Novikov Conjecture
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== Introduction == | == Introduction == | ||
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| − | Let $M$ be a closed oriented $n$-dimensional smooth manifold with a map $f : M \to B\pi$ for some discrete group $\pi$ and let $\alpha \in H^{n-4*}(B\pi; \Qq)$ be a rational cohomology class. The higher signature of $M$ | + | Let $M$ be a closed oriented $n$-dimensional smooth manifold with a map $f : M \to B\pi$ for some discrete group $\pi$ and let $\alpha \in H^{n-4*}(B\pi; \Qq)$ be a rational cohomology class. The ''higher signature'' of $M$ defined by $(f, \alpha)$ is the rational number |
$$ \sigma_\alpha(M, f) = \langle L_M \cup f^*\alpha, [M] \rangle \in \Qq $$ | $$ \sigma_\alpha(M, f) = \langle L_M \cup f^*\alpha, [M] \rangle \in \Qq $$ | ||
where $L_M \in H^{4*}(M; \Qq)$ is the Hirzebruch L-class of $M$. Let $h: N \to M$ be a homotopy equivalence of closed oriented smooth manifolds. The '''Novikov conjecture''' states | where $L_M \in H^{4*}(M; \Qq)$ is the Hirzebruch L-class of $M$. Let $h: N \to M$ be a homotopy equivalence of closed oriented smooth manifolds. The '''Novikov conjecture''' states | ||
| − | $$ \sigma_\alpha(N, f \circ h) = \sigma_\alpha(M, f).$$ | + | $$ \sigma_\alpha(N, f \circ h) = \sigma_\alpha(M, f)$$ |
| + | for all $\pi$, $f$, $\alpha$ and $h : N \to M$. For the trivial group $\pi$ the conjecture is true by Hirzebruch's signature theorem. | ||
The original 1969 statement of the Novikov conjecture may be found in \cite{Novikov1970} and \cite{Novikov1970a}: a history and survey including the original statement in Russian with a translation into English may be found in \cite{Ferry&Ranicki&Rosenberg1995b}. In the last 40 years the Novikov conjecture and the related conjectures of Borel and Farrell-Hsiang have been the subject of a great deal of research. In \cite{Novikov2010} Novikov described how he came to formulate the conjecture. | The original 1969 statement of the Novikov conjecture may be found in \cite{Novikov1970} and \cite{Novikov1970a}: a history and survey including the original statement in Russian with a translation into English may be found in \cite{Ferry&Ranicki&Rosenberg1995b}. In the last 40 years the Novikov conjecture and the related conjectures of Borel and Farrell-Hsiang have been the subject of a great deal of research. In \cite{Novikov2010} Novikov described how he came to formulate the conjecture. | ||
Revision as of 14:52, 6 October 2010
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Contents |
1 Introduction
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
![\displaystyle \sigma_\alpha(M, f) = \langle L_M \cup f^*\alpha, [M] \rangle \in \Qq](/images/math/9/1/4/9143205dcee6fec1293e14ba2f0abe7f.png)
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 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-Hsiang have been the subject of a great deal of research. In [Novikov2010] Novikov described how he came to formulate the conjecture.
2 Background
The following is a list of useful sources about the Novikov Conjecture.
[Ferry&Ranicki&Rosenberg1995a]
3 References
- [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) Zbl 0954.57017
4 External links
- The Wikipedia page about the Novikov conjecture.
