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Sullivan model of a spherical fibration

Contents

Contents

Idea

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

Properties

Rational Euler- and Pontryagin-class

Proposition

Let nn \in \mathbb{N} be a natural number, n1n \geq 1, let

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

be a spherical fibration of topological spaces such that XX admits a Sullivan model A XdgcAlgA_X \in dgcAlg. Then a Sullivan model A EA_E for the total space EE is of the following form:

nn odd

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

(2)A E=A X[ω 2k+1]/(dω 2k+1=c 2k+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+2A X c_{2k+2} \in A_X

being the rational Euler class of the spherical fibration.

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

[c 2k]=χ [c_{2k}] = \chi

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

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


nn even

If n=2kn = 2k is an even number, then the Sullivan model A EA_E for a rank-2k2k spherical fibration over some XX with Sullivan model A XA_X is

(5)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} evaluates 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 (5) 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(\widehat V) of the corresponding (…) rank reduction V^\widehat V of VV:

    [ω 2k]=12χ(V^)H 2k(X,) \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 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) \,.

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 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 (2) 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. 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 BAut(S n)B Aut(S^n) of the automorphism ∞-group Aut(S n)Maps(S n,S n)Aut(S^n) \hookrightarrow Maps(S^n, S^n) inside the mapping space from S nS^n to itself, which is those connected components corresponding to degree ±1\pm 1

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

Hence the spherical fibration is given by the homotopy pullback

(6)S n E S nAut(S n) (pb) X c BAut(S n) \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 cc.

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

Proposition

Let nn \in \mathbb{N} be a natural number and fcolonS nS nfcolon S^n \to S^n a continuous function from the n-sphere to itself. Then the connected component Maps f(S n,S n)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(S n,S n) {S n×S n1 | neven,deg(f)=0 S 2n1 | neven,deg(f)0 S n | nodd 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)deg(f) is the degree of ff.

(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+1n = 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+1S^{2k+1} in (7);

for n=2kn = 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 4k1S^{4k-1} in (7).

Examples of Sullivan models in rational homotopy theory:

References

Last revised on October 12, 2019 at 05:55:45. See the history of this page for a list of all contributions to it.