# On Macroscopic dimension of non-spin 4-manifolds

## TL;DR

This paper investigates the macroscopic dimension of certain 4-manifolds, establishing a relationship between the universal cover's properties and its macroscopic dimension, which advances understanding in geometric topology.

## Contribution

It proves that for 4-manifolds with residually finite fundamental groups and non-spin universal covers, a bound on the macroscopic dimension implies a tighter bound, revealing new geometric constraints.

## Key findings

- If $ ext{dim}_{mc} ilde{M} extless= 3$, then $ ext{dim}_{mc} ilde{M} extless= 2$ for the specified class of 4-manifolds.
- The result links the fundamental group properties to the macroscopic dimension of the universal cover.
- Provides new insights into the structure of non-spin 4-manifolds with residually finite fundamental groups.

## Abstract

We prove that for 4-manifolds $M$ with residually finite fundamental group and non-spin universal covering $\Wi M$, the inequality $\dim_{mc}\Wi M\le 3$ implies the inequality $\dim_{mc}\Wi M\le 2$.