Edwin Spanier - Algebraic Topology

The classic textbook by Spanier is infamously dense. On one hand, his insistence on spelling out in the most precise formality the details of proofs can be a benefit for a reader who gets frequently snagged on minor technical notes and needs them spelled out precisely and in detail, without any geometric handwaving. In a book like that of Hatcher, where such handwaving is frequent, it is easy to think one understands a proof or construction, only to realize later, when you need to use some detail of the construction, that you did not understand it at all. Spanier lets the reader have no such illusions to comprehension. On the other hand, opening up such a book can be quite intimidating, especially given his tendency to collect a wall of results together and leave their proofs as exercises in the reader, and express these statements in exceeding levels of generality.

I have spent some of the past few months studying Chapters 4 and 5 of his book, and taken some notes. In some places, Spanier's sometimes terse arguments are fleshed out. (For example, my commentary on his proof of the Leray-Hirsch theorem is substantially longer and slower paced than his, by an order of magnitude.) I am also proud of my own proof of the classical Acyclic Models theorem in these notes. I am not aware of any sources which point out that the argument hinges on the problem of Kan extension of a functor defined on the set of models to a functor defined on the whole category, so this contribution may be original. My hope is that readers find them a useful supplement. Previously, to quote a dead German philosopher, I had "abandoned these notes to the gnawing criticism of the mice all the more willingly since I had achieved my main purpose – self-clarification"; but I feel this is a mistake, and I would like to make them publicly available. So I am posting them here. I like to include my LaTeX, too, in case anyone wants to edit it themselves.

http://www.math.wisc.edu/~nicodemus/Algebraic_Topology_Edwin_Spanier.tex
http://www.math.wisc.edu/~nicodemus/Algebraic_Topology_Edwin_Spanier.pdf

Below I am attaching a review of Spanier's book I wrote as a letter to a friend, comparing it with Hatcher. I would say now with the benefit of hindsight that a better introduction to sheaf cohomology is the book by Godement, if you read any French. Spanier does not study sheaf cohomology or presheaf cohomology in any great amount of detail, it is only one section in an encyclopedic reference on general algebraic topology. If you are really just curious about sheaf cohomology, I would recommend getting some bare minimum understanding of homology and cohomology (say cohomology up to cup products and cap products) and picking up a book like that of Godement, "Manifolds, Sheaves and Cohomology" by Wedhorn, or "Cohomology of Sheaves" by Iversen. Bott and Tu's book on "Differential Forms in Algebraic Topology" would also be good for context.


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First of all, I recommend the book "Algebraic Topology" by Edwin Spanier. It strives to be categorical in style. For instance, early on in the book it defines the homotopy groups by proving that S^n, the n-dimensional circle is a group object in the opposite of the homotopy category, (hTop)^{op}. He then proves that, in any category C, if B is a group object, then Hom(A,B) is naturally endowed with a group structure for any A; so, in hTop^op, Hom(A, S^n) is a group; but this is the same as Hom(S^n, A) in hTop. This endows the set of homotopy classes of maps of S^n into A with a group structure, which gives the n-th homotopy group of A. Hatcher's presentation of the homotopy groups is much more geometric; I find Hatcher's arguments almost unreadable unfortunately. Have a look at the beginning of Hatcher's Chapter 4 right now, looking at his proof that the higher homotopy groups are Abelian;  it consists of a bizarre diagram with squares moving around each other. Spanier's proof spells out more algebraically what's going on, and shows it to be a consequence of the beautiful Eckmann-Hilton duality. (To be fair, Eckmann-Hilton is in Hatcher's book, too, in one of its dozens of appendices, but occurs only later on in the book. The same is true with its treatment of Leray-Hirsch, which can be treated without any homotopy theory.)

In defense of Hatcher, I suspect Hatcher is most effectively used as a supplement to another more formal book which is more algebraic, and you can use Hatcher for geometric intuition for the algebraic results; I have an extremely categorical friend (think homotopy type theory researcher) who swears by Hatcher's presentation. Indeed I have heard that Guillame Brunerie's proof in HoTT that the fourth homotopy group of S^3 is Z/2Z is based on a straight translation of Hatcher's proof, see here: https://arxiv.org/abs/1606.05916) I suppose after working with purely algebraic ideas all day it's nice to see a book that presents some of their topological content.

Spanier leans toward the formal side, with explicit walls of symbol-pushing when it is necessary for the sake of precision. This can occasionally be quite intimidating, as one will turn a page and see it absolutely covered in monstrous formulas. But once I get over this fear, I usually have no trouble at all reading the formulas and decoding what they are saying. (The proof that every fibre bundle is a fibration is an excellent illustration of a conceptually beautiful technique from manifold theory, which is to use partitions of unity to take a family of locally defined functions on a manifold  and tweak them all to the point where they agree pairwise and thus can be glued together into a global function. But Spanier's proof of this fact, in an effort to state everything explicitly, the formulas run everywhere and it can be difficult to follow.)  He also tries to spell out explicitly a lot of the naturality and functoriality conditions/results that come up in AT, which again results in an explosion of formulas. (You may briefly recall from us talking about the way homology arises in algebra; short exact sequences of chain complexes induce long exact sequences H_n(A) -> Hn(B) ->Hn(C) -> H_{n-1}(A) -> H_{n-1}(B)) ->... so the homology functors are woven together by these rather complex connecting homomorphisms Hn(C)->H_{n-1}(B). So one has to take these connecting homomorphisms into account too when talking about naturality.)

I find that the formality can occasionally make it a bit dry as he does not handwave away any of the homological algebra that comes up during homology theory. But it is better, perhaps, to have an intimidating wall of symbolism that can be worked through than to handwave it away and leave the reader high and dry when they can't figure out for themselves what exactly is being said or how it can be formalized.

Spanier covers roughly the same content as Hatcher. I would say Spanier is a bit less gentle early on; not being afraid to go a little bit farther for the sake of generality. For example, in Ch. 1 of Hatcher, he develops covering space theory with very heavy reliance on two facts about covering space maps: the homotopy lifting property, Prop 1.30, and its specialization to a point, the path lifting property. A map with the homotopy lifting property (dropping the uniqueness) is called a fibration, and Spanier shows that all these arguments go through for any fibration with a unique path lifting property. This is valuable because fibration maps have better categorical properties than covering spaces, as the composition of two fibrations is a fibration, but the composition of two covering space projections is not a covering space projection in general; likewise the product of two fibration maps is a fibration maps, but the product of covering space maps is not a covering space map.

Much like one can lay down a set of axioms for a topos, and show that any category which is a topos is a "good place to do set theory", one can lay down a set of axioms for a "Model category" and show that a great deal of homotopy theory can be reconstructed in any model category. Of course the category Top is a model category (maybe after restricting to a subcategory of well behaved spaces) but the category of chain complexes of R-modules is also a model category, and the development of homotopy theory in the model category of chain complexes turns out to be... precisely homological algebra. A beautiful explanation of why homological algebra and algebraic topology seem to share so much in common. Anyway, in this axiomatic development of homotopy theory in a model category, the notion of fibration is taken as one of the cornerstones of the theory, so it is good for it to be introduced early, even if model categories shouldn't be seen till much later.

Lastly, and this main reason I picked up Spanier's book to begin with, is that it covers sheaf cohomology and Cech cohomology for paracompact Hausdorff spaces (think spaces arising in differential geometry, like a space that is a manifold away from a small set of singularities), which I thought would be a nice and more accessible place to learn about sheaf cohomology and Cech cohomology before trying to understand their role in algebraic geometry out of a book like Hartshorne. After all if you're trying to learn about a complex topic like sheaf cohomology, wouldn't you rather assume (if only for the mental comfort) that the underlying spaces of the sheaves are manifolds or close to it, rather than schemes, which are not even generally Hausdorff?