How far from research frontier is Hatcher's book
After a student masters the entire Hatcher's book on Algebraic Topology, possibly including the additional chapter on Spectral Sequences, I am curious how far is he/she from the research frontier to do original research in Algebraic Topology?
What are some other books that will bring him/her closer to the research frontier?
For the sake of giving this question an answer, I think that in most ways Hatcher isn't particularly close to the cutting edge of algebraic topology. There are several reasons for this, and I give a few below:
- I think it's particularly important to note that that isn't the point of the book. The point of the book is to teach algebraic topology; thus the plethora of examples and exercises, and the large amount of exposition on each new idea.
- As manthanomen points out, there is very little category theory in Hatcher. This means that in the direction of model categories, the ``research frontier" is a very long way off.
- There are some aspects of Hatcher's book which are a little unusual. For example, his use of delta complexes in homology is non-standard (I don't think I've seen it anywhere else). This doesn't really take you any closer to or further away from the forefront of research, but it's just good to bear in mind.
In terms of recommendations for books which will advance you further, it depends on what you want to do.
- As manthanomen says, Peter May's book is a good reference, although I've heard that it's rather heavy going.
- If you're interested in Homotopy theory, I find Paul Selick's book very useful.
- If you're interested in Spectral Sequences then Mccleary's book is a nice one (although where that leaves one with regards to the ``research frontier" isn't known to me).
Having said all this, there's no reason why only having read Hatcher should mean that you can't be at the ``research frontier". Perhaps after carefully reading his exposition on cohomology theories, you invent a new cohomology theory, and use it to compute the cohomology of something which we've found difficult before? Perhaps you could adapt one of his numerous examples/exercises to the case of an object which hasn't been explored before - and end up computing something which isn't known?