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A geometrical approach to characteristic numbers. (English) Zbl 0573.57017

Geometry today, Int. Conf., Rome 1984, Prog. Math. 60, 21-38 (1985).
[For the entire collection see Zbl 0563.00006.]
Let \(MG_ n\) be the bordism group of smooth n-manifolds with a G- structure on their stable tangent bundle and \(\{\) \(M\}\) one of its elements. The authors write \(<M>\) for \((t_ M)_*[M]\) in \(H_ nBG\), where \(t_ M\) is a classifying map for the given G-structure on M. There is the following classical type of result of Milnor \(\{M\}=0\) \(\Leftrightarrow\) \(<M>=0.\)
In this paper the authors show how one can obtain this kind of result by geometrical methods, which work at least in the cases \(G=U\), \(G=O\), \(G=SO\) (away from the prime 2). They prove the following theorem and corollaries. Theorem: \(\bar P:\) \(\overline{MSO_*(pt)}\to \overline{K_*(pt)}\) is a monomorphism. - Corollary: The torsion subgroup of \(MSO_*=MSO_*(pt)\) is 2-primary. - Corollary: Let M be a closed SO-manifold of dimension n. If all Pontryagin numbers of M vanish, then \(2^ 0\{M\}=0\) in \(MSO_ n\).
Reviewer: M.Rassias

MSC:

57R75 \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
57R20 Characteristic classes and numbers in differential topology
57R77 Complex cobordism (\(\mathrm{U}\)- and \(\mathrm{SU}\)-cobordism)
57R90 Other types of cobordism

Citations:

Zbl 0563.00006