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 , the cone over is the space . The image of is the vertex of the cone and the (homeomorphic) image of is the base of the cone. Every cone is contractible (to the vertex point of the cone) and consequently has a trivial fundamental group.

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

**Definition:** Let be the usual Hawaiian earring space with basepoint . The *Griffiths twin cone*, denoted is the twin cone over the Hawaiian earring.

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

Notice that the bases of the adjoined cones form the one-point union of two copies of the Hawaiian earring which is clearly homeomorphic to itself. This means we can construct this space in a slightly different way that will help clarify the relationship between the fundamental groups of and .

**Alternative construction:** For each integer , let be the circle of radius centered at . Now view these circles and their union in the xy-plane of .

- Let and .
- If is odd, let be the union of all line segments from to .
- If is even, let be the union of all line segments from to .

In short, is the cone of in with vertex . The Griffiths twin cone is the union with basepoint .

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

Here are a few quick observations we can go ahead and make about the fundamental group :

- Every subspace is contractible since it is the cone of a circle (and thus homeomorphic to the unit disk). Consequently, if we let be the circle which traverses counterclockwise in the xy-plane, then is homotopic (in ) to the constant loop and is the identity element .
- The previous point implies that if is any finite set of integers and , then for any loop with image in .
- Suppose a given (finite or infinite) subset contains all even or all odd integers. Then is homeomorphic to the cone on the Hawaiian earring and is therefore contractible. This means we have for ANY based loop with image in , even if it has image in infinitely many distinct cones.
- What is less clear is what happens to the infinite concatenation defined as on and . 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 is homotopic to for any . But even though we can deform 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 and 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 is the union of two path-connected open sets such that is path connected and contains the basepoint . If is simply connected, then where is the conjugate closure of the image of the homomorphism induced by inclusion.

Let be the odd circle of the Hawaiian earring and be the even circles. Both of these subspaces are still homeomorphic to . By including these as subspaces of , we get two important subgroups of . Let

and

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

**Theorem**** 1:** The inclusion induces a surjection of fundamental groups. Moreover, is the conjugate closure of .

*Proof. *Define an open cover of as follows: Let , and .

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

- ,
- , , and are path connected,
- and are simply connected since each is a cone with deleted base and therefore contractible,
- deformation retracts onto . Therefore, we can identify ,
- is path connected and deformation retracts onto . This means the inclusion induces the canonical homomorphism .
- is path connected and deformation retracts onto . This means the inclusion induces the canonical homomorphism .

Now, we’re ready to apply the special case of van-Kampen Theorem – first to and then to .

*First application:* Since is simply connected, the van-Kampen Theorem implies that where , i.e. the conjugate closure of the image of the homomorphism on fundamental groups induced by the inclusion .

*Second application:* Notice . Since is simply connected, the van-Kampen Theorem implies that where is the conjugate closure of the image of the homomorphism induced by the inclusion .

Altogether, we see that both inclusions

induce a surjection of fundamental groups

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

Let be the conjugate closure of . We need to show . By our observations above, any loop with image entirely in or is null-homotopic in . Since is induced by inclusion, is trivial. This means that .

For the other inclusion, let and be the surjections induced by inclusion that compose to give . By our two applications of the van-Kampen theorem, and . Thus . Recall that is the conjugate closure of . Therefore if , then is of the form

where and . Notice lies in the conjugate closure of . Thus which is the conjugate closure of . Thus is an element of the coset . This concludes the proof that .

For each , we have a smaller copy of the Hawaiian earring that we can view as a subspace of . 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 is homotopic in to a based loop for every .

*Proof.* Fix . According to Lemma 1, is homotopic to a based loop . According to our study of the Hawaiian earring group (in the original post and also in Lemma 5 of this post) is homotopic to a finite concatenation of loops each of which has image in either or . But we observed above that any loop in which has image in is homotopically trivial. Therefore is homotopic in to a loop in .

**Another quick observation about **

I’ve already written a few posts about different properties of the Hawaiian earring group . Let’s use one of these oldies-but-goodies to prove something interesting about .

**Theorem 3:** . Consequently, cannot be a free group.

*Proof.* In Lemma 6 of this post, we decided that every homomorphism that sends to for all , must be the trivial homomorphism. But notice that for all . So if is any homomorphism, then must be trivial. Since is surjective, must be trivial.

Honestly, it can – for me – be tempting to think of as the quotient of by the conjugate closure of the countable subset , however, this quotient is “too big” since, to get , we must also kill the uncountable subgroups and . This, in my opinion, makes dealing with a little more complicated than it is to deal with the Harmonic archipelago group . For instance, if is any non-trivial finite group, there are uncountably many homomorphisms such that for all , however it does not follow immediately – as it did for – that 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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