Novikov conjecture

<h2 id="precise-formulation-of-the-conjecture">Precise formulation of the conjecture</h2>
<p>Let 
  
    
      
        G
      
    
    {\displaystyle G}
  
 be a <a href="/facts/Discrete_group/HIjkpJhS">discrete group</a> and 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
 its <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a>, which is an <a href="/facts/Eilenberg%25E2%2580%2593MacLane_space/mrux6oMJ">Eilenberg–MacLane space</a> of type 
  
    
      
        K
        (
        G
        ,
        1
        )
      
    
    {\displaystyle K(G,1)}
  
, and therefore unique up to <a href="/facts/Homotopy_equivalence/1nWrPxdj">homotopy equivalence</a> as a <a href="/facts/CW_complex/rN7buLMo">CW complex</a>. Let
</p>

f
        :
        M
        →
        B
        G
      
    
    {\displaystyle f\colon M\rightarrow BG}

<p>be a continuous map from a closed oriented 
  
    
      
        n
      
    
    {\displaystyle n}
  
-dimensional manifold 
  
    
      
        M
      
    
    {\displaystyle M}
  
 to 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
, and
</p>

x
        ∈
        
          H
          
            n
            −
            4
            i
          
        
        (
        B
        G
        ;
        
          Q
        
        )
        .
      
    
    {\displaystyle x\in H^{n-4i}(BG;\mathbb {Q} ).}

<p>Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the <a href="/facts/Fundamental_class/bqmdWEEc">fundamental class</a> 
  
    
      
        [
        M
        ]
      
    
    {\displaystyle [M]}
  
, and known as a higher signature:
</p>

⟨
          
            
              f
              
                ∗
              
            
            (
            x
            )
            ∪
            
              L
              
                i
              
            
            (
            M
            )
            ,
            [
            M
            ]
          
          ⟩
        
        ∈
        
          Q
        
      
    
    {\displaystyle \left\langle f^{*}(x)\cup L_{i}(M),[M]\right\rangle \in \mathbb {Q} }

<p>where 
  
    
      
        
          L
          
            i
          
        
      
    
    {\displaystyle L_{i}}
  
 is the 
  
    
      
        
          i
          
            
              t
              h
            
          
        
      
    
    {\displaystyle i^{\rm {th}}}
  
 <a href="/facts/Hirzebruch_polynomial/bUM7KXJ0">Hirzebruch polynomial</a>, or sometimes (less descriptively) as the 
  
    
      
        
          i
          
            
              t
              h
            
          
        
      
    
    {\displaystyle i^{\rm {th}}}
  
 
  
    
      
        L
      
    
    {\displaystyle L}
  
-polynomial. For each 
  
    
      
        i
      
    
    {\displaystyle i}
  
, this polynomial can be expressed in the Pontryagin classes of the manifold's <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a>. The Novikov conjecture states that the higher signature is an invariant of the oriented homotopy type of 
  
    
      
        M
      
    
    {\displaystyle M}
  
 for every such map 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and every such class 
  
    
      
        x
      
    
    {\displaystyle x}
  
, in other words, if 
  
    
      
        h
        :
        
          M
          ′
        
        →
        M
      
    
    {\displaystyle h\colon M'\rightarrow M}
  
 is an orientation preserving homotopy equivalence, the higher signature associated to 
  
    
      
        f
        ∘
        h
      
    
    {\displaystyle f\circ h}
  
 is equal to that associated to 
  
    
      
        f
      
    
    {\displaystyle f}
  
.
</p>
<h2 id="connection-with-the-borel-conjecture">Connection with the Borel conjecture</h2>
<p>The Novikov conjecture is equivalent to the rational injectivity of the <a href="/facts/Assembly_map/0KfeaLg4">assembly map</a> in <a href="/facts/L-theory/DjSy9S3p">L-theory</a>. The 
<a href="/facts/Borel_conjecture/hNYGPhKB">Borel conjecture</a> on the rigidity of aspherical manifolds is equivalent to the assembly map being an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>.
</p>

<ul><li>Davis, James F. (2000), <a href="https://jfdmath.sitehost.iu.edu/papers/d_manc.pdf">"Manifold aspects of the Novikov conjecture"</a> (PDF), in <a href="/facts/Sylvain_Cappell/w7RNoMpJ">Cappell, Sylvain</a>; <a href="/facts/Andrew_Ranicki/6jFoFDDA">Ranicki, Andrew</a>; <a href="/facts/Jonathan_Rosenberg_(mathematician)/9CY9YhDQ">Rosenberg, Jonathan</a> (eds.), <i>Surveys on surgery theory. Vol. 1</i>, Annals of Mathematics Studies, <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>, pp. 195–224, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-04937-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1747536">1747536</a></li>
<li><a href="/facts/John_Milnor/LTf24paN">John Milnor</a> and <a href="/facts/Jim_Stasheff/kceG3XPX">James D. Stasheff</a>, <i>Characteristic Classes,</i> Annals of Mathematics Studies 76, Princeton (1974).</li>
<li><a href="/facts/Sergei_Novikov_(mathematician)/hqMGDRul">Sergei P. Novikov</a>, <i>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</i>. 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.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://mathshistory.st-andrews.ac.uk/Biographies/Novikov_Sergi/">Biography of Sergei Novikov</a></li>
<li><a href="https://www.math.umd.edu/~jmr/NC.html">Novikov Conjecture Bibliography</a></li>
<li><a href="http://www.maths.ed.ac.uk/~aar/books/novikov1.pdf">Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 1</a></li>
<li><a href="http://www.maths.ed.ac.uk/~aar/books/novikov2.pdf">Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 2</a></li>
<li><a href="http://www.math.uni-muenster.de/u/lueck/publ/lueck/owsemfinalextract.pdf">2004 Oberwolfach Seminar notes on the Novikov Conjecture</a> (pdf)</li>
<li><a href="http://www.scholarpedia.org/article/Novikov_conjecture">Scholarpedia article by S.P. Novikov</a> (2010)</li>
<li><a href="http://www.map.him.uni-bonn.de/Novikov_Conjecture">The Novikov Conjecture</a> at the Manifold Atlas</li></ul>

Novikov conjecture open-in-new

Novikov conjecture