Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 18th 2013
• (edited Nov 18th 2013)

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 28th 2020

• Gereon Quick, The $e$-invariant (pdf)
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeNov 29th 2020

• Warren M. Krueger, Generalized Steenrod-Hopf Invariants for Stable Homotopy Theory, Vol. 39, No. 3 (Aug., 1973), pp. 609-615 (jstor:2039603)
• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 29th 2020
• (edited Nov 29th 2020)

added the original reference for the interpretation in bordism theory:

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 29th 2020
• (edited Nov 29th 2020)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 30th 2020
• (edited Nov 30th 2020)

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 30th 2020

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeNov 30th 2020

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 30th 2020

and this one:

• Yasumasa Hirashima, On the $BP_\ast$-Hopf invariant, Osaka J. Math., Volume 12, Number 1 (1975), 187-196 (euclid:ojm/1200757733)
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeDec 4th 2020

I have spelled out, in full detail, the definition of the e-invariant as a character $\pi_\bullet \to \mathbb{Q}/\mathbb{Z}$ (here)

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeDec 5th 2020

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

$\exp \left( 2 \pi \mathrm{i} \int_{C_f} ch\big( V_{2n} \big) \right) \;=\; \exp \left( 2 \pi \mathrm{i} \, { \color{blue} e(f) } \right) \;\;\; \in \mathrm{U}(1)$
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeDec 6th 2020
• (edited Dec 6th 2020)
• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeDec 18th 2020

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)

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJan 15th 2021
• (edited Jan 15th 2021)

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.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJan 16th 2021

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

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJan 18th 2021

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

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeJan 18th 2021

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

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJan 18th 2021

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.

• CommentRowNumber19.
• CommentAuthorDavid_Corfield
• CommentTimeJan 18th 2021

Ah yes, it works now.

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeFeb 19th 2021
• (edited Feb 19th 2021)

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

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeFeb 21st 2021

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.