After the statement/proof that the $e_{\mathbb{C}}$-invariant defined in terms of Adams operations equals the top degree coefficient of the Chern character

I have added a warning (now this Remark) that the analogous statement for $e_{\mathbb{R}}$ in general fails.

]]>I have fixed the statement in this Prop. of the entry (currently Prop. 5.6)

(i.e. the generalization to any multiplicative cohomology theory of the Conner-Floyd construction of a $U$-cobordism class with framed boundary from the trivialization of a d-invariant in $M U$-theory).

Namely, it used to say that the construction lifts through the boundary map as a *bijection*. But that’s evident nonsense:

It *is* a bijection *if* we retain the information of the 2-homotopy class of the homotopy involved in the *cone* shown in the first version of the proof. But as we don’t retain that information (one could, but it’s besides the point here) it’s just a map that lifts through $\partial$.

I have made a brief fix. But I realize there is room to beautify the statement of the proposition. Maybe later.

An analogous comment and fix applies to the corresponding proposition at *d-invariant*

Ah yes, it works now.

]]>The link in #16 is meant to go to theorem 5.11 in *Adams e-invariant*. It points to the anchor named `DiagrammaticeCInavriantReproducedClassicaleCInvariant`

, which is the label of that theorem.

I just checked again, clicking on it, and it works for me. What happens on your side when you click in #16?

Maybe this is a caching issue: If you had the page already opened, but with an earlier version loaded, then maybe your browser looks for the anchor in the old page without reloading, and then doesn’t find it.

]]>What’s that link in #16? I can’t see it even from the source code of the page.

]]>I have added more details to the proof that the “diagrammatic” e-invariant reproduces the classical construction (here).

]]>Now I have typed/drawn out the details of that “diagrammatic” construction of the e-invariant (here).

]]>Am starting a new experimental section “Construction via unit cofiber cohomology theories” (here) meant to lay out another approach to constructing the Adams e-invariant, more abstractly homotopy theoretic and maybe not considered in the literature (?).

If it works out, this is such that it makes various facts immediately manifest, notably the equality between Adams’ construction via the Chern character on KU with Conner-Floyd’s construction via the Todd character on MUFr.

So far the section contains one Lemma, identifying the “unit cofiber cohomology” of the cofiber space under even-periodic ordinary cohomology.

From this, the whole story should follow from looking at a single homotopy pasting diagram, to be included in a moment.

]]>I have added a brief mentioning also of the version for $E = KO$ (here) and of the example applied to the third stable stem (here)

(for the time being, both additions are not much more than glorified cross-links, for completeness)

]]>added the statement (here) that the e-invariant is Todd class of cobounding (U,fr)-manifolds

]]>I have added statement and proof (here) of the e-invariant as the top degree component of the Chern character on the cofiber space:

$\exp \left( 2 \pi \mathrm{i} \int_{C_f} ch\big( V_{2n} \big) \right) \;=\; \exp \left( 2 \pi \mathrm{i} \, { \color{blue} e(f) } \right) \;\;\; \in \mathrm{U}(1)$ ]]>I have spelled out, in full detail, the definition of the e-invariant as a character $\pi_\bullet \to \mathbb{Q}/\mathbb{Z}$ (here)

]]>and this one:

- Yasumasa Hirashima,
*On the $BP_\ast$-Hopf invariant*, Osaka J. Math., Volume 12, Number 1 (1975), 187-196 (euclid:ojm/1200757733)

added this pointer:

- Michael Atiyah, Vijay Patodi, Isadore Singer,
p. 18 onwards in:
*Spectral asymmetry and Riemannian geometry. II*, Volume 78, Issue 3 November 1975 , pp. 405-432 (doi:10.1017/S0305004100051872)

added pointer to:

- Martin Bendersky,
*The BP Hopf Invariant*, American Journal of Mathematics, Vol. 108, No. 5 (Oct., 1986) (jstor:2374595)

added this pointer:

- Robert Switzer, Section 19.19 in:
*Algebraic Topology - Homotopy and Homology*, Grundlehren der Mathematischen Wissenschaften, Vol. 212, Springer, 1975 (doi:10.1007/978-3-642-61923-6)

added pointer to:

- Michael Hopkins (notes by Akhil Mathew), Lecture 11 in:
*Spectra and stable homotopy theory*, 2012 (pdf)

added the original reference for the interpretation in bordism theory:

- Pierre Conner, Edwin Floyd, Section 16 of:
*The relation of cobordism to $K$-theories*, Lecture Notes in Mathematics**28**Springer 1966 (doi:10.1007/BFb0071091, MR216511)

added this pointer:

- Warren M. Krueger,
*Generalized Steenrod-Hopf Invariants for Stable Homotopy Theory*, Vol. 39, No. 3 (Aug., 1973), pp. 609-615 (jstor:2039603)

added this pointer:

- Gereon Quick,
*The $e$-invariant*(pdf)

added rough description and original citation to *Adams e-invariant*