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We prove that the stable moduli space of $(n-1)$-connected,$n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to aninfinite loopspace for $n \geq 4, n \neq 7$. The main novel ingredient is aversion of the cobordism category incorporating surgery data in the form ofLagrangian subspaces.

Cobordism categories and moduli spaces of odd dimensional manifolds