# Book for Algebraic Topology- Spanier vs Tom Dieck

A number of times, questions have been asked on this website about good books on Algebraic Topology and the responses have been very valuable. However I need some more specific advice in this matter. I have studied basic point-set topology (first few chapter of Munkres's Topology) and basic algebraic topology (all of part II of Munkres's book). Now I wish to learn more algebraic topology from a categorical viewpoint. I am aware of the books by Hatcher and Bredon, but they are more geometrically flavored. I have heard that Spanier is a very nice book and meets the criterion of being categorical. But it looks to be very old and I am afraid it could be outdated. I wish to ask :

Is it true that the book Algebraic Topology by E.H.Spanier now outdated or is it still advisable for a person with taste for category theory to study Algebraic Topology from this book ?

From the answers to other questions on this site (as well as MO), I learnt about the book 'Algebraic Topology' by Tammo tom Dieck. It appears to be very attractive and sort of modern version of Spanier. However from a review here I learn that this book is recommended exclusively for brightest students. So I wish to ask :

Are there any supplements which can be used alongside Tom Dieck's book as and when one gets stuck ? Can Spanier be used as a supplement to this book, or the approach/organizational differences will be hindrances ?

How does Tom Dieck's book compare with Spanier's in readability ?

Two more books which do not hesitate to use category theory are Homology Theory by James Vick and Algebraic Topology by J.Rotman. However Vick's book does not cover cohomology and homotopy theories and the book by Rotman looks nice but sort of intermediate between Massey and Spanier while I am looking for a comprehensive graduate level book.

Are there any other comprehensive, categorically flavored books on the subject at the same level as Spanier or Tom Dieck but that could be easier to read for self study ?

Edit : Just wish to add that I have had graduate level courses in algebra including category theory and homological algebra.

• Read Peter May.
– user158047
Dec 29, 2014 at 15:05
• Tammo tom Dieck's book assumes a very basic understanding of category right from the get go. Whilst he does not define what categories are, what you actually need from the beginning can be taught in a lecture or two (just basic definitions and some examples of categories, morphisms, and functors will do.). On the other hand, I do recommend Peter May's notes. Dec 29, 2014 at 15:25
• @fixedp Thanks for your comment, I had forgotten to mention my algebra background and have edited the question now. It would be very nice of you if you could advise, how May compares with Spanier/Tom Dieck in coverage and readablility. Thanks Dec 29, 2014 at 16:00
• @JakobWerner Thanks for the recommendation. Would be very nice of you if could elaborate as to why you recommend Peter May Dec 29, 2014 at 16:01
• I can recommend Tammo tom Dieck's book very highly. I do not think that it is only suitable for brilliant students ; in fact I believe it is easier to read than the book of May. Given what you have written about your background I think it would be at the right level for you. Dec 29, 2014 at 17:37

## 2 Answers

In one place, the theory of fibrations, I feel tom Dieck's book is superior. Thus in the first edition of Spanier, in the proof of Lemma 11, of Chapter 2, Section 7, Spanier writes down an extended lifting function $\Lambda$, but he does not prove it is continuous. I did not manage to prove it was continuous, and in fact found the function was not well defined. I sent a correction of the definition to Spanier, and this appeared in the second edition, but I still do not know how to prove $\Lambda$ is continuous. Have I missed something?

On the other hand Spanier's ideas on the construction of covering spaces are still referred today by experts, with the notion of what they call the Spanier group.

I find Section 3 of Chapter 7 of Spanier on "Change of base point" rather hard work, and I feel it can all be much easier done, and more generally, by using fibrations of groupoids. But tom Dieck's book does not use this method either.

Spanier gives van Kampen's theorem for the fundamental group of a simplicial complex as an exercise, while tom Dieck's book does give the statement of the theorem for a union of two spaces. Hatcher gives a more general theorem, for a union of many spaces, but none of these mention the fundamental groupoid on a set of base points.

I tend to agree with 313's answer that readers should look around, and find what is easier for them, in different aspects.

My copy of Spanier, dated 1966, had a price of \$15, but when I checked for inflation that was equivalent to \$111 today.

March 13: I add that neither book develops the algebraic theory of groupoids. For a relevant discussion on this, see https://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one/46808#46808

Spanier is not outdated. I read about half of it, and it never felt like an old book. Actually I think that in spirit it is more "modern" than many of the so-called modern books. Apart from that when Spanier's book was written, the foundations of algebraic topology were already laid down. Of course it does not include some of the new developments, but these are anyway too advanced to be included in an introduction to the subject. [But if you really want to read about them, see Switzer's book.] That being said, Spanier's book is more sophisticated than e.g. Hatcher, because Spanier includes e.g. spectral sequences.

I do not consider myself very bright, but I must say I feel that Tom Dieck is easier to read than Hatcher, because it is written more clearly and more carefully. Also Tom Dieck is very systematic and does include e.g. the method of acyclic models, the Eilenberg-Zilber theorem (as does Spanier), which Hatcher doesn't. I don't think that Tom Dieck is much more modern than Spanier, also TD does not include spectral sequences. I think reading both Spanier and Tom Dieck is a good idea, because their approaches are often similar. I once read that Spanier "was written for a computer, not for a human", but for me it is very readable. I recommend to supplement your books by a book on classical homological algebra, say the book by Weibel. (You say that you had graduate level courses in algebra including CT and HA, but from this it is not clear whether you really know some deep stuff, or merely diagram chasing and a bit of talking.) A nice book which is somewhat similar to Spanier and TD is Dold's Lectures on algebraic topology

• Thanks for this very helpful answer. Just one more question. Tammy Tom Dieck's book appears to emphasize homotopy approach, but I would like to learn Homology theory too. Should I supplement it with Vick's book or does TTD deal with homology theory in satisfactory detail ? Thanks Dec 31, 2014 at 18:46
• tom Dieck has a more than satisfactory treatment of both homotopy and homology. Dec 31, 2014 at 19:09
• @user90041: prefaisceau is right, there is no need to worry about this. Actually when I looked at the contents of Vick's book it was immediately clear to me that probably both Spanier and TD treat more material than Vick does (in homology as well), which is probably not too surprising as Spanier and TD are more advanced Jan 1, 2015 at 10:46