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nLab > Latest Changes: Adams e-invariant

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

    added rough description and original citation to Adams e-invariant

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2020

    added this pointer:

    diff, v3, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2020

    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)

    diff, v5, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2020
    • (edited Nov 29th 2020)

    added the original reference for the interpretation in bordism theory:

    diff, v5, current

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

    added pointer to:

    diff, v8, current

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

    added this pointer:

    diff, v9, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2020

    added pointer to:

    diff, v10, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2020

    added this pointer:

    diff, v11, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2020

    and this one:

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

    diff, v11, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2020

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

    diff, v17, current

    • 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(2πi∫ C fch(V 2n))=exp(2πie(f))∈U(1) \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)

    diff, v25, current

    • 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=KOE = 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)

    diff, v32, current

    • 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.

    diff, v38, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2021

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

    diff, v44, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2021

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

    diff, v52, current

    • 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 UU-cobordism class with framed boundary from the trivialization of a d-invariant in MUM 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

    diff, v55, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2021

    After the statement/proof that the e ℂe_{\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 ℝe_{\mathbb{R}} in general fails.

    diff, v57, current

  22. notation fix in the definition of Z(n) (Example 3.2)

    Anon.

    diff, v58, current

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