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1. • CommentRowNumber1.
   • CommentAuthorjcmckeown
   • CommentTimeNov 8th 2012
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexA 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. Cheers

   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.

   Cheers

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 8th 2012
   • (edited Nov 8th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks!! 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

   Thanks!!

   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
   

3. • CommentRowNumber3.
   • CommentAuthorjcmckeown
   • CommentTimeNov 9th 2012
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexCoThanks, too. also, the definition now makes sense.

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

4. • CommentRowNumber4.
   • CommentAuthorjcmckeown
   • CommentTimeNov 9th 2012
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexOK, 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 : X \to K_i $ where the $K_i$ are suitable Eilenberg-Mac Lane spaces, one has a sequence of maps of pointed-function spaces $$ K_1^X \to (K_1 \otimes K_2)^X \cdots \to (K_1\otimes \cdots K_{n+2})^X $$ representing the particular cup products $\bullet\smallsmile u_i$, as well as a map $$ \mathbb{S}^0 \to K_1^X $$ adjoint to $u_1$. Then the bracket machinery highlights a family of maps $$ \mathbb{S}^{n} \to (K_1 \otimes \cdots K_{n+2})^X $$ adjoint to $$ \Sigma^n X \to K_1 \otimes \cdots \otimes K_{n+2} .$$ 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.)

   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 $u_i : X \to K_i$ where the $K_i$ are suitable Eilenberg-Mac Lane spaces, one has a sequence of maps of pointed-function spaces

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

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

   $$\mathbb{S}^0 \to K_1^X$$

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

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

   adjoint to

   $$\Sigma^n X \to K_1 \otimes \cdots \otimes K_{n+2} .$$

   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.)

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeNov 9th 2012
   • (edited Nov 9th 2012)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexAdded 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](http://link.springer.com/article/10.1023/A%3A1007796409912) * Boryana Dimitrova, _Universal Toda brackets of commutative ring spectra_, poster, Bonn 2010, [pdf](http://www.math.uni-bonn.de/people/grk1150/YWT2010/YWT_Dimitrova.pdf) * C. Roitzheim, S. Whitehouse, _Uniqueness of $A_\infty$-structures and Hochschild cohomology_, [arxiv/0909.3222](http://arxiv.org/abs/0909.3222) * Steffen Sagave, _Universal Toda brackets of ring spectra_, Trans. Amer. Math. Soc., 360(5):2767-2808, 2008, [math.KT/0611808](http://arxiv.org/abs/math/0611808)

   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 $A_\infty$-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