# On the Alexandroff-Borsuk problem

## TL;DR

This paper explores whether non-triangulable manifolds can be approximated by finite polyhedra via small maps that preserve homotopy type, addressing a classical topological problem.

## Contribution

It investigates the existence of epsilon-maps from non-triangulable manifolds onto finite polyhedra that induce homotopy equivalences, extending the understanding of the Alexandroff-Borsuk problem.

## Key findings

- Results on the existence or non-existence of such epsilon-maps.
- Insights into the structure of non-triangulable manifolds.
- Connections between non-triangulability and homotopy approximation.

## Abstract

We investigate the classical Alexandroff-Borsuk problem in the category of non-triangulable manifolds: Given an $n$-dimensional compact non-triangulable manifold $M^n$ and $\varepsilon > 0$, does there exist an $\varepsilon$-map of $M^n$ onto an $n$-dimensional finite polyhedron which induces a homotopy equivalence?