Manifolds admitting a continuous cancellative binary operation are orientable

TL;DR

This paper proves that topological manifolds with a continuous cancellative binary operation must be orientable, demonstrating that the Möbius band cannot admit such an operation, thus answering a longstanding open question.

Contribution

It establishes a fundamental topological property linking cancellative binary operations to orientability of manifolds, resolving a question from 2010.

Findings

Manifolds with a continuous cancellative binary operation are orientable

The Möbius band cannot admit a cancellative continuous binary operation

Answers a question posed in 2010 about manifold operations

Abstract

We prove that a topological manifold (possibly with boundary) admitting a continuous cancellative binary operation is orientable. This implies that the M\"obius band admits no cancellative continuous binary operation. This answers a question posed by the second author in 2010.