The Griffiths Twin Cone

This post is about an important wild space which, in many ways, is similar to the harmonic archipelago space that I posted about a few weeks ago. The Griffiths twin cone (or Griffiths space) was first studied by H.B. Griffiths in the 1950’s [1]. he showed that despite being the union of two contractible subspaces, this beast has a non-trivial – in fact, uncountable – fundamental group.

Constructing the Griffiths twin cone

For any space X, the cone over X is the space \displaystyle CX=\frac{X\times [0,1]}{X\times \{1\}}. The image of X\times \{1\} is the vertex v of the cone and the (homeomorphic) image of X\times\{0\} is the base of the cone. Every cone CX is contractible (to the vertex point of the cone) and consequently has a trivial fundamental group.

Now suppose X has basepoint x and \ast is the image of (x,0) in the base of the cone CX. Now join two copies of the cone together at a single point to get the “wedge” space TC(X,x)=(CX,\ast)\vee (CX,\ast). Let’s call this space the twin cone over X. We’ve joined two contractible spaces together so this twin cone must be contractible right? Well… not exactly. If we had formed the wedge (CX,v)\vee (CX,v) by adjoining two cones at their vertices, then yes, we’d get back a contractible space. But the cone does not necessarily contract onto \ast if X is “wild” at x. The Griffiths twin cone is an example of a non-contractible twin cone.

Definition: Let \mathbb{H}\subset\mathbb{R}^2 be the usual Hawaiian earring space with basepoint x=(0,0). The Griffiths twin cone, denoted \mathbb{G} is the twin cone TC(\mathbb{H},x)=(C\mathbb{H},\ast)\vee(C\mathbb{H},\ast) over the Hawaiian earring.

In other words, we construct \mathbb{G} by taking two copies of the cone of the Hawaiian earring and pasting them together at the wild points on the base.

Griffiths twin cone

Griffiths twin cone

Notice that the bases of the adjoined cones form the one-point union of two copies of the Hawaiian earring (\mathbb{H},x)\vee(\mathbb{H},x) which is clearly homeomorphic to \mathbb{H} itself. This means we can construct this space in a slightly different way that will help clarify the relationship between the fundamental groups of \mathbb{H} and \mathbb{G}.

Alternative construction: For each integer n\geq 1, let C_n\subset\mathbb{R}^2 be the circle of radius \frac{1}{n} centered at \left(\frac{1}{n},0\right). Now view these circles and their union \mathbb{H}=\bigcup_{n\geq 1}C_n in the xy-plane of \mathbb{R}^3.

  • Let v_1=(0,0,1) and v_0=(0,0,-1).
  • If n is odd, let A_n be the union of all line segments from C_n to v_1.
  • If n is even, let A_n be the union of all line segments from C_n to v_0.

In short, A_n is the cone of C_n in \mathbb{R}^3 with vertex v_{n\text{ mod}2}. The Griffiths twin cone is the union \mathbb{G}=\bigcup_{n\geq 1}A_n with basepoint x_0=(0,0,0).

Alternative construction of the Griffiths twin cone

Alternative construction of the Griffiths twin cone

Notice the intersection of \mathbb{G} and the xy-plane is now the standard construction of the Hawaiian earring. From now on, we’ll identify \mathbb{H} as a subspace of \mathbb{G} in this way.

Here are a few quick observations we can go ahead and make about the fundamental group \pi_1(\mathbb{G}):

  • Every subspace A_n is contractible since it is the cone of a circle (and thus homeomorphic to the unit disk). Consequently, if we let \ell_n:[0,1]\to C_n\subset\mathbb{G} be the circle which traverses C_n counterclockwise in the xy-plane, then \ell_n is homotopic (in \mathbb{G}) to the constant loop and [\ell_n] is the identity element 1\in\pi_1(\mathbb{G}).
  • The previous point implies that if F\subset\{1,2,...\} is any finite set of integers and A_F=\bigcup_{n\in F}A_n, then [\alpha]=1 for any loop \alpha:[0,1]\to\mathbb{G} with image in A_F.
  • Suppose a given (finite or infinite) subset S\subset \{1,2,...\} contains all even or all odd integers. Then A_{S}=\bigcup_{n\in S}A_n\cong C\mathbb{H} is homeomorphic to the cone on the Hawaiian earring and is therefore contractible. This means we have [\alpha]=1 for ANY based loop \alpha:[0,1]\to\mathbb{G} with image in A_S, even if it has image in infinitely many distinct cones.
  • What is less clear is what happens to the infinite concatenation \alpha=\ell_1\ell_2\ell_3\ell_4\cdots defined as \ell_n on \left[\frac{n-1}{n},\frac{n}{n+1}\right] and \alpha(1)=x_0. This loop winds around infinitely many even circles and infinitely many odd circles in an alternating fashion. Obviously, we can contract any finite number of these loops, showing that \alpha is homotopic to \ell_n\ell_{n+1}\ell_{n+1}\cdots for any n\geq 1. But even though we can deform \alpha arbitrarily close to the basepoint, it seems unlikely that this loop is homotopically trivial since we’d have to contract it all the way up (and down) to the two vertices infinitely many times.

The fundamental group of the Griffiths twin cone

To understand the relationship between \pi_1(\mathbb{H}) and \pi_1(\mathbb{G}) in elementary terms, we’ll use the following specialized case of the van Kampen theorem – one of the most useful computational results for fundamental groups.

van-Kampen Theorem (special case): Suppose X is the union of two path-connected open sets U_1,U_2 such that U_1\cap U_2 is path connected and contains the basepoint x_0. If U_1 is simply connected, then \pi_1(X)\cong \pi_1(U_2)/N where N is the conjugate closure of the image Im(\pi_1(U_1\cap U_2)\to \pi_1(U_2)) of the homomorphism induced by inclusion.

Let \displaystyle\mathbb{H}_{odd}=\bigcup_{n\text{ even}}C_n be the odd circle of the Hawaiian earring and \displaystyle\mathbb{H}_{even}=\bigcup_{n\text{ even}}C_n be the even circles. Both of these subspaces are still homeomorphic to \mathbb{H}. By including these as subspaces of \mathbb{H}, we get two important subgroups of \pi_1(\mathbb{H}). Let

G_{odd}=Im(k_{odd}:\pi_1(\mathbb{H}_{odd})\to\pi_1(\mathbb{H})) and G_{even}=Im(k_{even}:\pi_1(\mathbb{H}_{even})\to\pi_1(\mathbb{H}))

be the respective images of the group monomorphisms induced by inclusion.

Theorem 1: The inclusion i:\mathbb{H}\to\mathbb{G} induces a surjection \phi:\pi_1(\mathbb{H})\to\pi_1(\mathbb{G}) of fundamental groups. Moreover, \ker\phi is the conjugate closure of G_{odd}\cup G_{even}.

Proof. Define an open cover of \mathbb{G} as follows: Let U=\{(x,y,z)\in\mathbb{G}|-2/3<z<2/3\}, V_{odd}=\{(x,y,z)\in\mathbb{G}|z>1/3\} and V_{even}=\{(x,y,z)\in\mathbb{G}|z<-1/3\}.

Open cover of the Griffiths twin cone

Let’s make a few elementary observations about these open sets and their fundamental groups:

  • \mathbb{G}=V_{odd}\cup U\cup V_{even},
  • U, V_{odd}, and V_{even} are path connected,
  • V_{odd} and V_{even} are simply connected since each is a cone with deleted base and therefore contractible,
  • U\cong\mathbb{H}\times(-2/3,2/3) deformation retracts onto \mathbb{H}. Therefore, we can identify \pi_1(U)=\pi_1(\mathbb{H}),
  • V_{odd}\cap U\cong\mathbb{H}_{odd}\times (1/3,2/3) is path connected and deformation retracts onto \mathbb{H}_{odd}. This means the inclusion V_{odd}\cap U\to U induces the canonical homomorphism k_{odd}:\pi_1(\mathbb{H}_{odd})\to\pi_1(\mathbb{H}).
  • V_{even}\cap U\cong\mathbb{H}_{even}\times (-1/3,-2/3) is path connected and deformation retracts onto \mathbb{H}_{even}. This means the inclusion V_{even}\cap U\to U induces the canonical homomorphism k_{even}:\pi_1(\mathbb{H}_{even})\to\pi_1(\mathbb{H}).

Now, we’re ready to apply the special case of van-Kampen Theorem – first to X=V_{odd}\cup U and then to \mathbb{G}=X\cup V_{even}.

First application: Since V_{odd} is simply connected, the van-Kampen Theorem implies that \pi_1(X)=\pi_1(V_{odd}\cup U)\cong\pi_1(\mathbb{H})/K_1 where K_1, i.e. the conjugate closure of the image G_{odd}=Im(\pi_1(\mathbb{H}_{odd})\to\pi_1(\mathbb{H})) of the homomorphism on fundamental groups induced by the inclusion V_{odd}\cap U\to U.

Second application: Notice V_{even}\cap X=V_{even}\cap U. Since V_{even} is simply connected, the van-Kampen Theorem implies that \pi_1(\mathbb{G})=\pi_1(V_{even}\cup X)\cong\pi_1(X)/K_2 where K_2 is the conjugate closure of the image of the homomorphism \pi_1(\mathbb{H}_{even})\to\pi_1(X) induced by the inclusion V_{even}\cap U=V_{even}\cap X\to X.

Altogether, we see that both inclusions

i:\mathbb{H}\to X\to\mathbb{G}

induce a surjection of fundamental groups

\phi:\pi_1(\mathbb{H})\to\pi_1(X)\to\pi_1(\mathbb{G})

This proves the first statement of the theorem – that \phi is surjective.

Let N be the conjugate closure of G_{odd}\cup G_{even}. We need to show N=\ker\phi. By our observations above, any loop with image entirely in \mathbb{H}_{odd} or \mathbb{H}_{even} is null-homotopic in \mathbb{G}. Since \phi is induced by inclusion, \phi(G_{odd})=\phi(G_{even})=1 is trivial. This means that N\subseteq\ker\phi.

For the other inclusion, let a:\pi_1(\mathbb{H})\to\pi_1(X) and b:\pi_1(X)\to\pi_1(\mathbb{H}) be the surjections induced by inclusion that compose to give \phi. By our two applications of the van-Kampen theorem, \ker(a)=K_1 and \ker(b)=K_2. Thus \ker\phi=a^{-1}(K_2). Recall that K_2 is the conjugate closure of Im(\pi_1(\mathbb{H}_{even})\to\pi_1(X))=a(G_{even}). Therefore if g\in\ker\phi, then a(g) is of the form

a(\ell_{1}^{-1}k_1\ell_1\ell_{2}^{-1}k_2\ell_2...\ell_{n}^{-1}k_n\ell_n)=[h_{1}^{-1}a(k_1)h_{1}][h_{2}^{-1}a(k_2)h_{2}]...[h_{n}^{-1}a(k_n)h_{n}]

where k_j\in G_{even} and a(\ell_j)=h_j. Notice k=\ell_{1}^{-1}k_1\ell_1\ell_{2}^{-1}k_2\ell_2...\ell_{n}^{-1}k_n\ell_n lies in the conjugate closure of G_{even}. Thus gk^{-1}\in\ker(a)=K_1 which is the conjugate closure of G_{odd}. Thus g is an element of the coset K_1k\subset N. This concludes the proof that \ker\phi=N. \square

For each m\geq 1, we have a smaller copy \mathbb{H}_{\geq m}=\bigcup_{n\geq m}C_n of the Hawaiian earring that we can view as a subspace of \mathbb{G}. The next Corollary basically says the we can continuously deform any loop as close as we want to the basepoint.

Corollary 2: Every based loop \alpha:[0,1]\to\mathbb{G} is homotopic in \mathbb{G} to a based loop \beta:[0,1]\to\mathbb{H}_{\geq m} for every m\geq 1.

Proof. Fix m\geq 1. According to Lemma 1, \alpha is homotopic to a based loop \gamma:[0,1]\to\mathbb{H}. According to our study of the Hawaiian earring group (in the original post and also in Lemma 5 of this post) \gamma is homotopic to a finite concatenation of loops each of which has image in either \mathbb{H}_{\geq m} or \bigcup_{1\leq k<m}C_k. But we observed above that any loop in \mathbb{G} which has image in \bigcup_{1\leq k<m}C_k is homotopically trivial. Therefore \gamma is homotopic in \mathbb{G} to a loop in \mathbb{H}_{\geq m}. \square

Another quick observation about \pi_1(\mathbb{G})

I’ve already written a few posts about different properties of the Hawaiian earring group \pi_1(\mathbb{H}). Let’s use one of these oldies-but-goodies to prove something interesting about \pi_1(\mathbb{G}).

Theorem 3: Hom(\pi_1(\mathbb{G}),\mathbb{Z})=0. Consequently, \pi_1(\mathbb{G}) cannot be a free group.

Proof. In Lemma 6 of this post, we decided that every homomorphism \pi_1(\mathbb{H})\to\mathbb{Z} that sends [\ell_n] to 0 for all n\geq 1, must be the trivial homomorphism. But notice that \phi([\ell_n])=1 for all n\geq 1. So if f:\pi_1(\mathbb{G})\to\mathbb{Z} is any homomorphism, then f\circ\phi:\pi_1(\mathbb{H})\to\mathbb{Z} must be trivial. Since \phi is surjective, f must be trivial.

Honestly, it can – for me – be tempting to think of \pi_1(\mathbb{G}) as the quotient of \pi_1(\mathbb{H}) by the conjugate closure of the countable subset \{[\ell_n]\in\pi_1(\mathbb{H})|n\geq 1\}\subset G_{odd}\cup G_{even}, however, this quotient is “too big” since, to get \pi_1(\mathbb{G}), we must also kill the uncountable subgroups G_{odd}=Im(\pi_1(\mathbb{H}_{odd})\to\pi_1(\mathbb{H})) and G_{even}=Im(\pi_1(\mathbb{H}_{even})\to\pi_1(\mathbb{H})). This, in my opinion, makes dealing with \pi_1(\mathbb{G}) a little more complicated than it is to deal with the Harmonic archipelago group \pi_1(\mathbb{HA}). For instance, if G is any non-trivial finite group, there are uncountably many homomorphisms f:\pi_1(\mathbb{H})\to G such that f([\ell_n])=1 for all n\geq 1, however it does not follow immediately – as it did for \pi_1(\mathbb{HA}) – that Hom(\pi_1(\mathbb{G}),G) is uncountable (though I have heard by word of mouth that this is true).

 

References.

[1] H.B. Griffiths, The fundamental group of two spaces with a common point, Quart. J. Math. Oxford (2) 5 (1954) 175-190.

[2] K. Eda, A locally simply connected space and fundamental groups of one point unions of cones, Proc. Amerc. Math. Soc. 116 no. 1 (1992) 239-249.

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One Response to The Griffiths Twin Cone

  1. Pingback: Homotopically Hausdorff Spaces I | Wild Topology

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