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Integrality relations on smooth manifolds. (English) Zbl 0356.57006

F. Hirzebruch [Topological methods in algebraic geometry. Berlin etc.: Springer-Verlag (1966; Zbl 0138.42001)] defined multiplicative sequences \(K(\xi)\) \((\xi\) a vector bundle over a complex \(X)\) with values in the cohomology \(H(X;A)\) where \(A\) is a fixed ring with unit. Let \(M\) be a smooth closed manifold with tangent bundle \(\tau_M\). Then one defines \(K(M)=K(\tau_M)\), whenever this is possible. Next, consider a subring \(S\subset A\), a fixed space \(Y\) and a multiplicative sequence \(K\). Put \(S(Y,K)\) equal to all classes \(\theta \in H(Y,A)\) such that for any \(M\) as above and any \(f: M\to Y\) \(\{f^*\theta \cdot K(M)\}[M]\in S\subset A\), where \([M]\) is the homology orientation class of \(M\). In this paper, \(S(Y,K)\) is calculated for \(Y\) either a classifying space or the Thom complex of the universal bundle over a classifying space. Characteristic classes for multiplicative sequences are studied and used.
In one of the appendices, the theory is carried over to PL, and topological manifolds. In another one, an interpretation via bordism is given.

MSC:

57R20 Characteristic classes and numbers in differential topology
57N65 Algebraic topology of manifolds
55R40 Homology of classifying spaces and characteristic classes in algebraic topology

Citations:

Zbl 0138.42001
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