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nLab > Latest Changes: rational model of mapping space

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2019

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2019
    • (edited Mar 5th 2019)

    some minimum, for the moment just so as to record the fact that

    Maps f(S n,S n)≃ ℚ{S n×S n−1 | neven,deg(f)=0 S 2n−1 | neven,deg(f)≠0 S n | nodd Maps_f\big( S^n, S^n\big) \;\simeq_{\mathbb{Q}}\; \left\{ \array{ S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0 \\ S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0 \\ S^n &\vert& n\, \text{odd} } \right.

    v1, current

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 5th 2019

    Something seems missing. Should the third line include the condition deg(f)=0deg(f) = 0? Because we can’t have both nodd,deg(f)≠0⇒nodd⇒Maps f(S n,S n)≃ ℚS nn\; odd, deg(f) \neq 0 \Rightarrow n\; odd \Rightarrow Maps_f(S^n, S^n) \simeq_\mathbb{Q} S^n and nodd,deg(f)≠0⇒Maps f(S n,S n)≃ ℚS 2n−1n\; odd, deg(f) \neq 0 \Rightarrow Maps_f(S^n, S^n) \simeq_\mathbb{Q} S^{2n-1}.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2019
    • (edited Mar 5th 2019)

    Thanks for catching this. The first “odd” should have been the second “even”. Fixed now.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 5th 2019

    Fixed typo and added link to published version of Buijs–Félix–Murillo.

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2019

    Thanks.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2019
    • (edited Sep 12th 2019)

    added pointer to

    • Samuel B. Smith, A based Federer spectral sequence and the rational homotopy of function spaces, Manuscripta Math (1997) 93: 59 (doi:10.1007/BF02677458)

    based on

    • Herbert Federer, A Study of Function Spaces by Spectral Sequences, Transactions of the American Mathematical Society Vol. 82, No. 2 (Jul., 1956), pp. 340-361 (jstor:1993052)

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 6th 2019

    added statement of the rational cohomology of Ω DS n\Omega^D S^n, for 1≤D<n=2k1 \leq D \lt n = 2k (here)

    am adding this statement also to iterated loop space

    diff, v10, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 7th 2019

    added a bunch of further references

    diff, v15, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019

    Corrected a mistake in the discussion of the rational model for Ω DS D\Omega^D S^D. To make up for it, I have now spelled out the first few cases explicitly, here

    diff, v17, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 12th 2021

    added pointer to:

    diff, v19, current

1 to 11 of 11

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