Novikov conjecture

In mathematics , the Novikov conjecture is a generally open conjecture about the topology of differentiable manifolds with a fundamental group that has been proven for numerous groups . ${\displaystyle \pi }$ ${\displaystyle \pi }$

It has numerous applications in surgical theory in the classification of differential structures for a given type of homotopy .

It makes a statement about the homotopy invariance of certain combinations of rational Pontryagin classes . Rational Pontrjagin classes are invariants of differentiable manifolds, which, according to Novikov's theorem, are invariant under homeomorphisms, but in general not invariant under homotopy equivalences . For the L-class formed from the Pontrjagin classes , the homotopy-invariant signature is according to Hirzebruch's signature theorem . The Novikov conjecture gives (depending on the fundamental group) further homotopy-invariant combinations. It is assumed that all homotopy-invariant combinations of rational Pontryagin classes result from the higher signatures considered in the Novikov conjecture. ${\displaystyle \langle L(M),\left[M\right]\rangle }$

It would follow from the Baum-Connes conjecture or the Borel conjecture .

Formulation of the presumption

Let be a closed , orientable , -dimensional differentiable manifold, its fundamental group and its classifying mapping . At any cohomology to define a higher signature by ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle \pi :=\pi _{1}M}$ ${\displaystyle f\colon M\to B\pi }$ ${\displaystyle \alpha \in H^{*}(B\pi ;\mathbb {Q} )}$ ${\displaystyle \sigma _{\alpha }(M)}$

${\displaystyle \sigma _{\alpha }(M):=\langle L(M)\cup f^{*}\alpha ,\left[M\right]\rangle }$,

where the L class of , the cup product , the fundamental class and the Kronecker pairing . ${\displaystyle L(M)}$ ${\displaystyle M}$ ${\displaystyle \cup }$ ${\displaystyle \left[M\right]}$ ${\displaystyle \langle .,.\rangle }$

The Novikov conjecture says that for any given the higher signature is a homotopy invariant of closed, orientable manifolds with a fundamental group , i.e. H. homotopy-equivalent , closed, orientable manifolds have the same higher signatures. ${\displaystyle \alpha \in H^{*}(B\pi ;\mathbb {Q} )}$ ${\displaystyle \pi }$

Proven cases

The Novikov conjecture is said to be proven for a group if it has been proved for all manifolds with a fundamental group . ${\displaystyle \pi }$ ${\displaystyle \pi }$

• Novikov proved his conjecture for Abelian groups .
• Kasparov proved by KK-theory which Novikov's conjecture for groups which have a real effect as isometrics a simply connected manifold of non-positive curvature have, especially so discrete subgroups of a Lie group with finitely many connected components .
• Connes and Moscovici proved the Novikov conjecture for Gromov hyperbolic groups . The surjectivity of homomorphism from bounded cohomology into group cohomology used in the proof was proved by Mineyev .
• Higson and Kasparow proved the Novikov conjecture for groups that actually have an effect as isometrics of the Hilbert space , especially for indirect groups .
• Yu proved the Novikov conjecture for groups of finite asymptotic dimension and, more generally, for groups that can roughly be embedded in the Hilbert space. The latter applies to all linear groups and to subgroups of . ${\displaystyle Out(F_{n})}$
• Hamenstädt proved that mapping class groups are exact (“boundary amenable”), from which the Novikov conjecture follows for these and all their subgroups.

literature

• SP Novikov: Analogues hermitiens de la K-théorie. In: Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 39-45. Gauthier-Villars, Paris (1971)
• S. Ferry , A. Ranicki , J. Rosenberg : A history and survey of the Novikov conjecture. In: Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser., Vol. 226, pp. 7-66. Cambridge Univ. Press, Cambridge (1995).
• M. Kreck , W. Lück : The Novikov conjecture, Geometry and Algebra , Oberwolfach Seminars, vol. 33.Birkhäuser Verlag, Basel (2005)
• J. Rosenberg: Novikov's conjecture , "Open Problems in Mathematics", JF Nash, Jr., and M. Th. Rassias, eds, Springer, 2016, pp. 377-402
• G. Yu: The Novikov conjecture , Russian Mathematical Surveys 2019

Web links

• SP Novikov: Novikov conjecture (Scholarpedia)
• Novikov Conjecture (Manifold Atlas)

Individual evidence

1. ^ G. Kasparov: Equivariant KK theory and the Novikov conjecture. Invent. Math. 91 (1988) no. 1, 147-201.
2. ↑ A. Connes, H. Moscovici: Cyclic cohomology, the Novikov conjecture and hyperbolic groups. Topology, 29, no. 3, 345-388 (1990).
3. ↑ I. Mineyev: Straightening and bounded cohomology of hyperbolic groups. GAFA, Geom. Funct. Anal. 11: 807-839 (2001).
4. ↑ N. Higson, G. Kasparov: E-theory and KK-theory for groups which act properly and isometrically on Hilbert space. Invent. Math. 144 (2001), no. 1, 23-74.
5. ^ G. Yu: The Novikov conjecture for groups with finite asymptotic dimension. Ann. of Math. (2) 147 (1998) no. 2, 325-355.
6. ^ G. Yu: The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math. 139 (1): 201-240 (2000).
7. ^ E. Guentner , N. Higson, S. Weinberger : The Novikov Conjecture for Linear Groups. Publ. Math. Inst. Hautes Etudes Sci. No. 101: 243-268 (2005).
8. ↑ M. Bestvina , K. Bromberg , K. Fujiwara : Constructing group actions on quasi-trees and applications to mapping class groups. Publ. Math. Inst. Hautes Etudes Sci. 122 (2015), 1-64.
9. ↑ U. Hamenstädt: Geometry of the mapping class groups. I. Boundary amenability. Invent. Math. 175 (2009), no. 3, 545-609.