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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 4th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexPage created, but author did not leave any comments. <a href="https://ncatlab.org/nlab/revision/Sullivan+model+of+a+spherical+fibration/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration">current</a>

   Page created, but author did not leave any comments.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 4th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave recorded the form of Sullivan models of spherical fibrations, and their relation to the rational homotopy type of the mapping space $Maps(S^n, S^n)$ <a href="https://ncatlab.org/nlab/revision/Sullivan+model+of+a+spherical+fibration/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration">current</a>

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

   v1, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 5th 2019
   • (edited Mar 5th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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: $$ c_{4k} \;=\; \tfrac{1}{4} p_{k} \,. $$ <a href="https://ncatlab.org/nlab/revision/diff/Sullivan+model+of+a+spherical+fibration/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Sullivan+model+of+a+spherical+fibration/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration">current</a>

   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:

   $$c_{4k} \;=\; \tfrac{1}{4} p_{k} \,.$$

   diff, v4, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 6th 2019
   • (edited Mar 6th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThere is more interesting stuff shown in the proof by F&eacute;lix-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 [[generalized the|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 [[n-sphere|2k-sphere]] [[fibers]] $S^{2k} \simeq E_x \hookrightarrow E$ over each point $x \in X$: $$ \big\langle \omega_{2k}, [S^{2k}] \big\rangle \;=\; 1 $$ 1. $c_{4k} \in A_X$ is some element in the base algebra, which by (eq:FibS2kModel) is closed and represents the rational [[cohomology class]] of the [[cup product|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 $V$: $$ \big[ \omega_{2k} \big] \;=\; \tfrac{1}{2}\chi(V) \;\in\; H^{2k}\big( X, \mathbb{Q} \big) $$ 1. the class of $c_{4k}$ is $1/4$th the rationalized $k$th [[Pontryagin class]] $p_k(V)$ of $V$: $$ \big[ c_{4k} \big] \;=\; \tfrac{1}{4} p_k(V) \;\in\; H^{4k}\big( X, \mathbb{Q}\big) \,. $$ <a href="https://ncatlab.org/nlab/revision/diff/Sullivan+model+of+a+spherical+fibration/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/Sullivan+model+of+a+spherical+fibration/5">v5</a>, <a href="https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration">current</a>

   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 $V$:

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

   diff, v5, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 28th 2019
   • (edited Apr 28th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded previously missing statement about the fiberwise normalization of the generators in the case of odd rank ([here](https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration#eq:FiberIntegrationOfOddGenerator)) by pointer to [this Prop.](https://ncatlab.org/nlab/show/Euler+class#TrivializationOfEulerFormOnUnitSphereBundle) <a href="https://ncatlab.org/nlab/revision/diff/Sullivan+model+of+a+spherical+fibration/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/Sullivan+model+of+a+spherical+fibration/9">v9</a>, <a href="https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration">current</a>

   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

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 5th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a warning ([here](https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration#NonMinimalityOfSullivanModels)): *** 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 subgroup|torsion]] [[relative homotopy groups]], but not in general non-torsion absolute homotopy groups. <a href="https://ncatlab.org/nlab/revision/diff/Sullivan+model+of+a+spherical+fibration/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/Sullivan+model+of+a+spherical+fibration/11">v11</a>, <a href="https://ncatlab.org/nlab/show/Sullivan+model+of+a+spherical+fibration">current</a>

   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 $B SO(8)$ 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.

   diff, v11, current