and
A minimal model for a spherical fibration in rational homotopy theory.
Let $n \in \mathbb{N}$ be a natural number, $n \geq 1$, let
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
for some
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
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:
$n$ 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
where
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 $x \in X$:
$c_{4k} \in A_X$ is some element in the base algebra, which by (4) is closed and represents the rational cohomology class of the cup square of the class of $\omega_{2k}$:
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
the class of $\omega_{2k}$ is $1/2$ the rationalized Euler class $\chi(\widehat V)$ of the corresponding (…) rank reduction $\widehat V$ of $V$:
the class of $c_{4k}$ is $1/4$th the rationalized $k$th Pontryagin class $p_k(V)$ of $V$:
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 (3) follows by this Prop..
By general facts (see at ∞-action) a spherical fibration as in (5) 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 $\pm 1$
Hence the spherical fibration is given by the homotopy pullback
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:
Let $n \in \mathbb{N}$ be a natural number and $fcolon S^n \to S^n$ 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:
where $deg(f)$ is the degree of $f$.
(Møller-Raussen 85, Example 2.5, Cohen-Voronov 05, Lemma 5.3.5)
Here Prop. and Prop. are two aspects of the same situation:
For $n = 2k+1$ an odd number the rational Euler class (2) of the spherical fibration is the class of the rational classifying map to the shift of $S^{2k+1}$ in (6);
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 (6).
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)
Last revised on April 28, 2019 at 13:31:29. See the history of this page for a list of all contributions to it.