Generalization of Hopf invariant

Question

This may be a dumb question, but I ask it here.

In ordinary cohomology, we can construct a Hopf invariant for $f \colon S^{2n-1} \to S^{n}$ by applying $H^{*}(- \colon \mathbb{F}_p)$ to the cofibre sequence, so that the Hopf invariant measures the non-triviality of $$ 0 \to H^{*}(S^{2n}) \to H^*(C_f) \to H^{*}(S^n) \to 0 \in \mathrm{Ext}_A^1(H^{*}(S^n), H^{*}(S^{2n})) $$ where $A$ is the Steenrod algebra. Similarly, Adams constructed the $e$-invariant by doing an analogy of Hopf invariant in complex K-theory and Adams operation and he computed the image of the $J$-homomorphism. My question is:

1. Is there a generalization of this method to another cohomology theory (e.g. $MU$, Morava K-theory, $tmf$, etc.)? If there was, what kind of elements in $\pi_i^{S}$ could be detected in this invariant?

2. Can we view this invariant in another way? I am not sure there must be a modern view of this or not, but I would like to know it if there was a more way to view Hopf invariant than just apply a cohomology and measure how non trivial the cohomology operation is.

Answer 1

The $E$-based Adams spectral sequence is the homotopy spectral sequence associated to the tower of spectra $\dots \to Y_2 \to Y_1 \to Y_0 = S$ with $Y_{s+1} \to Y_s \to E \wedge Y_s$ a homotopy fiber sequence for each $s\ge0$. The edge homomorphism to filtration $s=0$ detects the Hurewicz image of $\pi_*(S)$ in $\pi_*(E) = E_*$. On its kernel there is defined a homomorphism to filtration $s=1$, which specializes to the Hopf-Steenrod invariant when $E$ is mod $p$ homology. I believe Chapter 19 of Switzer's book "Algebraic Topology -- Homotopy and Homology" gives a classical introduction. In particular, for $E = KU$ and $E = KO$ he connects this generalization to Adams' $e$-invariant (in the complex and real cases).

For $E = ko$ there is work by Mahowald at $p=2$. His Bulletin of the AMS announcement (1970) discusses a splitting of $ko \wedge ko$ (closely related to $E \wedge Y_1$ in this case), which was improved by Milgram and Carlsson, and Mahowald's Pacific J. Math. (1981) paper titled "$bo$-resolutions" uses this approach to determine the Bott periodic homotopy of the mod $2$ Moore spectrum.

For $E = MU$ (or $E = BP$) the $E$-based Adams spectral sequence is the Adams-Novikov spectral sequence. The $s=1$-line was calculated by Miller, Ravenel and Wilson (Annals of Math., 1977), and detects the image-of-J, much like topological $K$-theory does.

For $E = \ell$ (the Adams summand of $p$-local connective $K$-theory, with $p$ odd), the cooperations $\ell \wedge \ell$ were studied by Kane (1981) and Lellmann (1984), while Miller (JPAA, 1981) determined the periodic homotopy of the mod $p$ Moore spectrum.

For $E = Ell$ a form of elliptic cohomology, an early reference is Clarke-Johnson (Adams memorial symposium, 1992).

For $E = tmf$ (topological modular forms), the Behrens-Ormsby-Stapleton-Stojanoska paper (J. of Topology 2019) is a good place to look. The $tmf$-Hurewicz image and the $tmf$-based $e$-invariant detect significantly more of $\pi_*(S)$ than the image-of-J. The Behrens-Mahowald-Quigley preprint (and my forthcoming book with Bruner) also contain more about this.