Reference for secondary cohomology operations

Question

I am learning some homotopy theory and am currently reading Mosher and Tangora.

I love the content of this book, it's very terse and comes straight to the point. At the same time I find it very abstruse and the notation never makes it through my thick skull. For the first part of the book I found lecture notes by Mike Hopkins on Steenrod squares very useful as a supplement and I could do the computations myself without having to look at the book for specifics.

Now I am reading about secondary cohomology and this part of the book is again unwieldy. I would really appreciate it if someone could give me a reference for secondary cohomology operations, hopefully with lots of applications.

And secondly what would be a good book to continue with after I'm done Mosher and Tangora.

Thanks.

Answer 1

Wikipedia gives a few references. In particular this book:

• Harper, John R., Secondary cohomology operations. Graduate Studies in Mathematics, 49. American Mathematical Society, Providence, RI, 2002. xii+268 pp. ISBN: 0-8218-3198-4

To quote the MathSciNet review of Lionel Schwartz:

In all, this book gives an excellent introduction, and more than that in fact, to a technical, but extremely important, field in homotopy theory. This book is highly recommended for both beginners and experts.

There's also Adams' original article, On the non-existence of elements of Hopf invariant one, and it's readable if you're willing to put in a lot of effort (though he uses this unusual convention of writing function application on the right like $x f$, supposedly to help with signs -- then all the applications in the paper are modulo 2...)

Answer 2

First the basic idea of secondary cohomology operations is to define homomorphisms from a certain subset of $H^n(X,G)$ to a certain quotient $H^l(X,H)$ that are natural in X; it will be called a secondary operation because one gets this operation from another cohomology operation $H^n(-,G) \to H^m( -,\pi)$.

To get the secondary operation, one needs some relation between the $K(G,n)$ and the $K(H,l)$. One can do this by taking a map(cohomology class) $\phi: \text{ a certain fibration over }K(n,G) \to K(H,l)$. The primary operation $\theta: K(G,n) \to K(\pi,m)$ influences the choice of the secondary operation by making the map $\phi$ be from the pullback of the path space fibration over $K(\pi,m)$ under $\theta$.

Biting the bullet: The 'certain subset' will be all the cohomology classes $u: X \to K(G,n)$ such the induced map $\theta(u)=0$. The operation is defined by sending any such $u$ to a lift of $u$ to the pullback fibration $\theta^*(P(K(\pi,m)))$ composed with the cohomology class $\phi: \theta^*(P(K(\pi,m))) \to K(H,l)$. The lift won't be unique so there is only a well defined map to a quotient of $H^l(X,H)$.

Look at mosher and tangora(since I know that you know this book well) pp 159-160 to learn about secondary cohomology operations. Some good applications of concepts learned earlier in mosher and tangora(discovered after the publication of mosher and tangora) include the fact that they can be defined in terms of k-postnikov invariants. For this you should see Thomas' paper. (Of course you can learn about k-invariants from Hatcher first).