The minimal dimension of a sphere with an equivariant embedding of the bouquet of $g$ circles is $2g-1$

TL;DR

This paper determines the smallest sphere dimension into which a bouquet of g circles can be embedded equivariantly, extending symmetry actions to orthogonal actions, and answers a question posed by Zimmermann.

Contribution

It establishes that the minimal dimension for such equivariant embeddings of the bouquet of g circles is 2g-1, providing a definitive answer to Zimmermann's question.

Findings

Minimal embedding dimension is 2g-1 for bouquet of g circles.

Equivariant embeddings extend to orthogonal actions on spheres.

Answers a previously open question by Zimmermann.

Abstract

To embed the bouquet of $g$ circles $B_g$ into the $n$-sphere $S^n$ so that its full symmetry group action extends to an orthogonal actions on $S^n$, the minimal $n$ is $2g-1$. This answers a question raised by B. Zimmermann.