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onefishtwofish
onefishtwofish
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Variant of Leray-Hirsch for multiplicativecomplex-oriented cohomology theory

I am interested in a variant of the Leray-Hirsch theorem. Let $\mathbb{E}^*$ be a multiplicativecomplex-oriented cohomology theory and let $\mathbb{E}_*$ the coefficient ring. Suppose that $F \xrightarrow{i} P \to B$ is a fiber bundle with $F,P,B$ finite CW complexes. I assume that $B$ has only even dimensional cells (though I expect the claim to be true more generally). Suppose I have a splitting $\mathbb{E}^*(F) \to \mathbb{E}^*(P)$ of the restriction $i^*: \mathbb{E}^*(P) \to \mathbb{E}^*(F).$

Is the resulting map $$\mathbb{E}^*(F)\otimes_{\mathbb{E}_{*}}\mathbb{E}^*(B) \to \mathbb{E}^*(P) $$ an isomorphism? I suspect that if this is true, it can be proven with standard spectral sequence arguments. Is there a reference for the result in this form?

Variant of Leray-Hirsch for multiplicative cohomology theory

I am interested in a variant of the Leray-Hirsch theorem. Let $\mathbb{E}^*$ be a multiplicative cohomology theory and let $\mathbb{E}_*$ the coefficient ring. Suppose that $F \xrightarrow{i} P \to B$ is a fiber bundle with $F,P,B$ finite CW complexes. I assume that $B$ has only even dimensional cells (though I expect the claim to be true more generally). Suppose I have a splitting $\mathbb{E}^*(F) \to \mathbb{E}^*(P)$ of the restriction $i^*: \mathbb{E}^*(P) \to \mathbb{E}^*(F).$

Is the resulting map $$\mathbb{E}^*(F)\otimes_{\mathbb{E}_{*}}\mathbb{E}^*(B) \to \mathbb{E}^*(P) $$ an isomorphism? I suspect that if this is true, it can be proven with standard spectral sequence arguments. Is there a reference for the result in this form?

Variant of Leray-Hirsch for complex-oriented cohomology theory

I am interested in a variant of the Leray-Hirsch theorem. Let $\mathbb{E}^*$ be a complex-oriented cohomology theory and let $\mathbb{E}_*$ the coefficient ring. Suppose that $F \xrightarrow{i} P \to B$ is a fiber bundle with $F,P,B$ finite CW complexes. I assume that $B$ has only even dimensional cells (though I expect the claim to be true more generally). Suppose I have a splitting $\mathbb{E}^*(F) \to \mathbb{E}^*(P)$ of the restriction $i^*: \mathbb{E}^*(P) \to \mathbb{E}^*(F).$

Is the resulting map $$\mathbb{E}^*(F)\otimes_{\mathbb{E}_{*}}\mathbb{E}^*(B) \to \mathbb{E}^*(P) $$ an isomorphism? I suspect that if this is true, it can be proven with standard spectral sequence arguments. Is there a reference for the result in this form?

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onefishtwofish
onefishtwofish
  • 959
  • 4
  • 7

Variant of Leray-Hirsch for multiplicative cohomology theory

I am interested in a variant of the Leray-Hirsch theorem. Let $\mathbb{E}^*$ be a multiplicative cohomology theory and let $\mathbb{E}_*$ the coefficient ring. Suppose that $F \xrightarrow{i} P \to B$ is a fiber bundle with $F,P,B$ finite CW complexes. I assume that $B$ has only even dimensional cells (though I expect the claim to be true more generally). Suppose I have a splitting $\mathbb{E}^*(F) \to \mathbb{E}^*(P)$ of the restriction $i^*: \mathbb{E}^*(P) \to \mathbb{E}^*(F).$

Is the resulting map $$\mathbb{E}^*(F)\otimes_{\mathbb{E}_{*}}\mathbb{E}^*(B) \to \mathbb{E}^*(P) $$ an isomorphism? I suspect that if this is true, it can be proven with standard spectral sequence arguments. Is there a reference for the result in this form?