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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
Zbl 0321.55017
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##### References:
 [1] Adams, J.F.: Stable Homotopy and Generalized Cohomology, Chicago: University of Chicago Press 1974 · Zbl 0309.55016 [2] Adams, J.F.: Infinite Loop Spaces. (Ann. Math. Stud., vol. 90) Princeton: Princeton University Press 1978 · Zbl 0398.55008 [3] Boyer, C.P., Lawson, H.B., Lima-Filho, P., Mann, B.M., Michelsohn, M.-L.: Algebraic cycles and infinite loop spaces. Invent. Math. (to appear). · Zbl 0797.55006 [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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