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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 28th 2018

starting something, not done yet

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 2nd 2018

added pointer to the original article Snaith 74 and to the interpretation of the stable splitting as the Goodwillie-Taylor tower of the mapping space by Arone 99

• CommentRowNumber3.
• CommentAuthorGuest
• CommentTimeNov 2nd 2018
Should that be $\Sigma^{+}_{\infty}Maps(X,A)$ or is that in based CW complexes?
• CommentRowNumber4.
• CommentAuthorGuest
• CommentTimeNov 2nd 2018

Sorry, I meant $\Sigma^{\infty}_{+}Maps(X,A)$ of course

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeNov 2nd 2018
• (edited Nov 2nd 2018)

I think the mapping space is/must be regarded as being pointed already, so that $\Sigma^\infty Maps(X,A)$ is correct.

Abstractly, the stable splitting is the Goodwillie derivative of the functor $Maps(X,-)$, and that is a functor from and to pointed topological spaces.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeNov 3rd 2018
• (edited Nov 3rd 2018)

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeNov 3rd 2018

added now detailed definitions and statements up to Snaith 74 theorem 1.1 and Boedigheimer 87, Examples 2 and 5.

So far, this material is duplicated also at configuration space of points. I think that’s okay. Once the discussion here enters the generalization as in Arone 99, the two entries will diverge.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeNov 3rd 2018

There are diffeomorphic-invariant results on stable splitting here if needed:

• Richard Manthorpe, Ulrike Tillmann, Tubular configurations: equivariant scanning and splitting, (arXiv:1307.5669)
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeNov 3rd 2018

Thanks! I’ll have a look.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeNov 6th 2018
• (edited Nov 6th 2018)

I have now added a section (here) indicating how the stable splitting of mapping spaces is the limiting Goodwillie-Taylor tower of the mapping space functor.

I am just following the basic observations about excisiveness of the stages in the splitting formula, from the second page of Arone 99, but regarding assumptions on the space that we are homming out, and the notation and conventions on the mapping space, I am sticking with what I already had in the entry. (See the tables there for matching notation to the literature.)

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeNov 16th 2018
• (edited Nov 16th 2018)

Fixed the notation from the point on where the decomposition $Conf_n(X,Y) \simeq Conf_n^{ord}(X) \wedge_{\Sigma_n} (Y/\partial Y)^{\wedge_n}$ is used (the superscriped for the ordered configuration space had been missing)

• CommentRowNumber12.
• CommentAuthorDavidRoberts
• CommentTimeNov 16th 2018

$Comp=Conf$ in #11?

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeNov 16th 2018

Yes, thanks, fixed.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 28th 2019

• Lauren Bandklayder, Stable splitting of mapping spaces via nonabelian Poincaré duality (arxiv:1705.03090)

and cross-linked with nonabelian Poincaré duality

1. The reference does not deal with general n-connective spaces.

Anonymous

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJul 2nd 2021