This post will be a guide on how to calculate Homology Groups, focusing on the example of the Klein Bottle. Homology groups can be quite difficult to grasp (it took me quite a while to understand it). Hope this post will help readers to get the idea of Homology. Our reference book will be Hatcher’s Algebraic Topology (Chapter 2: Homology). I will elaborate further on the Hatcher’s excellent exposition on Homology.
This is also Exercise 5 in Chapter 2, Section 2.1 of Hatcher.
The first step to compute Homology Groups is to construct a -complex of the Klein Bottle.
One thing to note for -complexes, is that the vertices cannot be ordered cyclically, as that would violate one of the requirements which is to preserve the order of the vertices.
The key formula for Homology is: .
We have , the free group generated by the vertex , because there is only one vertex!
Next, we have . Thus .
Next, we have . , . To learn more about calculating , check out the diagram on page 105 of Hatcher.
We then have , where we got from adding the two previous generators .
To intuitively understand the above working, we need to use the idea that elements in the quotient are “zero”. Hence , implies that , thus can be expressed as a linear combination of , thus is not a generator of . implies that , which gives us the part.
Finally we note that , and also for , since there are no simplices of dimension greater than or equal to 3. Thus, the second homology group onwards are all zero.
In conclusion, we have