A remark on $\varepsilon$-maps in dimension 1

TL;DR

This paper investigates whether certain surjective maps from the circle to a graph, with small pre-image diameters, can be decomposed as free factors in the fundamental group, exploring properties of epsilon-maps in one dimension.

Contribution

It examines conditions under which epsilon-maps from the circle to a graph split as free factors in the fundamental group, providing insights into their algebraic and topological structure.

Findings

Identifies conditions for epsilon-maps to split as free factors.

Establishes bounds on epsilon for such splittings.

Provides examples and counterexamples related to epsilon-maps.

Abstract

Let $f\colon \mathbb{S}^1\rightarrow G$ be a surjective map from the standard unit circle to a graph $G$ such that the pre-image of each point has diameter less than $\varepsilon$. If $\varepsilon$ is small enough, does $f$ split as a free factor in $\pi_1(G)$?

Equations

$$w_k=a_1{a}_2{a}_1{a}_2{a}_3{a}_2{a}_3{a}_4{a}_3...a_{2k-1}{a}_{2k}{a}_{2k-1}{a}_{2k}(\gamma a_1{\gamma }^{-1})a_{2k}\gamma .$$

$$f_k:({\mathbb{S}}^1,x)\to (G_k,y_1),$$

$$...{}^{-}{a}_i^{+}{}^{-}{a}_j^{+}{}^{-}(a_k^{-1}{)}^{+}...$$

$$...{}^{-}{a}_i^{+}{}^{-}{a}_j^{+}{}^{+}{a}_k^{-}...$$

Full text

1. Introduction

Bestvina-Brady’s paper “Morse theory and finiteness propeties of groups” ended with the following conjecture ([1]).

Conjecture 1** (Bestvina-Brady).**

Let $L$ be a finite 2-complex. Fix a metric on $L$. There is $\varepsilon>0$ such that if $g\colon L\rightarrow K$ is a surjective, PL $\varepsilon$-map111A map is an $\varepsilon$-map if pre-images of points have diameters less than $\varepsilon$., then $K$ is homotopy equivalent to $L$ with $1$-cells and $2$-cells attached.

They had a specific complex $L$ in the paper and it is a spine of the Poincare homology sphere, and they remarked that if the conjecture is true for this 2-complex, then the Eilenberg-Ganea conjecture is false. In this paper we will take the point of view that Conjecture 1 is clearly more interesting.

While Conjecture 1 might sound “obviously true”, it remains open. If we reduce the dimension of $L$ from $2$ to $1$, then Conjecture 1, in its literal form, is true for a rather dull reason, which is that such $PL$-quotients of a circle by $\varepsilon$-maps are graphs that are not trees, and such graphs are always circles with 1-cells attached.

A more interesting version of Conjecture 1 in dimension $1$ is the following.

Question 2**.**

Is there $\varepsilon>0$ small enough such that if $f\colon\mathbb{S}^{1}\rightarrow G$ is a surjective, PL $\varepsilon$-map onto a graph $G$, then the loop given by $f$ splits as a generator in a free basis in $\pi_{1}(G)$?

The answer to this question should be “yes, of course”. However, a second glance at it might change the “yes, of course” to “maybe not”. Experience shows that one easily gets confused when trying to prove this. As usual, these confusions revolve around a base point change and the affirmative answer to Question 2 starts out being obvious and then it becomes not so obvious and then it becomes just wrong.

This note is to remark that the answer to Question 2 is no.

Theorem 3**.**

For each $\varepsilon>0$, there is a surjective $\varepsilon$-map $f\colon\mathbb{S}^{1}\rightarrow G$ to a graph $G$ such that $f$ is NOT a generator in a free basis in $\pi_{1}(G)$.

This sheds no light on Conjecture 1 but we find it rather surprising. Maybe the reason why this is surprising is because if $\varepsilon$ is small enough, then the loop $f$ is a primitive element in homology. We will leave the proof of this to the readers.

Acknowledgement**.**

I would like to thank Grigori Avramidi and Max Forester for interesting conversations. I learned the Whitehead algorithm from the second lecture of a three-lecture series by Henry Wilton at the Hausdorff Institute in Fall 2018. I wish the video of his second lecture was available. Finally, I am thankful to the Max Planck Institute for Mathematics for the perfect working environment.

2. Proof of Theorem 3

For each positive integer $k$, take the following word.

[TABLE]

The word $w_{k}$ is an element of the free group $F_{2k+1}$ on $(2k+1)$ generators $a_{1},a_{2},...,a_{2k}$ and $\gamma$. Realize $F_{2k+1}$ as the fundamental group of a wedge of $(2k+1)$ circles that are in one-to-one correspondence with $a_{1},a_{2},...,a_{2k},\gamma$. The Whitehead graph222The definition of the Whitehead graph is recalled in the Appendix. of $w$ is given in Figure 1,

and does not have a cut vertex333A vertex $v$ of a graph is a cut vertex if the graph consists of two subgraphs with a single vertex in common that is $v$.. It follows from the Whitehead algorithm ([2]) that $w_{k}$ is not a generator of $F_{2k+1}$. We recall the theorem by Whitehead.

Theorem 4** (Whitehead).**

Let $F_{n}$ be the free group with $n$ generators and let $w$ be a word in the generators. If $w$ is a generator in a free basis of $F_{n}$, then the Whitehead graph of $w$ has a cut vertex.

Next, we explain how to realize $w_{k}$ as a surjective map

[TABLE]

for a particular graph $G_{k}$, such that for a fixed $\varepsilon$, the map $f_{k}$ is an $\varepsilon$-map if $k$ is large enough.

The graph $G_{k}$ is defined as in Figure 2. The base point is $y_{1}$. The large middle loop, oriented clockwise, is $\gamma$. Each small (embedded) loop $a_{i}$ based at $y_{i}$ has length $\pi/(6k)$. To make them based at $y_{1}$, use the arc going clockwise from $y_{1}$ to $y_{i}$ for a change of base point. Each segment $y_{i}y_{i+1}$ also has length $\pi/(6k)$.

The map $f_{k}$ is constructed as follows. Divide $\mathbb{S}^{1}$ into $(6\times 2k)$ equal segments. As we go around $\mathbb{S}^{1}$ starting at the base point $x$, the segments are mapped to $G_{k}$ in the following order.

• .

Loop $a_{1}$

• .

Segment $y_{1}y_{2}$

• .

Loop $a_{2}$

• .

Segment $y_{2}y_{1}$

• .

Loop $a_{1}$

• .

Segment $y_{1}y_{2}$ (we are done with the segment $a_{1}a_{2}a_{1}$ of $w_{k}$)

• .

Loop $a_{2}$

• .

Segment $y_{2}y_{3}$

• .

Loop $a_{3}$

• .

Segment $y_{3}y_{2}$

• .

Loop $a_{2}$

• .

Segment $y_{2}y_{3}$ (we are done with the segment $a_{2}a_{3}a_{2}$ of $w_{k}$)

• …
• …
• .

Loop $a_{2k}$

• .

Segment $y_{2k}y_{1}$

• .

Loop $a_{1}$

• .

Segment $y_{1}y_{2k}$

• .

Loop $a_{2k}$

• .

Segment $y_{2k}y_{1}$ (we are done with $w_{k}$).

Remark**.**

The loop we have just defined has cyclic symmetries. However, the element of $\pi_{1}(G_{k})=F_{2k+1}$ does not look like it has cyclic symmetries. This is because when the loop $a_{1}$ occurs at the end of the journey, we have already basically gone around $\gamma$, so going around $a_{1}$ then means that $\gamma a_{1}\gamma^{-1}$, instead of $a_{1}$, should be added to the $w$.

Appendix: The Whitehead graph

Let $F_{n}$ be the free group on $n$ generators $a_{1},a_{2},...,a_{n}$. Let $w$ be word in the generators.

The Whitehead graph of $w$ has $2n$ vertices called $a_{1}^{+},a_{1}^{-},a_{2}^{+},a_{2}^{-},...,a_{n}^{+},a_{n}^{-}$. To draw the edges of the Whitehead graph, first we put “$+$” on the right and “$-$” on the left of each letter that appears in $w$ so that it looks like

[TABLE]

For each inverse element $a_{k}^{-1}$ that occurs in $w$, we change ${}^{-}(a^{-1}_{k})^{+}$ to ${}^{+}a_{k}^{-}$. Then the chain of symbols we created above becomes

[TABLE]

Each pairs of adjacent signs corresponds to an edge in the Whitehead graph. For example, with the above labeling, there is an edge from $a_{i}^{+}$ to $a_{j}^{-}$ and there is an edge from $a_{j}^{+}$ to $a_{k}^{+}$.

One thing one should not forget is that $w$ should be written as a circular word, so there is also an edge from $a_{\text{last}}^{+}$ to $a_{\text{first}}^{-}$. That is how to draw the Whitehead graph of $w$.

Remark**.**

The Whitehead graph of $w$ is obtained by first drawing a wedge if $n$ circles intersecting at a point $x_{0}$ and then drawing the loop given by $w$ in a different color (say, blue) and then cutting out a small neighborhood of $x_{0}$. The edges are the blue segments of $w$ in this neighborhood and the vertices are where the blue curves exit the neighborhood. This explains why the word should be treated as circular.