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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

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

(2)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:

(3)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

(4)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 (4) 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 (3) follows by this Prop..

Relation to rational mapping space of spheres

By general facts (see at ∞-action) a spherical fibration as in (5) 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

(5)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:

(6)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 (2) of the spherical fibration is the class of the rational classifying map to the shift of S 2k+1S^{2k+1} in (6);

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 (6).

References

Last revised on April 28, 2019 at 13:31:29. See the history of this page for a list of all contributions to it.