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    • CommentRowNumber1.
    • CommentAuthorjcmckeown
    • CommentTimeNov 8th 2012

    A stub Massey product and a longer Toda bracket (still plenty gaps of reference, many many unlinked words). No promises w.r.t. spellings.

    I now see I’ve missed the convention for capitalization. Will fix that now… done.


    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2012
    • (edited Nov 8th 2012)


    I did some editing, as usual: added hyperlinks, a table of contents, a Referemces-section, pointers to the References, etc.

    Notice that top-level sections need two hash-signs to appear in the TOC correctly:

    ## Idea
    ## Definition
    ### General
    ### Some special case
    • CommentRowNumber3.
    • CommentAuthorjcmckeown
    • CommentTimeNov 9th 2012

    CoThanks, too. also, the definition now makes sense.

    • CommentRowNumber4.
    • CommentAuthorjcmckeown
    • CommentTimeNov 9th 2012

    OK, maybe I’ll take this next question to MO, but:

    the Toda Bracket should directly give something like the Massey Product, because if you have a bunch of maps u i:XK i u_i : X \to K_i where the K iK_i are suitable Eilenberg-Mac Lane spaces, one has a sequence of maps of pointed-function spaces

    K 1 X(K 1K 2) X(K 1K n+2) X K_1^X \to (K_1 \otimes K_2)^X \cdots \to (K_1\otimes \cdots K_{n+2})^X

    representing the particular cup products u i\bullet\smallsmile u_i, as well as a map

    𝕊 0K 1 X \mathbb{S}^0 \to K_1^X

    adjoint to u 1u_1. Then the bracket machinery highlights a family of maps

    𝕊 n(K 1K n+2) X \mathbb{S}^{n} \to (K_1 \otimes \cdots K_{n+2})^X

    adjoint to

    Σ nXK 1K n+2. \Sigma^n X \to K_1 \otimes \cdots \otimes K_{n+2} .

    It seems intuitive to me (don’t ask why) that the construction of Massey products should at least give a subset of these Toda brackets, but I don’t feel so clear about the vice-versa. Does anyone know if this is at least spelled-out somewhere? (McCleary’s book on SpSeqs, e.g., is perfectly vague about these brackets, at least in a neighborhood of index entries.)

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeNov 9th 2012
    • (edited Nov 9th 2012)

    Added references

    • Hans-Joachim Baues, On the cohomology of categories, universal Toda brackets and homotopy pairs, K-Theory 11:3, April 1997, pp. 259-285 (27) springerlink
    • Boryana Dimitrova, Universal Toda brackets of commutative ring spectra, poster, Bonn 2010, pdf
    • C. Roitzheim, S. Whitehouse, Uniqueness of A A_\infty-structures and Hochschild cohomology, arxiv/0909.3222
    • Steffen Sagave, Universal Toda brackets of ring spectra, Trans. Amer. Math. Soc., 360(5):2767-2808, 2008, math.KT/0611808