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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 4th 2019

Page created, but author did not leave any comments.

• 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 $Maps(S^n, S^n)$

• 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 $n$ odd or Pontryagin class fo $n$ 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 $n$ even this holds up to a remarkable factor of $1/4$:

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

$c_{4k} \;=\; \tfrac{1}{4} p_{k} \,.$
• 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 $n = 2k$ is an even number, then the Sullivan model $A_E$ for a rank-$n$ spherical fibration over some $X$ with Sullivan model $A_X$ is

$A_E \;=\; A_X \otimes \mathbb{Q} \Big[ \omega_{2k} , \omega_{4k-1} \Big] / \left( \array{ d \, \omega_{2k} &=& 0 \\ d \omega_{4k-1} & =& \omega_{2k} \wedge \omega_{2k} + c_{4k} } \right)$

where

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

$\big\langle \omega_{2k}, [S^{2k}] \big\rangle \;=\; 1$
2. $c_{4k} \in A_X$ 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 $\omega_{2k}$:

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

and this class classifies the spherical fibration, rationally.

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

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

$\big[ \omega_{2k} \big] \;=\; \tfrac{1}{2}\chi(V) \;\in\; H^{2k}\big( X, \mathbb{Q} \big)$
2. the class of $c_{4k}$ is $1/4$th the rationalized $k$th Pontryagin class $p_k(V)$ of $V$:

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

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 5th 2019

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 $B SO(8)$ of SO(8) with indecomposable Euler class generator $\chi_8$ the equation $d \omega_7 = \chi_8$ (eq:SullivanModelForOddDimensionalSphericalFibration) for the univeral 7-sperical fibration $S^7 \sslash SO(8) \to B SO(8)$ 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 $\omega_7$ 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.