Hey all! Today’s post is written by an awesome genius friend of mine, Nina Anikeeva. Without further ado…
We discuss what happens in terms of fundamental groups when we combine two spaces together in various ways. Unless otherwise stated, we will assume that our spaces are path-connected.
The Fundamental Group of a product
Let’s start with an example. Consider the two-dimensional torus What might its fundamental group be? We know that is generated by a loop going around the circle, so is probably generated by a loop going around the equator and a loop going around a meridian. Are there any relations between these loops? It’s not hard to see that . One can also notice this by considering the torus as a quotient of a square. So it seems like
One might guess that the fundamental group of the -torus is
generated by loops going around each factor. We show that this pattern holds in general:
The intuition is that the two factors don’t really interact with each other. This is the essence of the proof; to make it rigorous, recall that the product topology has the property that if one writes a continuous function as , then is continuous iff both and are. In particular, loop at is the same as a pair of loops at and .
A homotopy of loops at is also therefore the same as a homotopy of loops (in ) at and (in ) . Thus, we consider the map given by . The above discussion tells us that this is a bijection. Obviously, this is a group homomorphism, and, consequently, and isomorphism.
What else can we do with circles? The other obvious thing to try is to glue them together at a point. This is the wedge sum As before, the obvious guess is that is generated by a loop going around the first circle and a loop going around the second. But this time, there shouldn’t be any relations between and . If denotes the free group on letters, then our guess is
One might guess that
If and are spaces in general with chosen basepoints and , respectively, then one can define to be the quotient space of the disjoint union by identifying the basepoints and . More generally, it seems plausible that consists of all strings of paths in and with no relations between the paths in and the paths in : form something by concatenating and without any relations. To be formal, the free product of two groups and is presented by
So it seems likely that
Here’s another example. Start with the Möbius band . The boundary of is a circle, so we can glue a disc to it. Call the result . What does look like? Well, is homotopy equivalent to , so . We also know that . So it seems like we can get by taking and destroying some stuff. In particular, the loops in should be trivial. Since the boundary component of is represented by twice the generator, we see that
You should convince yourself that is homeomorphic to . If we view as a quotient of a sphere, then the nontrivial element of is represented by a path going from the North pole to the South pole.
What happens if we glue together two Möbius bands together along their boundaries? Call this space . As above, ot seems like is generated by a loop corresponding to a generator of the first band and a loop corresponding to a generator of the other. But these loops are related. Since the boundaries of the two Möbius bands have been identified, . Hence,
The second presentation is obtained from the first by setting and exhibits as a semi-direct product. It turns out that is homeomorphic to the Klein bottle . This is easily seen by drawing some pictures.
That concludes part I! Until next time,