Adams e-invariant

Question

In "On the Groups J(X) - IV", Adams introduces the $e$-invariant, which turns out to be closely connected to the image of the $J$-homomorphism, but he introduces it in a more general setting. He has $d$- and $e$-invariants of a map $X\rightarrow Y$ with respect to some exact functor $k$ from Top into an abelian category. They have the property that $d\in\text{Ext}^0(kY,kX)$ and $e\in\text{Ext}^1(kY,kX)$.

He then mentions (page 27) that we might "hope" to be able to construct further invariants in higher Ext groups, but it's not really clear to me how to go about doing this. How do you construct these "higher $e$-invariants" and has anyone put them to any use?

Answer 1

Yes, this has been done. Laures has defined an f-invariant using elliptic cohomology in The topological q-expansion principle. The explanation in Section 2 of his more recent paper on Toda brackets might be a bit easier to read.

Essentially the idea is the following: Look at the Adams-Novikov spectral sequence. A stable homotopy element $x$ has non-vanishing degree iff it is in exact Adams-Novikov filtration 0, i.e. detected in the 0-line. Suppose the degree vanishes. Then $x$ is in Adams-Novikov filtration (at least) 1. The most naive e-invariant is just the reduction of $x$ to the 1-line of the Adams-Novikov SS (which is an $\mathrm{Ext}^1$-term). To make it a bit easier, we map it into the 1-line of the Adams-Novikov spectral sequence with respect K-theory; this does not loose any information, essentially because K-theory carries a height-1 formal group law. Then we go one step further and map it into $\mathbb{Q}/\mathbb{Z}$. This is the Adams e-invariant.

If the e-invariant of $x$ vanishes, $x$ has Adams-Novikov filtration 2. (This happens automatically if the degree of $x$ is positive and even). We map now $x$ into the second line of the Adams-Novikov $E^2$-term for elliptic cohomology -- this does not loose information (essentially) because the formal group of elliptic cohomology has height 2. Laures maps this then further into a much more complicated analogue of $\mathbb{Q}/\mathbb{Z}$, essentially a quotient by the ring of divided congruences (of modular forms) by (certain) integral modular forms.

For concrete calculations with the $f$-invariant, von Bodecker's papers (e.g. as referenced in the Toda bracket paper by Laures) contain a wealth of information.

I guess, there are at least two reasons why the f-invariant is less known than the e-invariant: Firstly, the 1-line of the ANSS has a (very important!) geometric interpretation as the image of $J$. Secondly, the Adams-Novikov spectral sequence was not known at the times Adams invented the e-invariant; in particular, its first line was not computed yet. In contrast, since 1981 (by work of Miller, Ravenel, Wilson and Shimomura) the Adams-Novikov 2-line is completely computed. The f-invariant can therefore not longer help with its computation. On the other hand, the f-invariant can be still useful: Say you want to compute a Toda bracket (or a product) whose result lies in Adams-Novikov filtration 2. Then you can use the f-invariant to identify it with other classes already known. This is what happens in Laures' paper referenced above.