# Are the higher homotopy groups of the Hawaiian earring trivial?

The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not have a universal cover, the standard approach to showing that higher homotopy groups of graphs vanish does not apply.

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Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the Hawaiian earring. It is actually quite useful for understanding the structure of the fundamental group as a subgroup of the inverse limit $\varprojlim F_n$ of free groups. The "generalized universal covering" still provides all the same unique path lifting properties for maps out of locally path connected spaces that one might want. The study of these particular coverings for wild spaces is mainly due to Hanspeter Fischer and Andreas Zastrow. I think you should be able to use this to mimic the argument for graphs.

Let $\mathbb{H}$ be the Hawaiian earring and $\widetilde{\mathbb{H}}$ be the set of homotopy classes of paths starting at the basepoint. Take $p:\widetilde{\mathbb{H}}\to \mathbb{H}$, $p([\alpha])=\alpha(1)$ to be the endpoint projection and give $\widetilde{\mathbb{H}}$ the usual "whisker" topology that you do to construct universal covers: the basic neighborhoods are $B([\alpha],U)=\{[\alpha\cdot\epsilon]|\epsilon([0,1])\subset U\}$ where $U$ is open in $\mathbb{H}$. It turns out that $p:\widetilde{\mathbb{H}}\to \mathbb{H}$ is an open map which has unique lifting of all paths and homotopies of paths. Moreover, if $Y$ is path connected, locally path connected, and simply connected, then any based map $f:Y\to \mathbb{H}$ has a unique lift $\tilde{f}:Y\to \widetilde{\mathbb{H}}$ such that $p\tilde{f}=f$. Consequently, $p$ induces isomorphisms on higher homotopy groups.

It also turns out that $\widetilde{\mathbb{H}}$ has the structure of an $\mathbb{R}$-tree (uniquely arcwise connected metric space where each arc is isometric to a sub-arc of the reals), and these are known to be contractible (this is due to J. Morgan I believe).

Now you should have all the usual ingredients to mimic the usual argument for graphs. In fact, the Hawaiian earring is not special here. This should work out for all 1-dimensional Peano continua.

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I guess I never appreciated the importance of the condition "locally simply connected" in the construction of the universal cover. – Lior Silberman Apr 21 '14 at 0:48
@LiorSilberman : $\:$ I thought that standard facts about covering spaces only $\hspace{1.67 in}$ needed "semi-locally simply connected". $\;\;\;\;$ – Ricky Demer Apr 21 '14 at 3:00

I did not read this paper, but it claims that theanswer is yes: One-dimensional sets and planar sets are aspherical by J.W. Cannon, G.R. Conner and Andreas Zastrow.

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Also available publicly via an author's website – Noah Snyder Apr 20 '14 at 1:46
It is actually available publicly from the publisher's website. – Alexey Muranov Apr 20 '14 at 11:02

Here is a short natural argument that planar continua are aspherical, different from the technique of Cannon/Conner/Zastrow, and straight forwardly applied to the Hawaiian earring.

The Hawaiian earring (and more generally any planar compactum $X$) is the nested intersection of planar polyhedra $X_n$.

The planar Euclidean metric naturally induces a length structure on $X_n$ so that $X_n$ is locally CAT(0).

In particular there is a natural proof $X_n$ is aspherical, since inessential loops in $X_n$ can be canonically extended to maps of the disk, and since the m-ball $B_{m}$ is naturally fibred by disks, so that the boundaries of the disks fibre the m-sphere $S_{m}$.

To see $X$ is aspherical take a map $f: S_{m} \rightarrow X$, and obtain the natural extensions $F_{n}: B_{m} \rightarrow X_{n}$.

Ascoli's theorem ensures the existence of subsequential limit and hence $X$ is aspherical.

In Cannon/Conner/Zastrow, inessential loops in $X$ are contracted internally, within their own image. In contrast, in the argument at hand, roughly speaking, inessential loops in $X$ bound disks fibred by limits of external geodesic chords.

To apply Ascoli's theorem we must gain global control of the equicontinuity data.

It is useful to show there exists a universal constant K>0 so that if P is a planar polyhedron, if $f: [0,1] \rightarrow P$ is a path, if C is the local geodesic path homotopic to f, (and if g is the parameterization of C induced by projection of f), then if (d,e) is uniform continuity data for f with respect to absolute value, then (d,Ke) is uniform continuity data for g (also with respect to absolute value). K=1.5 is apparently the worst case.

However, in the special case of the Hawaiian earring we may use K=1, since $X_n$ is the union of a finite bouquet of loops and a convex disk.

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Nice argument; so you claim that in general planar continua (i.e., not for Hawaiian earring) there is a gap; can you fix it? – Anton Petrunin Apr 21 '14 at 0:18
There is a genuine gap since K=1.5 can actually happen. Place the letter V in the upper half plane with vertex at (0,0) and angles 60 degrees with w.r.t. x axis. Now place a different V with origin vertex, with angles 30 degrees and very long edges. Now project orthogonally the first V into the 2nd V while fixing the origin. This is purportedly the worst case, yielding a kind of poor man's Gehring-Hayman Theorem. – Paul Fabel Apr 21 '14 at 4:37
It is more straightforward to see K=3 is a universal upper bound. Let $Q$ denote the universal cover of $P$. Note $Q$ is CAT(0). Take any two geodesics segments A and B in $Q$. In $Q$, naturally project A into B. Call the image C. Now map A and C naturally into the plane and compare the ratios of the their diameters, using absolute value. – Paul Fabel Apr 21 '14 at 14:19
If A and C are disjoint, K =1 suffices, and this is provable by induction on the number of segments comprising A and B. If A and C are not disjoint, let D denote the intersection. Let A=A1*DA2 and C=C1*DC2. The diameter of the planar projection of Ai is at most that of Ci. Hence K=3 is an upper bound on the ratio of diameters of the planar projections of A and C. – Paul Fabel Apr 21 '14 at 14:29