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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2019

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2019

    have recorded the form of Sullivan models of spherical fibrations, and their relation to the rational homotopy type of the mapping space Maps(S n,S n)Maps(S^n, S^n)

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 5th 2019
    • (edited Mar 5th 2019)

    I have added the statements that the unique rational classes of these spherical fibrations – Euler class for nn odd or Pontryagin class fo nn even – are indeed the corresponding classes of the underlying vector bundles, if the fibrations arise as unit sphere bundles.

    Or rather, for the case of nn even this holds up to a remarkable factor of 1/41/4:


    In particular if E=S(V)E = S(V) is the unit sphere bundle of a real vector bundle VXV \to X, then the rational class of the spherical fibration is 1/41/4th of the rational Pontryagin class of that vector bundle:

    c 4k=14p k. c_{4k} \;=\; \tfrac{1}{4} p_{k} \,.

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 6th 2019
    • (edited Mar 6th 2019)

    There is more interesting stuff shown in the proof by Félix-Halperin-Thomas than is actually stated in their theorem. I have now made made it all explicit in the statement in the entry:


    If n=2kn = 2k is an even number, then the Sullivan model A EA_E for a rank-nn spherical fibration over some XX with Sullivan model A XA_X is

    A E=A X[ω 2k,ω 4k1]/(dω 2k = 0 dω 4k1 = ω 2kω 2k+c 4k) A_E \;=\; A_X \otimes \mathbb{Q} \Big[ \omega_{2k} , \omega_{4k-1} \Big] / \left( \array{ d \, \omega_{2k} &=& 0 \\ d \omega_{4k-1} & =& \omega_{2k} \wedge \omega_{2k} + c_{4k} } \right)

    where

    1. the new generator ω 2k\omega_{2k} restricts to unity on the fundamental classes of the 2k-sphere fibers S 2kE xES^{2k} \simeq E_x \hookrightarrow E over each point xXx \in X:

      ω 2k,[S 2k]=1 \big\langle \omega_{2k}, [S^{2k}] \big\rangle \;=\; 1
    2. c 4kA Xc_{4k} \in A_X is some element in the base algebra, which by (eq:FibS2kModel) is closed and represents the rational cohomology class of the cup square of the class of ω 2k\omega_{2k}:

      [c 4k]=[ω 2k] 2H 4k(X,) \big[ c_{4k} \big] \;=\; \big[ \omega_{2k} \big]^2 \;\in\; H^{4k} \big( X, \mathbb{Q} \big)

      and this class classifies the spherical fibration, rationally.

    Moreover, if the spherical fibration EXE \to X happens to be the unit sphere bundle E=S(V)E = S(V) of a real vector bundle VXV \to X, then

    1. the class of ω 2k\omega_{2k} is 1/21/2 the rationalized Euler class χ(V)\chi(V) of VV:

      [ω 2k]=12χ(V)H 2k(X,) \big[ \omega_{2k} \big] \;=\; \tfrac{1}{2}\chi(V) \;\in\; H^{2k}\big( X, \mathbb{Q} \big)
    2. the class of c 4kc_{4k} is 1/41/4th the rationalized kkth Pontryagin class p k(V)p_k(V) of VV:

      [c 4k]=14p k(V)H 4k(X,). \big[ c_{4k} \big] \;=\; \tfrac{1}{4} p_k(V) \;\in\; H^{4k}\big( X, \mathbb{Q}\big) \,.

    diff, v5, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2019
    • (edited Apr 28th 2019)

    added previously missing statement about the fiberwise normalization of the generators in the case of odd rank (here) by pointer to this Prop.

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2019

    added a warning (here):


    Beware that the Sullivan models for spherical fibrations in Prop. \ref{SullivanModelForSphericalFibration} are not in general minimal Sullivan models.

    For example over the classifying space BSO(8)B SO(8) of SO(8) with indecomposable Euler class generator χ 8\chi_8 the equation dω 7=χ 8d \omega_7 = \chi_8 (eq:SullivanModelForOddDimensionalSphericalFibration) for the univeral 7-sperical fibration S 7SO(8)BSO(8)S^7 \sslash SO(8) \to B SO(8) violates the Sullivan minimality condition (which requires that the right hand side is at least a binary wedge product of generators, or equivalently that the degree of the new generator ω 7\omega_7 is greater than that of any previous generators).

    But the Sullivan models in Prop. \ref{SullivanModelForSphericalFibration} are relative minimal models, relative to the Sullivan model for the base.

    This means in particular that the new generators of these models reflect non-torsion relative homotopy groups, but not in general non-torsion absolute homotopy groups.

    diff, v11, current

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