Stable moduli spaces of odd-dimensional manifold triads

TL;DR

This paper provides a homotopy-theoretic framework to understand the homology of stable moduli spaces of odd-dimensional manifold triads with fixed boundary components, extending previous results for even-dimensional cases.

Contribution

It introduces a new homotopy-theoretic description for the homology of stable moduli spaces of odd-dimensional manifold triads, generalizing earlier work and connecting to Kreck's classification.

Findings

Homology description of stable moduli spaces for odd-dimensional triads.

Extension of Galatius and Randal-Williams' results to odd dimensions.

Analog of Kreck's classification for odd-dimensional triads.

Abstract

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N, \partial^h N, \partial^v N)$ with fixed $\partial^v N$, whenever $n \geq 3$ and $(N, \partial^h N)$ 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.