Leray–Hirsch theorem

Leray–Hirsch theorem In mathematics, the Leray–Hirsch theorem[1] is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s. It can be thought of as a mild generalization of the Künneth formula, which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors. It is a very special case of the Leray spectral sequence.

Contents 1 Statement 1.1 Setup 1.2 The Leray–Hirsch isomorphism 1.3 Statement in coordinates 2 Notes Statement Setup Let {displaystyle pi colon Elongrightarrow B} be a fibre bundle with fibre {displaystyle F} . Assume that for each degree {displaystyle p} , the singular cohomology rational vector space {displaystyle H^{p}(F)=H^{p}(F;mathbb {Q} )} is finite-dimensional, and that the inclusion {displaystyle iota colon Flongrightarrow E} induces a surjection in rational cohomology {displaystyle iota ^{*}colon H^{*}(E)longrightarrow H^{*}(F)} .

Consider a section of this surjection {displaystyle scolon H^{*}(F)longrightarrow H^{*}(E)} , by definition, this map satisfies {displaystyle iota ^{*}circ s=mathrm {Id} } . The Leray–Hirsch isomorphism The Leray–Hirsch theorem states that the linear map {displaystyle {begin{array}{ccc}H^{*}(F)otimes H^{*}(B)&longrightarrow &H^{*}(E)\alpha otimes beta &longmapsto &s(alpha )smallsmile pi ^{*}(beta )end{array}}} is an isomorphism of {displaystyle H^{*}(B)} -modules.

Statement in coordinates In other words, if for every {displaystyle p} , there exist classes {displaystyle c_{1,p},ldots ,c_{m_{p},p}in H^{p}(E)} that restrict, on each fiber {displaystyle F} , to a basis of the cohomology in degree {displaystyle p} , the map given below is then an isomorphism of {displaystyle H^{*}(B)} modules.

{displaystyle {begin{array}{ccc}H^{*}(F)otimes H^{*}(B)&longrightarrow &H^{*}(E)\sum _{i,j,k}a_{i,j,k}iota ^{*}(c_{i,j})otimes b_{k}&longmapsto &sum _{i,j,k}a_{i,j,k}c_{i,j}wedge pi ^{*}(b_{k})end{array}}} where {displaystyle {b_{k}}} is a basis for {displaystyle H^{*}(B)} and thus, induces a basis {displaystyle {iota ^{*}(c_{i,j})otimes b_{k}}} for {displaystyle H^{*}(F)otimes H^{*}(B).} Notes ^ Hatcher, Allen (2002), Algebraic Topology (PDF), Cambridge: Cambridge University Press, ISBN 0-521-79160-X Categories: Fiber bundlesTheorems in algebraic topology