Stable moduli spaces of odd-dimensional manifold triads

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N,{\mathrm{\partial }}^{h}N,{\mathrm{\partial }}^{v}N)$ with fixed ${\mathrm{\partial }}^{v}N$, whenever $n\ge 3$ and $(N,{\mathrm{\partial }}^{h}N)$ is $1$-connected. Stabilization is performed by taking boundary connected sum with $S^n\times D^{n+1}$. This is an analog of earlier work of Galatius and Randal-Williams for even-dimensional manifolds with fixed boundary, and it extends a previous result by Botvinnik and Perlmutter. As a byproduct, we obtain an analog for odd-dimensional triads of Kreck’s stable diffeomorphism classification of even-dimensional manifolds.