## Celebrating the Career and Contributions of Katsuya Eda: Master of the Earring Group

Last week, I had the pleasure of speaking at the Arches Topology Conference in Moab, Utah. The conference was in honor of the career and contributions of Katsuya Eda, who retired not too long ago.

K. Eda considers himself a logician and set theorist, however, most of his research has been related to topology, what we now consider to be wild algebraic topology. He developed many of the technologies and proved many of the foundational theorems being used today in wild algebraic topology.

Katsuya Eda giving a talk at the 2018 Arches Topology Conference

Katsuya Eda at the 2018 Arches Topology Conference

Although a few phenomena of wild algebraic topology had appeared in a disjointed fashion in literature in the 1950’s and 60’s (e.g. H.B. Griffiths Twin Cone, Curtis and Fort’s asphericity of 1-dimensional spaces and Barratt and Milnor’s Anomalous Singular Homology) and shape theory was fashionable in the 70’s, the first systematic study of wild algebraic topology began with Eda’s series of papers on the earring space and one-dimensional spaces in the 1990’s. In many ways, Katsuya Eda is the pioneer of what is now the developing field of wild algebraic topology.

While Eda was not the first to describe the fundamental group of the earring space, i.e. the earring group, he certainly was the first to offer new ways of thinking about it so that it could be used in serious ways.

Among Eda’s many contributions are:

1. The homotopy classification of one-dimensional Peano continua: one-dimensional Peano continua $X$ and $Y$ are homotopy equivalent if $\pi_1(X,x_0)\cong\pi_1(Y,y_0)$. In particular, the subspace of wild points is a homotopy invariant.
• This includes the earring space, Sierpinski Carpet, Menger Curve, etc. All are determined up to homotopy by their fundamental group. Absolutely remarkable!
• The proof is highly technical and is completed in a series of papers.
2. Automatic Continuity: In the process of 1. Eda proved that any self-homomorphism $\pi_1(\mathbb{E},b_0)\to\pi_1(\mathbb{E},b_0)$ of the earring group is induced (up to conjugation) by a continuous function $\mathbb{E}\to\mathbb{E}$. His No-Head-Tail Lemma needed to prove this is an example of truly great mathematics. He later extended this result to show that all homomorphisms between fundamental groups of one-dimensional Peano continua are induced (up to conjugation) by a continuous function.
3. Singular Homology: Computing the first singular homology group $H_1(\mathbb{E})$ of the earring $\mathbb{E}$. Eda did this by showing that $H_1(\mathbb{E})$ splits as a direct sum of the first Cech homology group $\mathbb{Z}^{\mathbb{N}}$ and an algebraically compact group abstractly isomorphic to $\mathbb{Z}^{\mathbb{N}}/\oplus_{\mathbb{N}}\mathbb{Z}$. Eda later showed that $H_1$ of any wild 1-dimensional Peano continuum has the same singular homology. This strange phenomenon is very much a result of the decomposition theory of infinite abelian groups.
4. Introduction of non-commutatively slender or n-slender groups, which is an analogue of “slenderness” for non-abelian groups. His seminal paper on Free $\sigma$-products and noncommutatively slender groups  introduces the notion of transfinite word, i.e. a word with letters indexed by a countable linear order so that no single letter appears more than finitely many times. Although the term “transfinite word” was later by Jim Cannon and Greg Conner, Eda used this idea to allow the application of linear order theory to study the earring group and to use it as a non-commutative analogue of the Baer-Specker group.
5. Higher homotopy groups: Proving $\pi_n(\mathbb{E}_n)\cong \mathbb{Z}^{\mathbb{N}}$ where $\mathbb{E}_n$ is the n-dimensional earring. This is a lot harder than one might think. A huge insight here was that higher homotopy groups are “infinitely commutative” in a certain sense.
6. Much more: A huge list of other results relating to coverings/overlays of topological groups, the relationship between singular and Cech (co)homology, combinatorial aspects of the earring group and other infinitely generated non-abelian groups, and the introduction of many new examples and phenomena.

Katsuya Eda has a unique way of thinking about mathematics and I personally have been inspired by his remarkable work and ingenuity.

Jeremy Brazas and Katsuya Eda at the 2018 Arches Topology Conference

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