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An analytic proof of Novikov’s theorem on rational Pontrjagin classes. (English) Zbl 0531.58045
The authors give an analytic proof of topological invariance of rational Pontryagin classes of a compact smooth manifold. The proof is a consequence of the second author’s results on signature operators on Lipschitz manifolds (see the preceding review) combined with the first author’s theorem [Geometric topology, Proc. Conf., Athens/Ga. 1977, 543- 555 (1979; Zbl 0478.57007)] on existence of an essentially unique Lipschitz structure on any topological manifold of dimension 4.
Reviewer: J.Dodziuk

##### MSC:
 58J20 Index theory and related fixed point theorems (PDE on manifolds) 57R20 Characteristic classes and numbers (differential topology) 57N65 Algebraic topology of manifolds 58J22 Exotic index theories (PDE on manifolds) 57Q99 PL-topology 47A53 (Semi-)Fredholm operators; index theories
##### References:
 [1] M. F. Atiyah, I. M. Singer, The Index of Elliptic Operators, Part III:Annals of Math.,87 (1968), 546–604. · Zbl 0164.24301 · doi:10.2307/1970717 [2] J. W. Milnor, J. D. Stasheff,Characteristic Classes, Princeton, 1974. [3] S. P. Novikov, Topological Invariance of rational Pontrjagin Classes,Doklady A.N.S.S.S.R.,163 (2) (1965), 921–923. [4] I. M. Singer, Future Extension of Index Theory and Elliptic Operators, in Prospects in Mathematics,Annals of Math. Studies,70 (1971), 171–185. [5] D. Sullivan, Hyperbolic Geometry and Homeomorphisms, inGeometric Topology, Proc. Georgia Topology Conf. Athens, Georgia, 1977, 543–555, ed. J. C. Cantrell, Academic Press, 1979. [6] N. Teleman, The index of Signature Operators on Lipschitz Manifolds,Publ. Math. I.H.E.S., this volume, 39–78. [7] P. Tukia, J. Väisälä, Lipschitz and quasiconformal approximation and extension,Ann. Acad. Sci. Fenn. Ser. A,16 (1981), 303–342. [8] ———-, Quasiconformal extension from dimensionn ton + 1,Annals of Math.,115 (1982), 331–348. · Zbl 0484.30017 · doi:10.2307/1971394