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Leray-Hirsch theorem

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cohomology

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Idea

The Leray-Hirsch theorem states sufficient fiberwise condition for the ordinary cohomology of the total space of a fiber bundle with coefficients in a commutative ring to be free module over the cohomology ring of the base space.

An important consequence is the Thom isomorphism.

Statement

Let

(1)F ι Y p X \array{ F &\stackrel{\iota}{\hookrightarrow}& Y \\ && \downarrow^{\mathrlap{p}} \\ && X }

be an FF-fiber bundle (in Top) of topological spaces that admit the structure of finite CW-complexes.

In ordinary cohomology

Let RR be a commutative ring and write H (;R)H^\bullet(-;\,R) for the cohomology rings of ordinary cohomology with coefficients in RR.

If there exists

such that

  • for each point xXx \in X the restriction (pullback along ι\iota) of the α i\alpha_i to the fiber F xYF_x \hookrightarrow Y

    1. is RR-linearly independent

    2. their RR-linear span is isomorphic to the cohomology group H (F,R)H^\bullet(F,R) of the fiber

      H (F;R)Rι x *α 1,,ι x *α nRMod H^\bullet(F;\, R) \simeq R \langle \iota_x^\ast \alpha_1, \cdots, \iota_x^\ast \alpha_n \rangle \;\;\;\; \in R Mod

    (i.e. a free module over RR)

then:

  1. the {α 1,,α n}\{\alpha_1, \cdots, \alpha_n\} themselves are H (X;R)H^\bullet(X;\,R)-linearly independent,

  2. their H (X;R)H^\bullet(X;\,R)-linear span gives the cohomology group of the total space YY:

    H (Y;R)H (X;R)α 1,,α nH (X;R)Mod, H^\bullet(Y;\,R) \;\simeq\; H^\bullet(X;\,R) \langle \alpha_1, \cdots,\alpha_n \rangle \;\;\;\;\; \in \; H^\bullet(X;\, R) Mod \,,

    via the isomorphism

    H (X;R) RH (F;R)H (Y;R) H^\bullet(X;\, R) \otimes_R H^\bullet(F;\, R) \stackrel{\simeq}{\longrightarrow} H^\bullet(Y;\, R)

    given by pulling back classes from the base space and there forming their cup product with these generators on the total space:

    i,jc iι *(α j)i,jp *(c i)α j. \underset{i,j}{\sum} c_i \otimes \iota^\ast(\alpha_j) \mapsto \underset{i,j}{\sum} p^\ast(c_i) \cup \alpha_j \,.


In generalized cohomology

The statement generalizes verbatim from ordinary cohomology to any multiplicative Whitehead-generalized cohomology theory EE (Conner-Floyd 66, theorem 7,4, attributed there to Albrecht Dold, review in Tamaki-Kono 06, Section 3.1):

Let EE be a multiplicative Whitehead-generalized cohomology theory and write

If there exists

  • a finite set of elements

    (2)α iE (Y),i{1,2,,n} \alpha_i \;\in\; E^\bullet(Y)\;,\;\;\;\; i \in \{1, 2, \cdots, n\}

    in the ordinary cohomology of the total space YY,

such that

  • for each point xXx \in X the restriction (pullback along ι\iota) of the α i\alpha_i to the fiber F xYF_x \hookrightarrow Y

    1. is E E_{-\bullet}-linearly independent

    2. their E E_{-\bullet}-linear span is isomorphic to the cohomology group E (F)E^\bullet(F) of the fiber

      (3)E (F;)E ι x *α 1,,ι x *α nE Mod E_{-\bullet}(F;) \simeq E_{-\bullet} \langle \iota_x^\ast \alpha_1, \cdots, \iota_x^\ast \alpha_n \rangle \;\;\;\; \in E_{-\bullet} Mod

    (i.e. a free module over E E_{-\bullet})

then:

  1. the {α 1,,α n}\{\alpha_1, \cdots, \alpha_n\} themselves are E (X)E^\bullet(X)-linearly independent,

  2. their E (X)E^\bullet(X)-linear span gives the cohomology group of the total space YY:

    E (Y)E (X)α 1,,α nE (X)Mod, E^\bullet(Y) \;\simeq\; E^\bullet(X) \langle \alpha_1, \cdots,\alpha_n \rangle \;\;\;\;\; \in \; E^\bullet(X) Mod \,,

    via the isomorphism

    E (X) E E (F)E (Y) E^\bullet(X) \otimes_{E_{-\bullet}} E^\bullet(F) \stackrel{\simeq}{\longrightarrow} E^\bullet(Y)

    given by pulling back classes from the base space and there forming their cup product with these generators on the total space:

    i,jc iι *(α j)i,jp *(c i)α j. \underset{i,j}{\sum} c_i \otimes \iota^\ast(\alpha_j) \mapsto \underset{i,j}{\sum} p^\ast(c_i) \cup \alpha_j \,.

Examples

Complex-oriented cohomology of the twistor fibration

Let EE be a Whitehead-generalized cohomology theory equipped with complex orientation in the form of a first Conner-Floyd-Chern class

c 1 EE˜ 2(P )E˜ 2(P n). c^E_1 \;\in\; {\widetilde E}{}^2\big( \mathbb{C}P^\infty \big) \longrightarrow {\widetilde E}{}^2\big( \mathbb{C}P^n \big) \,.

Then, for nn \in \mathbb{N}, the EE-cohomology ring of the complex projective space P n\mathbb{C}P^n is (see there)

E (P n)E [c 1 E]/(c 1 E) n+1E Algebras, E^\bullet \big( \mathbb{C}P^n \big) \;\simeq\; E_{-\bullet} \big[ c^E_1 \big] \big/ \big( c^E_1 \big)^{n+1} \;\;\; \in \; E_{-\bullet} Algebras \,,

whence the cohomology group is

(4)E (P n)E 1,c 1 E,(c 1 E) 2,,(c 1 E) nE Modules. E^\bullet \big( \mathbb{C}P^n \big) \;\simeq\; E_{-\bullet} \big\langle 1,\, c^E_1,\, \big(c^E_1\big)^2,\, \cdots ,\, \big(c^E_1\big)^n \big\rangle \;\;\; \in \; E_{-\bullet} Modules \,.

For each n=2k+1n = 2 k + 1 these are Riemann sphere ×/ ×=P 1\mathbb{H}^\times/\mathbb{C}^\times = \mathbb{C}P^1-fiber bundles

P 1 2k+1 = ( 2k+2{0})/ × v ×v × P k = ( k+1{0})/ × \array{ \mathbb{C}P^1 & \longrightarrow & \mathbb{C}^{2k+1} & = & \big( \mathbb{C}^{2k+2} \setminus \{0\} \big) \big/ \mathbb{C}^\times \\ && \big\downarrow && \big\downarrow {}^{ \mathrlap{ v \cdot \mathbb{C}^\times \mapsto v \cdot \mathbb{H}^\times } } \\ && \mathbb{H}P^k & = & \big( \mathbb{H}^{k+1} \setminus \{0\} \big) \big/ \mathbb{H}^\times }

over quaternionic projective space P k\mathbb{H}P^{k}, whose fiber-inclusion is (homotopic to) the canonical inclusion P 1P n\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^n (see there).

E.g. for k=1k = 1 this is also known as the twistor fibration; while for k=k = \infty this is the fibration of classifying spaces

SU(2)/U(1) BU(1) B(zdiag(z,z *)) BSU(2). \array{ SU(2)/\mathrm{U}(1) &\longrightarrow& B \mathrm{U}(1) \\ && \big\downarrow {}^{\mathrlap{ B\big( z \mapsto diag(z,z^\ast) \big) }} \\ && B SU(2) \,. }

Therefore, by (4), the assumption (3) of the EE-Leray-Hirsch theorem (above) is met if we take the classes (2) to be the cup powers (c 1 E) n(c^E_1)^n. Now the EE-Leray-Hirsch theorem says that:

References

For ordinary cohomology

Review of the theorem for ordinary cohomology:

For generalized cohomology

Discussion for Whitehead-generalized multiplicative cohomology theories:

Last revised on January 23, 2021 at 04:28:23. See the history of this page for a list of all contributions to it.