Involutions in the topologists' orthogonal group

TL;DR

This paper classifies involutions in the orthogonal group over a field of two elements, revealing a duality and analyzing numerical invariants, especially in even-dimensional spaces.

Contribution

It provides a detailed classification of involutions in the orthogonal group over GF(2), including new insights into their duality and invariants in even dimensions.

Findings

Involutions satisfy a duality property.

Classification of conjugacy classes over GF(2).

Analysis of numerical invariants in even-dimensional cases.

Abstract

We classify conjugacy classes of involutions in the isometry groups of nondegenerate, symmetric bilinear forms over the field of two elements. The new component of this work focuses on the case of an orthogonal form on an even dimensional space. In this context we show that the involutions satisfy a remarkable duality, and we investigate several numerical invariants.