on sphere spaces

## TL;DR

This paper extends the vanishing of cup products with the orientation class from manifolds to sphere spaces by generalizing the loop coproduct, introducing new coproduct constructions for these spaces.

## Contribution

It introduces a new coproduct construction for sphere spaces and generalizes previous results on cup product vanishing to these spaces.

## Key findings

- Cup product with orientation class vanishes on sphere spaces with non-trivial Euler characteristic
- New coproduct construction for sphere spaces
- Generalization of Menichi's result to broader class of spaces

## Abstract

Using the loop coproduct, Menichi proved that the cup product with the orientation class vanishes for a closed connected oriented manifold with non-trivial Euler characteristic. We generalize this to the sphere spaces, i.e. the mapping spaces from spheres, using two generalizations of the loop coproduct to sphere spaces. One is constructed in this paper and the other in a previous paper of the author.