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Let be a closed oriented -dimensional smooth manifold with a map for some discrete group and let be a rational cohomology class. The higher signature of defined by is the rational number
where is the Hirzebruch L-class of . Let be a homotopy equivalence of closed oriented smooth manifolds. The Novikov conjecture states
for all , , and for all homotopy equivalences . For the trivial group the conjecture is true by Hirzebruch's signature theorem.
The original 1969 statement of the Novikov conjecture may be found in [Novikov1970] and [Novikov1970a]: a history and survey including the original statement in Russian with a translation into English may be found in [Ferry&Ranicki&Rosenberg1995b]. In the last 40 years the Novikov conjecture and the related conjectures of Borel and Farrell-Jones have been the subject of a great deal of research. In [Novikov2010] Novikov described how he came to formulate the conjecture.
 2 References
- [Davis2000] J. F. Davis, Manifold aspects of the Novikov conjecture, Surveys on surgery theory, Vol. 1, Princeton Univ. Press (2000), 195–224. MR1747536 (2002a:57037) Zbl 0948.57001
- [Ferry&Ranicki&Rosenberg1995] S. C. Ferry, A. A. Ranicki and J. Rosenberg, Novikov conjectures, index theorems and rigidity. Vol. 1. London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, Cambridge, 1995. MR1388294 (96m:57002) Zbl 0829.00027
- [Ferry&Ranicki&Rosenberg1995a] S. C. Ferry, A. A. Ranicki and J. Rosenberg, Novikov conjectures, index theorems and rigidity. Vol. 2, London Math. Soc. Lecture Note Ser., 227, Cambridge Univ. Press, Cambridge, 1995. MR1388306 (96m:57003) Zbl 0829.00028
- [Ferry&Ranicki&Rosenberg1995b] S. C. Ferry, A. A. Ranicki and J. Rosenburg, A history and survey of the Novikov conjecture in Ferry&Ranicki&Rosenberg1995 7–66, London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, Cambridge, 1995. MR1388295 (97f:57036) Zbl 0954.57018
- [Novikov1970] S. P. Novikov, Algebraic construction and properties of Hermitian analogs of -theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. I. II, Math. USSR-Izv. 4 (1970), 257–292; ibid. 4 (1970), 479–505; translated from Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 253–288; ibid. 34 (1970), 475. MR0292913 (45 #1994) Zbl 0216.45003 Zbl 0233.57009
- [Novikov1970a] S. P. Novikov, Pontrjagin classes, the fundamental group and some problems of stable algebra 1970 Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham) pp. 147–155 Springer, New York. MR0268907 (42 #3804)
- [Novikov2010] S. P. Novikov, Scholarpedia article on the Novikov Conjecture (2010)
- [Ranicki1995] A. A. Ranicki On the Novikov conjecture in [Ferry&Ranicki&Rosenburg1995], 272–337, London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, Cambridge, 1995. MR1388304 (97d:57045)
- The Wikipedia page about the Novikov conjecture.