Attaching a 2-cell to create a trivial fundamental group

Question

I'm attempting a past exam paper and I'm stuck on a question. We're provided with K, a cell-complex, with a single 0-cell b, two 1-cells labelled x and y and a 2-cell attached via the word $x^2$$y^5$.

The question is: "show how to attach a single 2-cell to K to create a cell complex L with trivial fundamental group".

I know if I can make K contractible then it will have trivial fundamental group, but I'm not sure if that line of thinking helps. I also know that the fundamental group of $S^n$ is trivial for n≥2, so I wondered if I attached an extra 2-cell along the word y then there would be 3 'y-spheres' and 1 'x-sphere' on either side of b that contract to just leave b - but I've no idea if that type of thinking makes sense as it's hard to visualise!

I'm also not sure if I'm meant to think about this algebraically or visually. I suspect the latter as the next question asks us to transform the presentation of the fundamental group of L to just have one generator and one relation using Tietze transformations.

In case it's helpful, the previous part of the question got us to show that the fundamental group of K is isomorphic to a push-out of cyclic groups. (As the fundamental group is < x , y |$x^2$$y^5$> I think it is isomorphic to the push-out of Z<-Z->Z with the x2, x5 homomorphisms).

Any help would be much appreciated! Thank you.

Answer 1

Currently, your space has a fundamental group which is presented by $\langle x,y \mid x^2y^5\rangle$ (to prove this, use Van Kampen)

Adding a two-cell along some map $S^1\to K$ kills the corresponding element in $\pi_1(K)$; but if you kill some element in $\pi_1(K)$, all its conjugates die too, and everything that they generate as well.

So you need to find some map $S^1\to K$ such that the conjugates of the corresponding element generate the whole group $\pi_1(K)$.

Now the algebra comes along : you need to find $w\in \langle x,y\mid x^2y^5\rangle$ such that the conjugates of $w$ generate the whole group. This essentially amounts to saying that $\langle x,y\mid x^2y^5, w\rangle = 0$.

Can you try to find such a $w$ ?