I am about halfway through Hatcher's Algebraic topology book, and have learnt something about homology and cohomology. However, there are a few things that I am looking forward to which doesn't appear yet.
- The relation between Euler characteristic and genus of a surface. The word genus has been used several times without a formal definition.
- The concept of orientablity. Again, this is used informally without definition.
Both of the above concepts are pretty famous, and they doesn't seem to be very difficult (high school students know it, although not rigorously).
I know that orientability is not far away in the book. But is there a reference that proves results about the relation between homology groups and genus?