nLab Sullivan model of a spherical fibration

Idea

A minimal model for a spherical fibration in rational homotopy theory.

Properties

Rational Euler- and Pontryagin-class

Proposition

Let $n \in \mathbb{N}$ be a natural number, $n \geq 1$, let

(1)$$\array{ S^n &\longrightarrow& E \\ && \big\downarrow \\ && X }$$

be a spherical fibration of topological spaces such that $X$ admits a Sullivan model $A_X \in dgcAlg$. Then a Sullivan model $A_E$ for the total space $E$ is of the following form:

$n$ odd

If $n = 2k+1$ is an odd number, then

(2)$$A_E \;=\; A_X \otimes \mathbb{Q}\big[ \omega_{2k+1} \big] / \big( d \omega_{2k+1} = c_{2k+2} \big)$$

for some

(3)$$c_{2k+2} \in A_X$$

being the rational Euler class of the spherical fibration.

In particular, if $E = S(V)$ is the unit sphere bundle of a real vector bundle $V \to X$, then

$$[c_{2k}] = \chi$$

is the Euler class of that vector bundle and $\omega_{2k+1}$ is a cochain on the unit sphere bundle $S(E)$ which on the fundamental class of any (2k+1)-sphere fiber evaluates to minus unity:

(4)$$\left\langle \omega_{2k+1}, \left[ S^{2k+1} \right] \right\rangle \;=\; -1 \,.$$

$n$ even

If $n = 2k$ is an even number, then the Sullivan model $A_E$ for a rank-$2k$ spherical fibration over some $X$ with Sullivan model $A_X$ is

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

where

1. the new generator $\omega_{2k}$ evaluates 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 (5) 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 $\chi(\widehat V)$ of the corresponding (…) rank reduction $\widehat V$ of $V$:

   $$\big[ \omega_{2k} \big] \;=\; \tfrac{1}{2}\chi\big( \widehat V \big) \;\in\; H^{2k}\big( X, \mathbb{Q} \big)$$

2. the class of $c_{4k}$ is $1/4$th the rationalized $k$th Pontryagin class $p_k(V)$ of $V$:

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

This may be found as Félix-Halperin-Thomas 00, 15, Example 4, p. 202, see also Félix-Oprea-Tanré 16, Prop. 2.3. The fiber integral (4) follows by this Prop..

Remark

Beware that the Sullivan models for spherical fibrations in Prop. 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 $d \omega_7 = \chi_8$ (2) 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. 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.

Relation to rational mapping space of spheres

By general facts (see at ∞-action) a spherical fibration as in (6) is classified by a map to the classifying space $B Aut(S^n)$ of the automorphism ∞-group $Aut(S^n) \hookrightarrow Maps(S^n, S^n)$ inside the mapping space from $S^n$ to itself, which is those connected components corresponding to degree $\pm 1$

$$Aut(S^n) \;=\; Maps_{\pm 1}\big( S^n, S^n\big) \,.$$

Hence the spherical fibration is given by the homotopy pullback

(6)$$\array{ S^n &\longrightarrow& E &\longrightarrow& S^{n}\sslash Aut(S^n) \\ && \big\downarrow &{}_{(pb)}& \big\downarrow \\ && X &\underset{c}{\longrightarrow}& B Aut(S^n) }$$

of the universal spherical fibration along a classifying map $c$.

The rational homotopy type of these connected components of the mapping space are given by Sullivan models of mapping spaces:

Proposition

Let $n \in \mathbb{N}$ be a natural number and $f \colon S^n \to S^n$ a continuous function from the n-sphere to itself. Then the connected component $Maps_f\big( S^n, S^n\big)$ of the mapping space which contains this map has the following rational homotopy type:

(7)$$Maps_f\big( S^n, S^n\big) \;\simeq_{\mathbb{Q}}\; \left\{ \array{ S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0 \\ S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0 \\ S^n &\vert& n\, \text{odd} } \right.$$

where $deg(f)$ is the degree of $f$.

(Møller-Raussen 85, Example 2.5, Cohen-Voronov 05, Lemma 5.3.5)

Remark

Here Prop. and Prop. are two aspects of the same situation:

For $n = 2k+1$ an odd number the rational Euler class (3) of the spherical fibration is the class of the rational classifying map to the shift of $S^{2k+1}$ in (7);

for $n = 2k$ an even number the rational Pontryagin class (?) of the spherical fibration is the class of the rational classifying map to the shift of $S^{4k-1}$ in (7).

Related concepts

Examples of Sullivan models in rational homotopy theory:

• Sullivan model of an n-sphere

• Sullivan model of a spherical fibration

• Sullivan model of complex projective space

• Sullivan model of a classifying space

• Sullivan model of mapping space

• Sullivan model of free loop space

• Sullivan model of a suspension

• Sullivan model of a finite G-quotient

References

• Yves Félix, Steve Halperin, J.C. Thomas, Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

• Jesper Møller, Martin Raussen, Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (jstor:2000242)

• Ralph Cohen, Alexander Voronov, Notes on string topology (arXiv:math/0503625)

• Yves Félix, John Oprea, Daniel Tanré, Prop. 2.3 in Lie-model for Thom spaces of tangent bundles, Proc. Amer. Math. Soc. 144 (2016), 1829-1840 (pdf, doi:10.1090/proc/12829)