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The total Chern class is not a map of multiplicative cohomology theories. (English) Zbl 0804.55005
For any topological space \(X\), the set \(\prod_{i\geq 0} H^{2i} (X;\mathbb{Z})\) can be made into a commutative ring, denoted \(M(X)\), such that the augmented total Chern class \(c: K^ 0(X)\to M(X)\) which sends a vector bundle \(E\) to \(( \text{rk } E, c_ 1(E), c_ 2(E),\dots)\) becomes a homomorphism of rings. Graeme Segal asked [Quart. J. Math., Oxford, II. Ser. 26, 289-293 (1975; Zbl 0321.55017)] if there exists a multiplicative cohomology theory \(M^*\) with \(M^ 0(X)\) naturally isomorphic to \(M(X)\). This question is answered in the negative in the present paper. The proof is very short: The author considers the two-sheeted cover \(f: BH\to BG\) with \(G= \mathbb{Z}/2\oplus \mathbb{Z}/2\) and \(H=\mathbb{Z}/2 \subset G\) and shows that there cannot exist a transfer \(f_ *: M^ 0 (BG)\to M^ 0(BG)\) which is a homomorphism of modules over \(M^ 0(BG)\).

MSC:
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
55R12 Transfer for fiber spaces and bundles in algebraic topology
Citations:
Zbl 0321.55017
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References:
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[2] Adams, J.F.: Infinite Loop Spaces. (Ann. Math. Stud., vol. 90) Princeton: Princeton University Press 1978 · Zbl 0398.55008
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[4] Grothendieck, A.: La théorie des classes de Chern. Bull. Soc. Math. Fr.86, 137–154 (1958) · Zbl 0091.33201
[5] Segal, G.: The multiplicative group of classical cohomology. Q. J. Math., Oxf.26, 289–293 (1975) · Zbl 0321.55017
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