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| :''This page concerns mathematician Sergei Novikov's topology conjecture. For astrophysicist Igor Novikov's conjecture regarding time travel, see [[Novikov self-consistency principle]].
| | The author's title is Christy Brookins. For many years he's been residing in Alaska and he doesn't strategy on altering it. It's not a common thing but what I like performing is to climb but I don't have the time recently. Distributing production has been his occupation for some time.<br><br>my web site ... [http://cartoonkorea.com/ce002/1093612 clairvoyant psychic] |
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| The '''Novikov conjecture''' is one of the most important unsolved problems in [[topology]]. It is named for [[Sergei Novikov (mathematician)|Sergei Novikov]] who originally posed the conjecture in 1965.
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| The Novikov conjecture concerns the [[homotopy]] invariance of certain [[polynomial]]s in the [[Pontryagin class]]es of a [[manifold (mathematics)|manifold]], arising from the [[fundamental group]]. According to the Novikov conjecture, the ''higher signatures'', which are certain numerical invariants of smooth manifolds, are homotopy invariants.
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| The conjecture has been proved for finitely generated [[abelian groups]]. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture.
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| ==Precise formulation of the conjecture==
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| Let ''G'' be a [[discrete group]] and ''BG'' its [[classifying space]], which is a [[Eilenberg–Maclane space|K(G,1)]] and therefore unique up to [[homotopy equivalence]] as a CW complex. Let
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| :<math>f: M\rightarrow BG</math>
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| be a continuous map from a closed oriented ''n''-dimensional manifold ''M'' to ''BG'', and
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| :<math>x \in H^{n-4i} (BG;\mathbb{Q} ).</math>
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| Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the [[fundamental class]] [''M''], and known as a '''higher signature''':
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| :<math>\left\langle f^*(x) \cup L_i(M),[M] \right\rangle \in \mathbb{Q}</math>
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| where ''L''<sub>i</sub> is the ''i''<sup>th</sup> [[Hirzebruch polynomial]], or sometimes (less descriptively) as the ''i''<sup>th</sup> ''L''-polynomial. For each ''i'', this polynomial can be expressed in the Pontryagin classes of the manifold's tangent bundle. The '''Novikov conjecture''' states that the higher signature is an invariant of the oriented homotopy type of ''M'' for every such map ''f'' and every such class ''x'', in other words, if <math>h: M' \rightarrow M</math> is an orientation preserving homotopy equivalence, the higher signature associated to <math>f \circ h</math> is equal to that associated to ''f''.
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| ==Connection with the Borel conjecture==
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| The Novikov conjecture is equivalent to the rational injectivity of the [[assembly map]] in [[L-theory]]. The
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| [[Borel conjecture]] on the rigidity of aspherical manifolds is equivalent to the assembly map being an isomorphism.
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| ==References==
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| *{{Citation | last1=Davis | first1=James F. | editor1-last=Cappell | editor1-first=Sylvain | editor2-last=Ranicki | editor2-first=Andrew | editor3-last=Rosenberg | editor3-first=Jonathan | editorlink3=Jonathan Rosenberg (mathematician) | title=Surveys on surgery theory. Vol. 1 | url=http://www.indiana.edu/~jfdavis/papers/d_manc.pdf | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-04937-3; 978-0-691-04938-0 |mr=1747536 | year=2000 | chapter=Manifold aspects of the Novikov conjecture | pages=195–224}}
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| *[[J. Milnor]] and [[Jim Stasheff|J. D. Stasheff]], ''Characteristic Classes,'' Ann. Math. Stud. 76, Princeton (1974).
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| *S. P. Novikov, ''Algebraic construction and properties of Hermitian analogs of k-theory over rings with involution from the point of view of Hamiltonian formalism. Some applications to differential topology and to the theory of characteristic classes''. Izv.Akad.Nauk SSSR, v. 34, 1970 I N2, pp. 253-288; II: N3, pp. 475-500. English summary in Actes Congr. Intern. Math., v. 2, 1970, pp. 39-45.
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| ==External links==
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| *[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Novikov_Sergi.html Biography of Sergei Novikov]
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| *[http://www.math.umd.edu/~jmr/NC.html Novikov Conjecture Bibliography]
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| *[http://www.maths.ed.ac.uk/~aar/books/novikov1.pdf Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 1]
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| *[http://www.maths.ed.ac.uk/~aar/books/novikov2.pdf Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 2]
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| *[http://www.math.uni-muenster.de/u/lueck/publ/lueck/owsemfinalextract.pdf 2004 Oberwolfach Seminar notes on the Novikov Conjecture] (pdf)
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| *[http://www.scholarpedia.org/article/Novikov_conjecture Scholarpedia article by S.P. Novikov] (2010)
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| *[http://www.map.him.uni-bonn.de/Novikov_Conjecture The Novikov Conjecture] at the Manifold Atlas
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| [[Category:Geometric topology]]
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| [[Category:Homotopy theory]]
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| [[Category:Conjectures]]
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| [[Category:Surgery theory]]
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The author's title is Christy Brookins. For many years he's been residing in Alaska and he doesn't strategy on altering it. It's not a common thing but what I like performing is to climb but I don't have the time recently. Distributing production has been his occupation for some time.
my web site ... clairvoyant psychic