Problem List

Here is my list of open problems in wild topology. Some of these problems are very hard and fairly well-known. Some of them are less well known but still hard and some are probably answerable with some effort. This list does not speak for “the field” or really anyone but myself. However, if you have an interesting problem that you think should be on here, feel free to send it (with some context) to me. For example, I’m confident that I’m leaving some relevant homology and geometric group theory problems off of this list. I am happy to receive any corrections or updates as well.

Fundamental Groups and First Singular Homology

* $\mathbb{R}^3$ Torsion Problem: Suppose $X\subset\mathbb{R}^3$ and $x_0\in X$ .

Is $\pi_{1}(X,x_0)$ torsion free? A stronger question would be: is $\pi_{1}(X,x_0)$ residually locally free, i.e. a subgroup of a product of locally free groups?
Is $H_1(X)$ torsion free?

*Problem (Eda, 1992): Is $H_1\left(\prod_{n=1}^{\infty}S^1\vee S^1\right)\cong Ab((F_{2})^{\mathbb{N}})$ torsion free?

Katsuya Eda asked the question in his 1992 paper Free sigma-products and noncommutatively slender groups and reminded us of it at the 2018 Arches Topology Conference. It was also asked later by other group theorists.

*Archipelago Group Problem (Conner-Hojka-Meilstrup): Consider the “archipelago group” $\mathcal{A}(G)$ of a non-trivial group $G$ constructed in

Conner, Hojka, Meilstrup, Archipelago groups, PAMS 143 no. 11 (2015) 4973-4988.

Are the isomorphism types groups $\mathcal{A}(G)$ independent of $G$ ?

*One-dimensional Surjectivity Problem: If $X$ is a Peano continuum, is there a one-dimensional metric space $Y$ (ideally a Peano continuum) and a map $f:Y\to X$ which induces a surjection $f_{\#}:\pi_1(Y,y)\to \pi_1(X,f(y))$ on fundamental groups?

I have asked several people about the above problem. Everyone thinks it’s true but no one seems to know how to prove it. I think having a positive answer would be very useful considering fundamental groups of 1-dimensional spaces have a usable calculus.

*One-dimensional Embedding Classification: Given a one-dimensional Peano continuum $X$ , characterize those one-dimensional Peano continua $Y$ for which $\pi_1(Y,y_0)$ embeds as a subgroup of $\pi_1(X,x_0)$ .

*One-point compactifications of infinite-type surfaces: suppose $X$ is an infinite type surface and $X^{\ast}=X\cup\{x_0\}$ is its one-point compactification. Does $\pi_1(X,x_0)$ canonically inject into the first Cech homotopy group $\check{\pi}_1(X,x_0)$ ? Is there a characterization of $\pi_1(X,x_0)$ as a quotient of the earring group?

*Fundamental Groups of Free Topological Algebraic Objects: If $X$ is a path-connected, locally path-connected, metric space with basepoint $e\in X$ . Let $H_{e}(X)$ denote the infinitary abelianization of $\pi_1(X,e)$ at $e$ as defined by Brazas-Gillespie. Consider the following spaces:

the James reduced product $J(X)$ ,
the infinite symmetric product $SP(X)$ ,
the path-component of the constant loop in the loop space $\Omega(\Sigma X)$ .

Do all of the above spaces have fundamental group isomorphic to $H_{e}(X)$ ?

The Infinite Earring $\mathbb{E}$

Let $\mathbb{E}_{\geq n}$ denote the smaller copies of the earring that consist of all circles with index $\geq n$ , i.e. radius $\leq\frac{1}{n}$ . Since this is a retract of $\mathbb{E}$ , we may identify $\pi_1(\mathbb{E}_{\geq n},b_0)$ as a subgroup of $\pi_1(\mathbb{E},b_0)$ .

Characterize n-slender Groups (Eda): A group $G$ is non-commutatively slender (or n-slender) if for every homomorphism $f:\pi_1(\mathbb{E},b_0)\to G$ , there exists a $n\in\mathbb{N}$ such that $f(\pi_1(\mathbb{E}_{\geq n},b_0))=\{1\}$ . Characterize n-slender groups.

*Pseudo-Earring Conjecture (Conner): If $G=f(\pi_1(\mathbb{E},b_0))$ is the uncountable image of an endomorphism $f:\pi_1(\mathbb{E},b_0)\to \pi_1(\mathbb{E},b_0)$ of the earring group, then $G$ is isomorphic to $\pi_1(\mathbb{E},b_0)$ .

Let $F\leq \pi_1(\mathbb{E},b_0)$ be the free subgroup generated by the homotopy classes of the loops $\ell_n$ , $n\in\mathbb{N}$ traversing the individual circles of $\mathbb{E}$ .

*Transfinite Closure of Pseudo-Earring Groups (Brazas-Fischer): Suppose $f:\mathbb{E}\to\mathbb{E}$ is a map and $H=f_{\#}(\pi_1(\mathbb{E},b_0))$ is the image of the induced homomorphism. If $g:\mathbb{E}\to\mathbb{E}$ is a map such that $g_{\#}(F)\leq H$ , then must we have $g_{\#}(\pi_1(\mathbb{E},b_0)\leq H$ .

*Scattered-Dense Gap Problem (Brazas-Fischer): Let $N$ be the smallest normal subgroup of $\pi_1(\mathbb{E},b_0)$ that contains the set $\{[\ell_{2n-1}\cdot\ell_{2n}^{-}]\mid n\in\mathbb{N}\}$ and is closed under transfinite products. Must $N$ contain the homotopy class of the loop shown below?

The Scattered-Dense Gap Problem is equivalent to the problem in my paper Scattered products in fundamental groupoids asking “Must every space with well-defined scattered $\pi_1$ -products have well-defined transfinite $\pi_1$ -products?

*Closures of Countable Subgroups Problem: Consider the $\pi_1$ -subgroup closure operators defined in the paper “Test-map characterizations of locally properties of fundamental groups.” Suppose $X$ is any space and $H\leq\pi_1(X,x_0)$ is a countable index $(C,c_{\infty})$ -closed subgroup. Must $H$ be closed according to other closure operators, e.g. must $H$ be $(C,c_{\tau})$ -closed, $(P,p_{\tau})$ -closed, $(W,w_{\infty})$ -closed, or $(D,d_{\infty})$ -closed?

Generalized covering maps and unique lifting

*Dydak’s Unique Path-Lifting Problem: Suppose $D^2$ is the closed unit disk and $p:E\to D^2$ is a map such that $E$ is path connected and locally path connected and all paths lift uniquely, i.e. for every path $\alpha:[0,1] \to D^2$ and point $e\in p^{-1}(\alpha(0))$ , there is a unique path $\widetilde{\alpha}:[0,1]\to E$ such that $\widetilde{\alpha}(0)=e$ and $p\circ \widetilde{\alpha}=\alpha$ . Must $p$ be a homeomorphism?

Note on Dydak’s Problem: I know of many related and partial results; however, Dydak’s Problem is a fundamental and general topological problem, which has broad consequences related to lifting properties. A positive answer essentially implies that unique lifting of paths alone implies all other unique lifting properties in covering space theory.

Overlays were introduced by R.H. Fox in the 1970’s as a refinement of the definition of covering map: an overlay is a covering map $p:X\to Y$ such that there is an open cover $\mathscr{U}$ of $Y$ consisting of neighborhoods evenly covered by $p$ such that if $U,V\in\mathscr{U}$ are intersecting neighborhoods, and $p^{-1}(U)=\coprod_{\alpha}A_{\alpha}$ and $p^{-1}(V)=\coprod_{\beta}B_{\beta}$ , then for every $\alpha$ there is a unique $\beta$ such that $A_{\alpha}\cap B_{\beta}\neq \emptyset$ .

Topological Group Overlay Problem (Eda): Let $G$ be a connected topological group and $p:\widetilde{G}\to G$ be an overlay. Does $\widetilde{G}$ inherit the structure of a topological group so that $p$ is a group homomorphism?

The following problems are related to generalized covering maps. An excellent paper on the basics of generalized universal covering spaces is

H. Fischer, A. Zastrow, Generalized universal covering spaces and the shape group, Fund. Math. 197 (2007) 167-196.

*Generalized Covering Space Problem for Peano Continua (Eda): Let $X$ be a Peano continuum. If $X$ is homotopically Hausdorff, does $X$ admit a generalized universal covering space?

*Fischer’s Problem: If $X$ is a planar set and $\widetilde{X}$ is the generalized universal covering space, does $\widetilde{X}$ admit a metric giving it the structure of a CAT(0) space?

Lifting Projection Problem: Give an example of a path-connected, locally path-connected metric space $X$ and a fibration $p:E\to X$ with the unique path-lifting property (sometimes called a lifting projection), which is not an inverse limit of covering maps.

*Missing Ends: Is there a suitable theory of ends for generalized covering spaces?

Higher Homotopy

*Higher Homotopy Groups of the n-dimensional earring: Characterize $\pi_m(\mathbb{E}_n)$ , $m>n$ in terms of the higher homotopy groups of $S^n$ .

The reality is that very little is known about these groups. The next problem is a special case of interest – the smallest unknown case. Although we may have an idea of what it should be, the tools to complete the proof are still missing.

*Problem: Characterize $\pi_3(\mathbb{E}_2)$ .

*Problem (Eda-Karimov-Repovs): Does there exist a finite-dimensional non-contractible Peano continuum for which all homotopy groups are trivial?

Problem (Eda-Karimov-Repovs): Let $\mathbb{W}_T$ be the shrinking wedge of tori (replace each $S^1$ in $\mathbb{E}$ with a torus). Is $H_3(\mathbb{W}_T)$ trivial?

Note: it is known that $H_2(\mathbb{W}_T)\cong \bigoplus_{\mathbb{N}}\mathbb{Z}$ .

Topological homotopy groups

The n-th quasitopological homotopy group $\pi_{n}^{qtop}(X,x_0)$ is the usual homotopy group endowed with the quotient topology with respect to the map $\pi:\Omega^n(X,x_0)\to\pi_{n}(X,x_0)$ sending a based map $S^n\to X$ to its homotopy class.

*Problem: Characterize the locally path-connected spaces $X$ with a single wild point $x_0$ for which $\pi_{1}^{qtop}(X,x_0)$ is a topological group.

*Problem: Find Peano continua $X,Y$ with isomorphic fundamental groups whose quasitopological fundamental groups are both non-discrete and non-isomorphic.

Based on Eda’s homotopy classification of one-dimensional spaces, at least one of the spaces in an answer to the previous problem must have dimension at least 2.

*Problem (Brazas-Fabel): If $X$ is a compact metric space and $\pi_{1}^{qtop}(X,x_0)$ satisfies the $T_1$ separation axiom, must $\pi_{1}^{qtop}(X,x_0)$ also satisfy the $T_4$ separation axiom?

*Problem: Are there one-dimensional conditions that would ensure $\pi_{n}^{qtop}(X,x_0)$ is a topological group for $n\geq 2$ ? For example, if $X$ is 1-connected and locally 1-connected, must $\pi_{n}^{qtop}(X,x_0)$ , $n\geq 2$ be a topological group?

Prizes:

I’m going to hide this down here for those dedicated enough to read the whole list.

For any person or group who provides/publishes a verifiable solution to a problem with a * next to it, I will happily buy a pizza and round of beer (or other beverage of preference) as a friendly gesture of appreciation and admiration. Since I am neither omnipresent nor looking for ways to give away pizza, you must contact me about your solution to redeem your refreshment prize. Pizzas redeemed in Philadelphia may be replaced with a Cheesesteak and soft pretzel. If I cannot join you in person, I will find a way to get you your prize.

Updates and Solved Problems:

Some of the problems in this list have been solved after being posted here. When a solution to a listed problem is published, I’ll move the problem down here and link to the solution.

*Archipelago-Twin Cone Problem: Are the fundamental groups of the harmonic archipelago and the Griffiths twin cone isomorphic?

Update: Sam Corson has posted a preprint proving that these two groups are isomorphic. This was a starred problem so once the solution is refereed and published, I will offer Sam a delicious prize.

Wild Fundamental Group Actions Problem: Consider the one-point union $\mathbb{E}\vee\mathbb{E}_n$ for $n\geq 2$ .

Characterize $\pi_n(\mathbb{E}\vee\mathbb{E}_n)$ .
Describe the action of $\pi_1(\mathbb{E}\vee\mathbb{E}_n)\approx \pi_1(\mathbb{E})$ on $\pi_n(\mathbb{E}\vee\mathbb{E}_n)$ . Does the image of $\pi_n(\mathbb{E}_n)$ under the action generate $\pi_n(\mathbb{E}\vee\mathbb{E}_n)$ ?

Update: This problem was solved (by me!) in 2021 in this paper. The group $\pi_n(\mathbb{E}\vee\mathbb{E}_n)$ naturally embeds into $\mathbb{Z}^{\mathbb{N}\times\pi_1(\mathbb{E})}$ . The characterization uses generalized universal covering maps and the Whitney covering lemma. The image of $\pi_n(\mathbb{E}_n)$ under the $\pi_1$ -action does NOT generate $\pi_n(\mathbb{E}\vee\mathbb{E}_n)$ .

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