Homological Stability For The Moduli Spaces of Products of Spheres

TL;DR

This paper establishes a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds when forming connected sums with products of spheres, extending previous results to new cases.

Contribution

It introduces a homological stability result for moduli spaces involving connected sums with $S^{p} imes S^{q}$ under specific dimension constraints, generalizing prior work.

Findings

Proves stability for moduli spaces under connected sums with sphere products

Extends homological stability results to new high-dimensional manifold classes

Provides a framework for understanding the topology of high-dimensional manifolds

Abstract

We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension $2n > 4$, with respect to forming connected sums with $S^{n}\times S^{n}$.