nLab Leray-Hirsch theorem

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

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

In ordinary cohomology

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

If there exists

• a finite set of elements

  $$\alpha_i \in H^\bullet(Y,R)\;,\;\;\;\; i \in \{1, 2, \cdots, n\}$$

  in the ordinary cohomology of $Y$ with coefficients in $R$,

such that

• for each point $x \in X$ the restriction (pullback along $\iota$) of the $\alpha_i$ to the fiber $F_x \hookrightarrow Y$

  1. is $R$-linearly independent

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

     $$H^\bullet(F;\, R) \simeq R \langle \iota_x^\ast \alpha_1, \cdots, \iota_x^\ast \alpha_n \rangle \;\;\;\; \in R Mod$$

  (i.e. a free module over $R$)

then:

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

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

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

   via the isomorphism

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

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

   $$\underset{i,j}{\sum} c_i \otimes \iota^\ast(\alpha_j) \mapsto \underset{i,j}{\sum} p^\ast(c_i) \cup \alpha_j \,.$$

In generalized cohomology

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

Let $E$ be a multiplicative Whitehead-generalized cohomology theory and write

• $E^\bullet(-)$ for its cohomology rings;

• $E_{-\bullet} \;\coloneqq \; E^\bullet(\ast)$ for its ground ring

If there exists

• a finite set of elements

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

  in the ordinary cohomology of the total space $Y$,

such that

• for each point $x \in X$ the restriction (pullback along $\iota$) of the $\alpha_i$ to the fiber $F_x \hookrightarrow Y$

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

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

     (3)$$E_{-\bullet}(F;) \simeq E_{-\bullet} \langle \iota_x^\ast \alpha_1, \cdots, \iota_x^\ast \alpha_n \rangle \;\;\;\; \in E_{-\bullet} Mod$$

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

then:

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

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

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

   via the isomorphism

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

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

   $$\underset{i,j}{\sum} c_i \otimes \iota^\ast(\alpha_j) \mapsto \underset{i,j}{\sum} p^\ast(c_i) \cup \alpha_j \,.$$

Examples

Complex-oriented cohomology of the twistor fibration

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

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

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

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

whence the cohomology group is

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

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

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

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

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

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

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

Related concepts

• Thom isomorphism

References

For ordinary cohomology

Review of the theorem for ordinary cohomology:

• Alan Hatcher, Algebraic topology, 2002, theorem 4D.1 on p. 432 (pdf)

• Johannes Ebert, section 2.3 of A lecture course on Cobordism Theory, 2012 (pdf)

For generalized cohomology

Discussion for Whitehead-generalized multiplicative cohomology theories:

• Pierre Conner, Edwin Floyd, Theorem 7.4 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28, Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• Dai Tamaki, Akira Kono, Section 3.1 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)