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Sur la non-invariance homotopique de la première classe de Chern. (On the homotopy non-invariance of the first Chern class). (French) Zbl 0592.55014

The variation of the first Chern classes of weakly complex manifolds in a given bordism class or homotopy type is studied. Especially, the non- invariance of the first Chern class under homotopy equivalence is shown. Let X be a finite polyhedron and \(\xi\) a complex vector bundle over X. We denote its i-th Chern class and Pontryagin class by \(c_ i(\xi)\) and \(p_ i(\xi)\) respectively. Let J(X) be the group of all stable fiber homotopy equivalence classes of the real vector bundles over X defined by Adams and J(\(\xi)\) the class in J(X) represented by \(\xi\). If two complex vector bundles \(\xi\) and \(\zeta\) are stably homotopy equivalent, then their first Chern classes \(c_ 1(\xi)\) and \(c_ 1(\zeta)\) are equal modulo 2, since the second Stiefel-Whitney class \(w_ 2\) is a homotopy invariant.
The following theorem shows that this is the only relation between the first Chern classes of two complex vector bundles of the same stable homotopy equivalence. Theorem 1. Let \(x\in H^ 2(X, {\mathbb{Z}})\) be a cohomology class of X with coefficients in the ring of integers \({\mathbb{Z}}\) such that its reduction modulo 2 is the second Stiefel-Whitney class \(w_ 2(\xi)\in H^ 2(X, {\mathbb{Z}}_ 2)\) of a complex vector bundle \(\xi\) over X. Then there exists a complex vector bundle \(\zeta\) over X such that (i) \(J(\xi)=J(\zeta)\) in J(X); (ii) \(p(\xi)=p(\zeta)\), where \(p=p_ 1+p_ 2+..\). is the total Pontryagin class; (iii) \(c_ 1(\xi)=x\). A USpin\({}^ c\)-manifold (resp. USpin-manifold) is a stable almost complex manifold (U-manifold) with a compatible \(Spin^ c\)-structure (Spin- structure). To a \(USpin^ c\)-manifold M a class \(x=x(M)\in H^ 2(M, {\mathbb{Z}})\) which is equal to \(c_ 1(M)\) modulo 2 is associated canonically. Every U-manifold M has a canonical \(USpin^ c\)-structure for which \(x(M)=c_ 1(M).\)
Similarly, every SU-manifold has a canonical USpin-structure. Theorem 2. Every \(USpin^ c\)-manifold is cobordant to a U-manifold with its canonical \(Uspin^ c\)-structure as a \(Spin^ c\)-manifold. Corollary. Every USpin-manifold is cobordant to an SU-manifold as a Spin-manifold. This is a generalization of a Stong’s result. Theorem 3. There exists a U-manifold N and a homotopy equivalence \(h: N\to M\) such that \(h*x(M)=c_ 1(N)\).
Reviewer: K.Shiraiwa

MSC:

55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57R20 Characteristic classes and numbers in differential topology
57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
57R85 Equivariant cobordism
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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