I think I'll extend Zack's thread a bit with a slightly different point of view:

We can understand X as a CW complex with one 0-cell, one 1-cell and four 2-cells, two of which are attached to via the identity map and two of which are attached via maps of degree 2.

(1/7) https://twitter.com/dzackgarza/status/1265096596888444930

How does this help us to compute pi_1(X)?

Intuitively, we can think of the fundamental group of a (connected) two-dimensional CW complex as the fundamental group of its 1-skeleton, with certain loops killed:

(2/7)

Namely, attaching a 2-cell allows you to nullhomotope all loops in the image of its attaching map through the interior. Furthermore, since the interior of the cell is contractible, any loop in the cell interior is nullhomotopic.

(3/7)

...which is to say that the cell itself doesn't contribute any new generators to the fundamental group; it only serves to quotient out a certain subgroup.

My account here is obviously not super rigorous, but it can be made so; Hatcher, for instance, has a proof.

(4/7)

So let's apply this reasoning to pi_1(X): its one skeleton is just S^1, so it has fundamental group isomorphic to Z. But now we attach two 2-cells via the identity, which kill _every_ loop. The other two 2-cells we attach don't matter, since they would kill fewer elements!

(5/7)

Hence pi_1(X) = 0.

But we can also use this consideration in the opposite direction, to construct spaces with arbitrary pi_1! Choose a presentation of a group G with n generators (possibly infinitely many), take X^1 = (S^1)^n and then glue a 2-cell for each relation.

(6/7)

Finally, we can also use the CW-structure on our original X to easily compute homology, without resorting to any fancy techniques: cellular homology allows a direct computation from knowing only the number of cells and their attaching maps!

(7/7)

I'll tweet about how to do that when Zack posts his homology computation.